Determine the total number of electrons that can be held in all orbitals having the same principal quantum number n when (a) n = 1, (b) n = 2, (c) n = 3, and (d) n = 4.

Question

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Answer 1

Textbook Question

Chapter 3, Problem 3.97QP

Determine the total number of electrons that can be held in all orbitals having the same principal quantum number n when (a) n = 1, (b) n = 2, (c) n = 3, and (d) n = 4.

(a)

Expert Solution

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_l values, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 1

Find the value of ‘l’ for n = 1

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 1, the angular momentum quantum number (l) value is 0. It corresponds to a s subshell.

Find the value of ‘m_l’ for n = 1

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 1 are 2 (a).

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 1. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when n = 1. It corresponds to 1s-atomic orbital.

Count the total number of electrons when n = 1

One 1s-atomic orbital has two electrons. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 1 are 2.

(b)

Expert Solution

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_lvalues, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same pr

Find the value of ‘l’ for n = 2

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 2, the angular momentum quantum number (l) values are 0 and 1. They correspond to s and p-subshells.

Find the value of ‘m_l’ for n = 2

Principal quantum number when n = 2

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 2 are 8 (b).

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 2. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when l = 0. It corresponds to 2s-atomic orbital.

If l = 1, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 2. Here, (2(1) + 1) = 3 results. Therefore, there are three orbitals present when l = 1. They correspond to 2p-atomic orbitals. Totally, 4 atomic orbitals are present when n = 2.

Count the total number of electrons when n = 2

One atomic orbital has two electrons. 4 atomic orbitals have 8 electrons. Here, 2 electrons are filled in 2s-atomic orbital whereas 6 electrons are filled in 2p-atomic orbitals. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 2 are 8.

(c)

Expert Solution

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_lvalues, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 3

Find the value of ‘l’ for n = 3

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 3, the angular momentum quantum number (l) values are 0, 1 and 2. They correspond to s, p and d-subshells.

Find the value of ‘m_l’ for n = 3

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 3 are 18 (c).

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 3. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when l = 0. It corresponds to 3s-atomic orbital.

If l = 1, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 3. Here, (2(1) + 1) = 3 results. Therefore, there are three orbitals present when l = 1. They correspond to 3p-atomic orbitals.

If l = 2, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 3. Here, (2(2) + 1) = 5 results. Therefore, there are five orbitals present when l = 2. They correspond to 3d-atomic orbitals. Totally, 9 atomic orbitals are present when n = 3.

Count the total number of electrons when n = 3

One atomic orbital has two electrons. 9 atomic orbitals have 18 electrons. Here, 2 electrons are filled in 3s-atomic orbital, 6 electrons filled in 3p-atomic orbitals and 10 electrons filled in 3d-atomic orbitals. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 3 are 18.

(d)

Expert Solution

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_l values, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 4

Find the value of ‘l’ for n = 4

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 4, the angular momentum quantum number (l) values are 0, 1, 2 and 3. They correspond to s, p, d and f-subshells.

Find the value of ‘m_l’ for n = 4

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 4 are 32 (d).

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when l = 0. It corresponds to 4s-atomic orbital.

If l = 1, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(1) + 1) = 3 results. Therefore, there are three orbitals present when l = 1. They correspond to 4p-atomic orbitals.

If l = 2, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(2) + 1) = 5 results. Therefore, there are five orbitals present when l = 2. They correspond to 4d-atomic orbitals.

If l = 3, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(3) + 1) = 7 results. Therefore, there are seven orbitals present when l = 3. They correspond to 4f-atomic orbitals. Totally, 16 atomic orbitals are present when n = 3.

Count the total number of electrons when n = 4

One atomic orbital has two electrons. 16 atomic orbitals have 32 electrons. Here, 2 electrons are filled in 4s-atomic orbital, 6 electrons filled in 4p-atomic orbitals, 10 electrons filled in 4d-atomic orbitals and 14 electrons filled in 4f-atomic orbitals. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 4 are 32.

