Chapter 12, Problem 145AE

(a)

Interpretation Introduction

Interpretation:Whether set of quantum numbers that include $n$ equal to 3, $l$ equal to 2, $m_l$ equal to 2is allowed set or not should be identified and incorrect set should be pointed out.

Concept introduction:Energy, size, shape, and orientation of atomic orbital are determined with help of some numbers. These numbers are called quantum numbers and are obtained from solution of Schrodinger equation of hydrogen atom by application of boundary conditions.

Below mentioned are four quantum numbers.

1. Principal Quantum Number

It is represented by $n$ . This quantum number is related to size and energy of different atomic orbitals. With increase in value of $n$ , size of orbital becomes larger and electron is present farther from atomic nucleus for longer time.

2. Angular Momentum Quantum Number

It is represented by $l$ . It describes angular momentum of electron in any specific atomic orbital. Shape of orbitals is determined by this quantum number. It can have integral values from 0 to $\left(n-1\right)$ .

Designation of orbitals on basis of different $l$ values is as follows:

  $\begin{array}{ccccc}\text{Value of }l& 0& 1& 2& 3\\ \text{Orbital}& s& p& d& f\end{array}$

3. Magnetic Quantum Number

This quantum number is denoted by $m_l$ . It describes orientation of angular momentum associated with atomic orbital. Its value ranges from $-l$ to $+l$ . Total values of magnetic quantum number is governed by $2l+1$

4. Spin Quantum Number

This quantum number is represented by $m_s$ . It indicates direction of electron spin. It can either $+\frac{1}{2}$ or $-\frac{1}{2}$ .

Explanation of Solution

For a given value of principal quantum number $n$ allowed values of $l$ range from 0 to $n-1$ , therefore for $n=3$ the permitted values of azimuthal quantum number must range from 0 to 2. Therefore $l=2$ is possible and allowed.

Since $m_l$ ranges from $-l$ to $+l$ so values of $m_l$ can range from $-2$ to $+2$ . Thus $m_l=2$ is also allowed and while set is correct.

(b)

Interpretation Introduction

Interpretation: Whether set of quantum numbers that include $n$ equal to 4, $l$ equal to 3, $m_l$ equal to 4 is allowed set or not should be identified and incorrect set should be pointed out.

Concept introduction:Energy, size, shape, and orientation of atomic orbital are determined with help of some numbers. These numbers are called quantum numbers and are obtained from solution of Schrodinger equation of hydrogen atom by application of boundary conditions.

Below mentioned are four quantum numbers.

1. Principal Quantum Number

It is represented by $n$ . This quantum number is related to size and energy of different atomic orbitals. With increase in value of $n$ , size of orbital becomes larger and electron is present farther from atomic nucleus for longer time.

2. Angular Momentum Quantum Number

It is represented by $l$ . It describes angular momentum of electron in any specific atomic orbital. Shape of orbitals is determined by this quantum number. It can have integral values from 0 to $\left(n-1\right)$ .

Designation of orbitals on basis of different $l$ values is as follows:

  $\begin{array}{ccccc}\text{Value of }l& 0& 1& 2& 3\\ \text{Orbital}& s& p& d& f\end{array}$

3. Magnetic Quantum Number

This quantum number is denoted by $m_l$ . It describes orientation of angular momentum associated with atomic orbital. Its value ranges from $-l$ to $+l$ . Total values of magnetic quantum number is governed by $2l+1$

4. Spin Quantum Number

This quantum number is represented by $m_s$ . It indicates direction of electron spin. It can either $+\frac{1}{2}$ or $-\frac{1}{2}$ .

Explanation of Solution

For a given value of principal quantum number $n$ allowed values of $l$ range from 0 to $n-1$ , therefore for $n=4$ the permitted values of azimuthal quantum number must range from 0 to 3. Therefore $l=3$ is possible and allowed.

