FindFind*arrow_forward*

14th Edition

Burdge

ISBN: 9781259327933

(a)

Interpretation Introduction

**Interpretation:**

The total number of electrons which can occupy in s, p, d and f-orbitals 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 n^{2}. 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_{s }values. 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 one s-orbital

Find the value of ‘l’ for one s-orbital

If the angular momentum quantum number (l) is 0, it corresponds to an s subshell for any value of the principal quantum number (n).

Find the value of ‘m_{l}’ for one s-orbital

If l = 0, the magnetic quantum number (m_{l}) has only one possible value which is again zero. It corresponds to an s orbital.

Count the electrons in one s-orbital

(b)

Interpretation Introduction

**Interpretation:**

The total number of electrons which can occupy in s, p, d and f-orbitals 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 n^{2}. 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_{s }values. 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 one p-orbital

Find the value of ‘l’ for one p-orbital

If the angular momentum quantum number (l) is 1, it corresponds to a p subshell for the principal quantum number (n) of 2 or greater values.

Find the value of ‘m_{l}’ for one p-orbital

If l = 1, the magnetic quantum number (m_{l}) has the three possible ways such as −1, 0 and +1 values. It corresponds to three p-subshells. They are labeled p_{x}, p_{y}, and p_{z} with the subscripted letters indicating the axis along which each orbital is oriented. The three p orbitals are identical in size, shape and energy; they differ from one another only in orientation.

Count the electrons in one p-orbital.

(c)

Interpretation Introduction

**Interpretation:**

The total number of electrons which can occupy in s, p, d and f-orbitals 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 n^{2}. 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_{s }values. 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 one d-orbital

Find the value of ‘l’ for one d-orbital

If the angular momentum quantum number (l) is 2, it corresponds to a d subshell for the principal quantum number (n) of 3 or greater values.

Find the value of ‘m_{l}’ for one d-orbital

If l = 2, the magnetic quantum number (m_{l}) has the five possible ways such as −2, −1, 0, +1 and +2 values. It corresponds to five d-subshells.

Count the electrons in one d-orbital.

(d)

Interpretation Introduction

**Interpretation:**

**Concept Introduction**

**Quantum Numbers**

_{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)**

**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 n^{2}. As the value of ‘n’ increases, the energy of the electron also increases.

**Angular Momentum Quantum Number (l)**

**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})**

_{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.

_{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})

**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_{s }values. 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 one f-orbital

Find the value of ‘l’ for one f-orbital

If the angular momentum quantum number (l) is 3, it corresponds to a f subshell for the principal quantum number (n) of 4 or greater values.

Find the value of ‘m_{l}’ for one f-orbital.