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Answer 2

Textbook Question

Chapter 3, Problem 3.97QP

Determine the total number of electrons that can be held in all orbitals having the same principal quantum number n when (a) n = 1, (b) n = 2, (c) n = 3, and (d) n = 4.

(a)

Expert Solution

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_l values, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 1

Find the value of ‘l’ for n = 1

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 1, the angular momentum quantum number (l) value is 0. It corresponds to a s subshell.

Find the value of ‘m_l’ for n = 1

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 1 are 2 (a).

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 1. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when n = 1. It corresponds to 1s-atomic orbital.

Count the total number of electrons when n = 1

One 1s-atomic orbital has two electrons. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 1 are 2.

(b)

Expert Solution

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_lvalues, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same pr

Find the value of ‘l’ for n = 2

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 2, the angular momentum quantum number (l) values are 0 and 1. They correspond to s and p-subshells.

Find the value of ‘m_l’ for n = 2

Principal quantum number when n = 2

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 2 are 8 (b).

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 2. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when l = 0. It corresponds to 2s-atomic orbital.

If l = 1, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 2. Here, (2(1) + 1) = 3 results. Therefore, there are three orbitals present when l = 1. They correspond to 2p-atomic orbitals. Totally, 4 atomic orbitals are present when n = 2.

Count the total number of electrons when n = 2

One atomic orbital has two electrons. 4 atomic orbitals have 8 electrons. Here, 2 electrons are filled in 2s-atomic orbital whereas 6 electrons are filled in 2p-atomic orbitals. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 2 are 8.

(c)

Expert Solution

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_lvalues, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 3

Find the value of ‘l’ for n = 3

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 3, the angular momentum quantum number (l) values are 0, 1 and 2. They correspond to s, p and d-subshells.

Find the value of ‘m_l’ for n = 3

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 3 are 18 (c).

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 3. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when l = 0. It corresponds to 3s-atomic orbital.

If l = 1, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 3. Here, (2(1) + 1) = 3 results. Therefore, there are three orbitals present when l = 1. They correspond to 3p-atomic orbitals.

If l = 2, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 3. Here, (2(2) + 1) = 5 results. Therefore, there are five orbitals present when l = 2. They correspond to 3d-atomic orbitals. Totally, 9 atomic orbitals are present when n = 3.

Count the total number of electrons when n = 3

One atomic orbital has two electrons. 9 atomic orbitals have 18 electrons. Here, 2 electrons are filled in 3s-atomic orbital, 6 electrons filled in 3p-atomic orbitals and 10 electrons filled in 3d-atomic orbitals. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 3 are 18.

(d)

Expert Solution

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_l values, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 4

Find the value of ‘l’ for n = 4

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 4, the angular momentum quantum number (l) values are 0, 1, 2 and 3. They correspond to s, p, d and f-subshells.

Find the value of ‘m_l’ for n = 4

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 4 are 32 (d).

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when l = 0. It corresponds to 4s-atomic orbital.

If l = 1, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(1) + 1) = 3 results. Therefore, there are three orbitals present when l = 1. They correspond to 4p-atomic orbitals.

If l = 2, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(2) + 1) = 5 results. Therefore, there are five orbitals present when l = 2. They correspond to 4d-atomic orbitals.

If l = 3, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(3) + 1) = 7 results. Therefore, there are seven orbitals present when l = 3. They correspond to 4f-atomic orbitals. Totally, 16 atomic orbitals are present when n = 3.

Count the total number of electrons when n = 4

One atomic orbital has two electrons. 16 atomic orbitals have 32 electrons. Here, 2 electrons are filled in 4s-atomic orbital, 6 electrons filled in 4p-atomic orbitals, 10 electrons filled in 4d-atomic orbitals and 14 electrons filled in 4f-atomic orbitals. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 4 are 32.

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Answer 3

Textbook Question

Chapter 3, Problem 3.97QP

Determine the total number of electrons that can be held in all orbitals having the same principal quantum number n when (a) n = 1, (b) n = 2, (c) n = 3, and (d) n = 4.