Since $m_l$ ranges from $-l$ to $+l$ so values of $m_l$ can range from $-3$ to $+3$ . Thus $m_l=4$ is not allowed in this set and is incorrect. The correct set should have $m_l=3$ or any value from $-3$ to $+3$ .

(c)

Interpretation Introduction

Interpretation: Whether set of quantum numbers that include $n$ equal to 0, $l$ equal to 0, $m_l$ equal to 0 is allowed set or not should be identified and incorrect set should be pointed out.

Concept introduction:Energy, size, shape, and orientation of atomic orbital are determined with help of some numbers. These numbers are called quantum numbers and are obtained from solution of Schrodinger equation of hydrogen atom by application of boundary conditions.

Below mentioned are four quantum numbers.

1. Principal Quantum Number

It is represented by $n$ . This quantum number is related to size and energy of different atomic orbitals. With increase in value of $n$ , size of orbital becomes larger and electron is present farther from atomic nucleus for longer time.

2. Angular Momentum Quantum Number

It is represented by $l$ . It describes angular momentum of electron in any specific atomic orbital. Shape of orbitals is determined by this quantum number. It can have integral values from 0 to $\left(n-1\right)$ .

Designation of orbitals on basis of different $l$ values is as follows:

  $\begin{array}{ccccc}\text{Value of }l& 0& 1& 2& 3\\ \text{Orbital}& s& p& d& f\end{array}$

3. Magnetic Quantum Number

This quantum number is denoted by $m_l$ . It describes orientation of angular momentum associated with atomic orbital. Its value ranges from $-l$ to $+l$ . Total values of magnetic quantum number is governed by $2l+1$

4. Spin Quantum Number

This quantum number is represented by $m_s$ . It indicates direction of electron spin. It can either $+\frac{1}{2}$ or $-\frac{1}{2}$ .

Explanation of Solution

The principle quantum number $n=0$ is not allowed since it may have values from $n=1,2,3$ and so on but not begin with zero. Therefore $n=0$ , $l=0$ , $m_l=0$ cannot be allowed or correct.

(d)

Interpretation Introduction

Interpretation: Whether set of quantum numbers that include $n$ equal to2, $l$ equal to $-1$ 0, $m_l$ equal to 1 is allowed set or not should be identified and incorrect set should be pointed out.

Concept introduction:Energy, size, shape, and orientation of atomic orbital are determined with help of some numbers. These numbers are called quantum numbers and are obtained from solution of Schrodinger equation of hydrogen atom by application of boundary conditions.

Below mentioned are four quantum numbers.

1. Principal Quantum Number

It is represented by $n$ . This quantum number is related to size and energy of different atomic orbitals. With increase in value of $n$ , size of orbital becomes larger and electron is present farther from atomic nucleus for longer time.

2. Angular Momentum Quantum Number

It is represented by $l$ . It describes angular momentum of electron in any specific atomic orbital. Shape of orbitals is determined by this quantum number. It can have integral values from 0 to $\left(n-1\right)$ .

Designation of orbitals on basis of different $l$ values is as follows:

  $\begin{array}{ccccc}\text{Value of }l& 0& 1& 2& 3\\ \text{Orbital}& s& p& d& f\end{array}$

3. Magnetic Quantum Number

This quantum number is denoted by $m_l$ . It describes orientation of angular momentum associated with atomic orbital. Its value ranges from $-l$ to $+l$ . Total values of magnetic quantum number is governed by $2l+1$

4. Spin Quantum Number

This quantum number is represented by $m_s$ . It indicates direction of electron spin. It can either $+\frac{1}{2}$ or $-\frac{1}{2}$ .

Explanation of Solution

For a given value of principal quantum number $n$ allowed values of $l$ range from 0 to $n-1$ , therefore for $n=2$ the permitted values of azimuthal quantum number must range from 0 to 1. Therefore $l=-1$ is not possible and is therefore not allowed.

Since $m_l$ ranges from $-l$ to $+l$ so values of $m_l$ can range from $-1$ to $+1$ . Thus $m_l=1$ is allowed in this set however whole set together is incorrect.