Chapter 7, Problem 7.84QP

(a)

Interpretation Introduction

Interpretation: The allowed combinations of quantum numbers from the given combinations are to be stated.

Concept introduction: There are four quantum numbers that indicates the size, energy and shape of an atomic orbital. The quantum number $n$ is the principal quantum number that indicates the energy level and size of the orbital and $l$ is the azimuthal quantum number that determines the shape and angular momentum of an orbital. The magnetic quantum number $\left(m_l\right)$ is an integer that takes the values from $\left(-l\text{ to }+l\right)$ and the spin quantum number $\left(m_s\right)$ indicates the spin of an electron $\left(m_s=+\frac{1}{2}\text{or}-\frac{1}{2}\right)$.

To determine: If the given combination of quantum numbers is allowed.

Answer to Problem 7.84QP

Solution

The combination of quantum number is allowed.

Explanation of Solution

Explanation

Explanation

The given combination of quantum numbers is $n=3$, $l=2$, $m_l=0$ and $m_s=-\frac{1}{2}$. In the given combination, the principal quantum number $\left(n\right)$ is $3$. For $n=3$ the possible value of $l$ from $l=0\text{to}n-1$ is,

$\begin{array}{c}l=n-1\\ =3-1\\ =2\end{array}$

The possible values of $l$ are $0,1$ and $2$. Therefore, given value of $l$ is allowed. For the given value of $l$, $m_l$ take values from $\left(-l\text{ to }+l\right)$. The value of $m_l$ and $m_s$ for the given value of $l$ are,

$m_l=-2,-1,0,+1,+2$

$m_s=+\frac{1}{2}\text{or}-\frac{1}{2}$

Therefore, the value of $m_l=0$ and $m_s=-\frac{1}{2}$ are also allowed. Hence, the given combination of quantum numbers is an allowed combination.

(b)

Interpretation Introduction

To determine: If the given combination of quantum numbers is allowed.

Answer to Problem 7.84QP

Solution

The combination of quantum number is allowed.

Explanation of Solution

Explanation

The given combination of quantum numbers is $n=5$, $l=4$, $m_l=4$ and $m_s=+\frac{1}{2}$. The principal quantum number is $5$. For $n=5$, angular momentum quantum number $\left(l\right)$ takes the value from $l=0\text{to}n-1$ which is,

$\begin{array}{c}l=n-1\\ =5-1\\ =4\end{array}$

The possible values of $l$ are $0,1,3$ and $4$ respectively. The given value of $l=4$ is an allowed value. For the given value of $l$, $m_l$ take values from $\left(-l\text{ to }+l\right)$. The value of $m_l$ and $m_s$ for the given value of $l$ are,

$m_l=-4,-3,-2,-1,0,+1,+2+3,+4$

$m_s=+\frac{1}{2}\text{or}-\frac{1}{2}$

Therefore, the value of $m_l=4$ and $m_s=+\frac{1}{2}$ are also allowed. Hence, the given combination of quantum numbers is an allowed combination.

(c)

Interpretation Introduction

To determine: If the given combination of quantum numbers is allowed.

Answer to Problem 7.84QP

Solution

The combination of quantum number is not allowed.

Explanation of Solution

Explanation

The given combination of quantum numbers is $n=3$, $l=0$, $m_l=1$ and $m_s=+\frac{1}{2}$. In the given combination, the principal quantum number $\left(n\right)$ is $3$. For $n=3$ the possible value of $l$ is,

$\begin{array}{c}l=n-1\\ =3-1\\ =2\end{array}$

The given value of $l$ is allowed. The possible value of $m_l$ is also zero for $l=0$ but in the given set the value of $m_l=1$ which is not allowed. Hence, the combination of quantum numbers is not allowed.

(d)

Interpretation Introduction

To determine: If the given combination of quantum numbers is allowed.

Answer to Problem 7.84QP

Solution

The combination of quantum number is not allowed.

Explanation of Solution

Explanation

The given combination of quantum numbers is $n=4$, $l=4$, $m_l=1$ and $m_s=-\frac{1}{2}$. The principal quantum number is $4$. For $n=4$, angular momentum quantum number $\left(l\right)$ takes the value from $l=0\text{to}n-1$ which is,

$\begin{array}{c}l=n-1\\ =4-1\\ =3\end{array}$

The possible values of $l$ are $0,1,2$ and $3$ respectively. The given value of $l=4$ is not an allowed value. Hence, the combination of quantum numbers is not allowed.

Conclusion

In the given combinations of quantum numbers, options (a) and (b) correspond to the allowed combinations.