(a)

Expert Solution

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_l values, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 1

Find the value of ‘l’ for n = 1

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 1, the angular momentum quantum number (l) value is 0. It corresponds to a s subshell.

Find the value of ‘m_l’ for n = 1

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 1 are 2 (a).

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 1. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when n = 1. It corresponds to 1s-atomic orbital.

Count the total number of electrons when n = 1

One 1s-atomic orbital has two electrons. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 1 are 2.

(b)

Expert Solution

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_lvalues, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same pr

Find the value of ‘l’ for n = 2

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 2, the angular momentum quantum number (l) values are 0 and 1. They correspond to s and p-subshells.

Find the value of ‘m_l’ for n = 2

Principal quantum number when n = 2

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 2 are 8 (b).

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 2. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when l = 0. It corresponds to 2s-atomic orbital.

If l = 1, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 2. Here, (2(1) + 1) = 3 results. Therefore, there are three orbitals present when l = 1. They correspond to 2p-atomic orbitals. Totally, 4 atomic orbitals are present when n = 2.

Count the total number of electrons when n = 2

One atomic orbital has two electrons. 4 atomic orbitals have 8 electrons. Here, 2 electrons are filled in 2s-atomic orbital whereas 6 electrons are filled in 2p-atomic orbitals. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 2 are 8.

(c)

Expert Solution

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_lvalues, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 3

Find the value of ‘l’ for n = 3

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 3, the angular momentum quantum number (l) values are 0, 1 and 2. They correspond to s, p and d-subshells.

Find the value of ‘m_l’ for n = 3

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 3 are 18 (c).

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 3. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when l = 0. It corresponds to 3s-atomic orbital.

If l = 1, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 3. Here, (2(1) + 1) = 3 results. Therefore, there are three orbitals present when l = 1. They correspond to 3p-atomic orbitals.

If l = 2, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 3. Here, (2(2) + 1) = 5 results. Therefore, there are five orbitals present when l = 2. They correspond to 3d-atomic orbitals. Totally, 9 atomic orbitals are present when n = 3.

Count the total number of electrons when n = 3

One atomic orbital has two electrons. 9 atomic orbitals have 18 electrons. Here, 2 electrons are filled in 3s-atomic orbital, 6 electrons filled in 3p-atomic orbitals and 10 electrons filled in 3d-atomic orbitals. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 3 are 18.

(d)

Expert Solution

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_l values, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 4

Find the value of ‘l’ for n = 4

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 4, the angular momentum quantum number (l) values are 0, 1, 2 and 3. They correspond to s, p, d and f-subshells.

Find the value of ‘m_l’ for n = 4

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 4 are 32 (d).

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when l = 0. It corresponds to 4s-atomic orbital.

If l = 1, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(1) + 1) = 3 results. Therefore, there are three orbitals present when l = 1. They correspond to 4p-atomic orbitals.

If l = 2, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(2) + 1) = 5 results. Therefore, there are five orbitals present when l = 2. They correspond to 4d-atomic orbitals.

If l = 3, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(3) + 1) = 7 results. Therefore, there are seven orbitals present when l = 3. They correspond to 4f-atomic orbitals. Totally, 16 atomic orbitals are present when n = 3.

Count the total number of electrons when n = 4

One atomic orbital has two electrons. 16 atomic orbitals have 32 electrons. Here, 2 electrons are filled in 4s-atomic orbital, 6 electrons filled in 4p-atomic orbitals, 10 electrons filled in 4d-atomic orbitals and 14 electrons filled in 4f-atomic orbitals. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 4 are 32.

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Answer 4

Textbook Question

Chapter 3, Problem 3.97QP

Determine the total number of electrons that can be held in all orbitals having the same principal quantum number n when (a) n = 1, (b) n = 2, (c) n = 3, and (d) n = 4.

(a)

Expert Solution

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_l values, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 1

Find the value of ‘l’ for n = 1

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 1, the angular momentum quantum number (l) value is 0. It corresponds to a s subshell.

Find the value of ‘m_l’ for n = 1

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 1 are 2 (a).

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 1. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when n = 1. It corresponds to 1s-atomic orbital.

Count the total number of electrons when n = 1

One 1s-atomic orbital has two electrons. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 1 are 2.

(b)

Expert Solution

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_lvalues, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same pr

Find the value of ‘l’ for n = 2

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 2, the angular momentum quantum number (l) values are 0 and 1. They correspond to s and p-subshells.

Find the value of ‘m_l’ for n = 2

Principal quantum number when n = 2

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 2 are 8 (b).

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 2. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when l = 0. It corresponds to 2s-atomic orbital.

If l = 1, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 2. Here, (2(1) + 1) = 3 results. Therefore, there are three orbitals present when l = 1. They correspond to 2p-atomic orbitals. Totally, 4 atomic orbitals are present when n = 2.

Count the total number of electrons when n = 2

One atomic orbital has two electrons. 4 atomic orbitals have 8 electrons. Here, 2 electrons are filled in 2s-atomic orbital whereas 6 electrons are filled in 2p-atomic orbitals. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 2 are 8.

(c)

Expert Solution

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_lvalues, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 3

Find the value of ‘l’ for n = 3

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 3, the angular momentum quantum number (l) values are 0, 1 and 2. They correspond to s, p and d-subshells.

Find the value of ‘m_l’ for n = 3

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 3 are 18 (c).

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 3. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when l = 0. It corresponds to 3s-atomic orbital.

If l = 1, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 3. Here, (2(1) + 1) = 3 results. Therefore, there are three orbitals present when l = 1. They correspond to 3p-atomic orbitals.

If l = 2, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 3. Here, (2(2) + 1) = 5 results. Therefore, there are five orbitals present when l = 2. They correspond to 3d-atomic orbitals. Totally, 9 atomic orbitals are present when n = 3.

Count the total number of electrons when n = 3

One atomic orbital has two electrons. 9 atomic orbitals have 18 electrons. Here, 2 electrons are filled in 3s-atomic orbital, 6 electrons filled in 3p-atomic orbitals and 10 electrons filled in 3d-atomic orbitals. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 3 are 18.

(d)

Expert Solution

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_l values, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 4

Find the value of ‘l’ for n = 4

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 4, the angular momentum quantum number (l) values are 0, 1, 2 and 3. They correspond to s, p, d and f-subshells.

Find the value of ‘m_l’ for n = 4

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 4 are 32 (d).

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when l = 0. It corresponds to 4s-atomic orbital.

If l = 1, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(1) + 1) = 3 results. Therefore, there are three orbitals present when l = 1. They correspond to 4p-atomic orbitals.

If l = 2, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(2) + 1) = 5 results. Therefore, there are five orbitals present when l = 2. They correspond to 4d-atomic orbitals.

If l = 3, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(3) + 1) = 7 results. Therefore, there are seven orbitals present when l = 3. They correspond to 4f-atomic orbitals. Totally, 16 atomic orbitals are present when n = 3.

Count the total number of electrons when n = 4

One atomic orbital has two electrons. 16 atomic orbitals have 32 electrons. Here, 2 electrons are filled in 4s-atomic orbital, 6 electrons filled in 4p-atomic orbitals, 10 electrons filled in 4d-atomic orbitals and 14 electrons filled in 4f-atomic orbitals. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 4 are 32.

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Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

(a)

Expert Solution

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_l values, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 1

Find the value of ‘l’ for n = 1

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 1, the angular momentum quantum number (l) value is 0. It corresponds to a s subshell.

Find the value of ‘m_l’ for n = 1

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 1 are 2 (a).

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 1. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when n = 1. It corresponds to 1s-atomic orbital.

Count the total number of electrons when n = 1

One 1s-atomic orbital has two electrons. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 1 are 2.

(b)

Expert Solution

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_lvalues, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same pr

Find the value of ‘l’ for n = 2

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 2, the angular momentum quantum number (l) values are 0 and 1. They correspond to s and p-subshells.

Find the value of ‘m_l’ for n = 2

Principal quantum number when n = 2

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 2 are 8 (b).

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 2. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when l = 0. It corresponds to 2s-atomic orbital.

If l = 1, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 2. Here, (2(1) + 1) = 3 results. Therefore, there are three orbitals present when l = 1. They correspond to 2p-atomic orbitals. Totally, 4 atomic orbitals are present when n = 2.

Count the total number of electrons when n = 2

One atomic orbital has two electrons. 4 atomic orbitals have 8 electrons. Here, 2 electrons are filled in 2s-atomic orbital whereas 6 electrons are filled in 2p-atomic orbitals. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 2 are 8.

(c)

Expert Solution

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_lvalues, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 3

Find the value of ‘l’ for n = 3

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 3, the angular momentum quantum number (l) values are 0, 1 and 2. They correspond to s, p and d-subshells.

Find the value of ‘m_l’ for n = 3

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 3 are 18 (c).

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 3. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when l = 0. It corresponds to 3s-atomic orbital.

If l = 1, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 3. Here, (2(1) + 1) = 3 results. Therefore, there are three orbitals present when l = 1. They correspond to 3p-atomic orbitals.

If l = 2, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 3. Here, (2(2) + 1) = 5 results. Therefore, there are five orbitals present when l = 2. They correspond to 3d-atomic orbitals. Totally, 9 atomic orbitals are present when n = 3.

Count the total number of electrons when n = 3

One atomic orbital has two electrons. 9 atomic orbitals have 18 electrons. Here, 2 electrons are filled in 3s-atomic orbital, 6 electrons filled in 3p-atomic orbitals and 10 electrons filled in 3d-atomic orbitals. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 3 are 18.

(d)

Expert Solution

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_l values, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 4

Find the value of ‘l’ for n = 4

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 4, the angular momentum quantum number (l) values are 0, 1, 2 and 3. They correspond to s, p, d and f-subshells.

Find the value of ‘m_l’ for n = 4

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 4 are 32 (d).

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when l = 0. It corresponds to 4s-atomic orbital.

If l = 1, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(1) + 1) = 3 results. Therefore, there are three orbitals present when l = 1. They correspond to 4p-atomic orbitals.

If l = 2, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(2) + 1) = 5 results. Therefore, there are five orbitals present when l = 2. They correspond to 4d-atomic orbitals.

If l = 3, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(3) + 1) = 7 results. Therefore, there are seven orbitals present when l = 3. They correspond to 4f-atomic orbitals. Totally, 16 atomic orbitals are present when n = 3.

Count the total number of electrons when n = 4

One atomic orbital has two electrons. 16 atomic orbitals have 32 electrons. Here, 2 electrons are filled in 4s-atomic orbital, 6 electrons filled in 4p-atomic orbitals, 10 electrons filled in 4d-atomic orbitals and 14 electrons filled in 4f-atomic orbitals. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 4 are 32.

Answer 8

(a)

Expert Solution

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_l values, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 1

Find the value of ‘l’ for n = 1

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 1, the angular momentum quantum number (l) value is 0. It corresponds to a s subshell.

Find the value of ‘m_l’ for n = 1

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 1 are 2 (a).

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 1. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when n = 1. It corresponds to 1s-atomic orbital.

Count the total number of electrons when n = 1

One 1s-atomic orbital has two electrons. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 1 are 2.

Answer 9

(a)

Expert Solution

Answer 10

(a)

Expert Solution

Answer 11

Expert Solution

Answer 12

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_l values, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 1

Find the value of ‘l’ for n = 1

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 1, the angular momentum quantum number (l) value is 0. It corresponds to a s subshell.

Find the value of ‘m_l’ for n = 1

Answer 13

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 1 are 2 (a).

Answer 14

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 1. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when n = 1. It corresponds to 1s-atomic orbital.

Count the total number of electrons when n = 1

One 1s-atomic orbital has two electrons. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 1 are 2.

Answer 15

(b)

Expert Solution

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_lvalues, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same pr

Find the value of ‘l’ for n = 2

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 2, the angular momentum quantum number (l) values are 0 and 1. They correspond to s and p-subshells.

Find the value of ‘m_l’ for n = 2

Principal quantum number when n = 2

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 2 are 8 (b).

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 2. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when l = 0. It corresponds to 2s-atomic orbital.

If l = 1, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 2. Here, (2(1) + 1) = 3 results. Therefore, there are three orbitals present when l = 1. They correspond to 2p-atomic orbitals. Totally, 4 atomic orbitals are present when n = 2.

Count the total number of electrons when n = 2

One atomic orbital has two electrons. 4 atomic orbitals have 8 electrons. Here, 2 electrons are filled in 2s-atomic orbital whereas 6 electrons are filled in 2p-atomic orbitals. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 2 are 8.

Answer 16

(b)

Expert Solution

Answer 17

(b)

Expert Solution

Answer 18

Expert Solution

Answer 19

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_lvalues, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same pr

Find the value of ‘l’ for n = 2

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 2, the angular momentum quantum number (l) values are 0 and 1. They correspond to s and p-subshells.

Find the value of ‘m_l’ for n = 2

Principal quantum number when n = 2

Answer 20

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 2 are 8 (b).

Answer 21

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 2. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when l = 0. It corresponds to 2s-atomic orbital.

If l = 1, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 2. Here, (2(1) + 1) = 3 results. Therefore, there are three orbitals present when l = 1. They correspond to 2p-atomic orbitals. Totally, 4 atomic orbitals are present when n = 2.

Count the total number of electrons when n = 2

One atomic orbital has two electrons. 4 atomic orbitals have 8 electrons. Here, 2 electrons are filled in 2s-atomic orbital whereas 6 electrons are filled in 2p-atomic orbitals. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 2 are 8.

Answer 22

(c)

Expert Solution

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_lvalues, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 3

Find the value of ‘l’ for n = 3

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 3, the angular momentum quantum number (l) values are 0, 1 and 2. They correspond to s, p and d-subshells.

Find the value of ‘m_l’ for n = 3

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 3 are 18 (c).

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 3. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when l = 0. It corresponds to 3s-atomic orbital.

If l = 1, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 3. Here, (2(1) + 1) = 3 results. Therefore, there are three orbitals present when l = 1. They correspond to 3p-atomic orbitals.

If l = 2, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 3. Here, (2(2) + 1) = 5 results. Therefore, there are five orbitals present when l = 2. They correspond to 3d-atomic orbitals. Totally, 9 atomic orbitals are present when n = 3.

Count the total number of electrons when n = 3

One atomic orbital has two electrons. 9 atomic orbitals have 18 electrons. Here, 2 electrons are filled in 3s-atomic orbital, 6 electrons filled in 3p-atomic orbitals and 10 electrons filled in 3d-atomic orbitals. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 3 are 18.

Answer 23

(c)

Expert Solution

Answer 24

(c)

Expert Solution

Answer 25

Expert Solution

Answer 26

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_lvalues, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 3

Find the value of ‘l’ for n = 3

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 3, the angular momentum quantum number (l) values are 0, 1 and 2. They correspond to s, p and d-subshells.

Find the value of ‘m_l’ for n = 3

Answer 27

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 3 are 18 (c).

Answer 28

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 3. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when l = 0. It corresponds to 3s-atomic orbital.

If l = 1, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 3. Here, (2(1) + 1) = 3 results. Therefore, there are three orbitals present when l = 1. They correspond to 3p-atomic orbitals.

If l = 2, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 3. Here, (2(2) + 1) = 5 results. Therefore, there are five orbitals present when l = 2. They correspond to 3d-atomic orbitals. Totally, 9 atomic orbitals are present when n = 3.

Count the total number of electrons when n = 3

One atomic orbital has two electrons. 9 atomic orbitals have 18 electrons. Here, 2 electrons are filled in 3s-atomic orbital, 6 electrons filled in 3p-atomic orbitals and 10 electrons filled in 3d-atomic orbitals. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 3 are 18.

Answer 29

(d)

Expert Solution

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_l values, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 4

Find the value of ‘l’ for n = 4

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 4, the angular momentum quantum number (l) values are 0, 1, 2 and 3. They correspond to s, p, d and f-subshells.

Find the value of ‘m_l’ for n = 4

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 4 are 32 (d).

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when l = 0. It corresponds to 4s-atomic orbital.

If l = 1, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(1) + 1) = 3 results. Therefore, there are three orbitals present when l = 1. They correspond to 4p-atomic orbitals.

If l = 2, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(2) + 1) = 5 results. Therefore, there are five orbitals present when l = 2. They correspond to 4d-atomic orbitals.

If l = 3, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(3) + 1) = 7 results. Therefore, there are seven orbitals present when l = 3. They correspond to 4f-atomic orbitals. Totally, 16 atomic orbitals are present when n = 3.

Count the total number of electrons when n = 4

One atomic orbital has two electrons. 16 atomic orbitals have 32 electrons. Here, 2 electrons are filled in 4s-atomic orbital, 6 electrons filled in 4p-atomic orbitals, 10 electrons filled in 4d-atomic orbitals and 14 electrons filled in 4f-atomic orbitals. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 4 are 32.

Answer 30

(d)

Expert Solution

Answer 31

(d)

Expert Solution

Answer 32

Expert Solution

Answer 33

Interpretation Introduction

Interpretation:

The total numbers of electrons which can occupy in all orbitals having the same principal quantum number with different integral values should be identified using the concept of quantum numbers.

Concept Introduction:

Quantum Numbers

Quantum numbers are explained for the distribution of electron density in an atom. They are derived from the mathematical solution of Schrodinger’s equation for the hydrogen atom. The types of quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l) and the electron spin quantum number (m_s). Each atomic orbital in an atom is categorized by a unique set of the quantum numbers.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_svalues. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

Pauli Exclusion Principle

No two electrons in an atom should have the four same quantum numbers. Two electrons are occupied in an atomic orbital because there are two possible values of m_s. As an orbital can contain a maximum of only two electrons, the two electrons must have opposing spins. If two electrons have the same values of n, l and m_l values, they should have different values of m_s.

To find: Count the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 4

Find the value of ‘l’ for n = 4

For a given value of n, the possible values of l range are from 0 to n – 1. When n = 4, the angular momentum quantum number (l) values are 0, 1, 2 and 3. They correspond to s, p, d and f-subshells.

Find the value of ‘m_l’ for n = 4

Answer 34

Answer to Problem 3.97QP

The total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 4 are 32 (d).

Answer 35

Explanation of Solution

If l = 0, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(0) + 1) = 1 results. Therefore, there is only one number of orbital present when l = 0. It corresponds to 4s-atomic orbital.

If l = 1, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(1) + 1) = 3 results. Therefore, there are three orbitals present when l = 1. They correspond to 4p-atomic orbitals.

If l = 2, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(2) + 1) = 5 results. Therefore, there are five orbitals present when l = 2. They correspond to 4d-atomic orbitals.

If l = 3, the number of possible magnetic quantum number (m_l) values are calculated using the formula (2l + 1) for n = 4. Here, (2(3) + 1) = 7 results. Therefore, there are seven orbitals present when l = 3. They correspond to 4f-atomic orbitals. Totally, 16 atomic orbitals are present when n = 3.

Count the total number of electrons when n = 4

One atomic orbital has two electrons. 16 atomic orbitals have 32 electrons. Here, 2 electrons are filled in 4s-atomic orbital, 6 electrons filled in 4p-atomic orbitals, 10 electrons filled in 4d-atomic orbitals and 14 electrons filled in 4f-atomic orbitals. Therefore, the total number of electrons which can occupy in all orbitals having the same principal quantum number when n = 4 are 32.