Chapter 2, Problem 81E

Give the maximum number of electrons in an atom that can have these quantum numbers:

a. n = 4

b. n = 5, m_l = + l

c. n = 5,m_s = $+\frac{1}{2}$

d. n = 3, l = 2

e. n = 2, l = 1

Interpretation Introduction

Interpretation:

The maximum number of electrons for the given quantum numbers to be predicted.

Concept Introduction:

In an electronic configuration, each subshell can only have a definite number of orbitals. In $s$ subshell, the number of orbitals is 1; in subshell $p$ the number of orbitals is $3$, in $d$ subshell the number of orbitals is $5$, in the $f$ subshell the number orbitals is 7 and in subshell $g$ the number of orbitals is $9$. The maximum number of electrons present in each orbital is 2.

To determine: The maximum number of electrons present in an atom having quantum number $n=4$.

Answer to Problem 81E

Answer

The maximum number of electrons present in an atom with quantum number $n=4$ are $\underset{\_}{32}$.

Explanation of Solution

The maximum number of electrons present in an atom with quantum number n is $2n^2$

Where,

• $n$ is the principal quantum number

For quantum number $n=4$, the sub-shells possible are,

$n=4s,4p,4d,4f$

Number of orbitals possible in subshell $4s$ is $1$.

Numbers of orbitals possible in subshell $4p$ are $3$.

Numbers of orbitals possible in subshell $4d$ are $5$.

Numbers of orbitals possible in subshell $4f$ are $7$.

Therefore, the total number of orbitals possible for quantum number, $n=4$ is

$1+3+5+7=16$

The maximum number of electrons present in each orbital is $2$.

Therefore, for $n=4$, the maximum number of electrons present is $16\times 2=\underset{\_}{32}$.

Conclusion

The maximum number of electrons present in an atom with quantum number, $n=4$ are $\underset{\_}{32}$.

Interpretation Introduction

Interpretation:

The maximum number of electrons for the given quantum numbers to be predicted.

Concept Introduction:

In an electronic configuration, each subshell can only have a definite number of orbitals. In $s$ subshell, the number of orbitals is 1; in subshell $p$ the number of orbitals is $3$, in $d$ subshell the number of orbitals is $5$, in the $f$ subshell the number orbitals is 7 and in subshell $g$ the number of orbitals is $9$. The maximum number of electrons present in each orbital is 2.

To determine: The maximum number of electrons present in an atom having quantum number $n=5\text{and}{m}_l=+1$.

Answer to Problem 81E

Answer

The maximum number of electrons present in an atom with quantum number $n=5\text{and}{m}_l=+1$ are $\underset{\_}{8}$.

Explanation of Solution

The number of subshell having the same quantum number $n=5\text{and}{m}_l=+1$,

Principal quantum number $\left(n\right)$	Orbital quantum number $\left(l=0\text{to}\left(n-1\right)\right)$	Magnetic quantum number $\left(m_l=-l\text{to}+l\right)$
5	0	0
5	1	$-1,0+1$
5	2	$-2,-1,0,+1,+2$
5	3	$-3,-2,-1,0,+1,+2,+3$
5	4	$-4,-3,-2,-1,0,+1,+2,+3,+4$

From the above table it can be stated that there are $4$ orbitals for which the value of quantum number, $n=5\text{and}{m}_l=+1$.

In each orbital the maximum number of electrons that can be present is $2$.

Therefore, the total number of electrons present in the orbital having quantum number $n=5\text{and}{m}_l=+1$ is $2\times 4=\underset{\_}{8}$.

Conclusion

The maximum number of electrons present in an atom with quantum number $n=5\text{and}{m}_l=+1$ is $\underset{\_}{8}$.

Interpretation Introduction

Interpretation:

The maximum number of electrons for the given quantum numbers to be predicted.

Concept Introduction:

In an electronic configuration, each subshell can only have a definite number of orbitals. In $s$ subshell, the number of orbitals is 1; in subshell $p$ the number of orbitals is $3$, in $d$ subshell the number of orbitals is $5$, in the $f$ subshell the number orbitals is 7 and in subshell $g$ the number of orbitals is $9$. The maximum number of electrons present in each orbital is 2.

To determine: The maximum number of electrons present in an atom having quantum number $n=5\text{and}{m}_s=+\frac{1}{2}$.

Answer to Problem 81E

Answer

The maximum number of electrons present in an atom with quantum number $n=5\text{and}{m}_s=+\frac{1}{2}$ is $\underset{\_}{25}$.

Explanation of Solution

The number of subshell having the same quantum number $n=5\text{and}{m}_l=+1$,

Principal quantum number $\left(n\right)$	Orbital quantum number $\left(l=0\text{to}\left(n-1\right)\right)$	Magnetic quantum number $\left(m_l=-l\text{to}+l\right)$
5	0	0
5	1	$-1,0+1$
5	2	$-2,-1,0,+1,+2$
5	3	$-3,-2,-1,0,+1,+2,+3$
5	4	$-4,-3,-2,-1,0,+1,+2,+3,+4$

The total number of orbitals for quantum number $n=5$ is,

$1+3+5+7+9=25$

In each orbital, the maximum number of electrons that can be present is $2$; one of the electron will revolve positively and have value of spin quantum number, $m_s=+\frac{1}{2}$ while the other electron will have the spin quantum number, $m_s=-\frac{1}{2}$. Therefore, there are total $\underset{\_}{25}$ orbitals for which the value of quantum number $n=5\text{and}{m}_s=+\frac{1}{2}$.

Conclusion

The possible number of electrons that can have the value of quantum number $n=5\text{and}{m}_s=+\frac{1}{2}$ is $\underset{\_}{25}$.

Interpretation Introduction

Interpretation:

The maximum number of electrons for the given quantum numbers to be predicted.

Concept Introduction:

In an electronic configuration, each subshell can only have a definite number of orbitals. In $s$ subshell, the number of orbitals is 1; in subshell $p$ the number of orbitals is $3$, in $d$ subshell the number of orbitals is $5$, in the $f$ subshell the number orbitals is 7 and in subshell $g$ the number of orbitals is $9$. The maximum number of electrons present in each orbital is 2.

To determine: The maximum number of electrons present in an atom having quantum number $n=3\text{and}l=2$.

Answer to Problem 81E

Answer

The maximum number of electrons present in an atom with quantum number $n=3\text{and}l=2$ is $\underset{\_}{10}$.

Explanation of Solution

The number of subshell having the same quantum number $n=3\text{and}l=2$,

Principal quantum number $\left(n\right)$	Orbital quantum number $\left(l=0\text{to}\left(n-1\right)\right)$	Magnetic quantum number $\left(m_l=-l\text{to}+l\right)$
3	0	0
3	1	$-1,0+1$
3	2	$-2,-1,0,+1,+2$
3	3	$-3,-2,-1,0,+1,+2,+3$

The total number of orbitals for quantum number $n=3\text{and}l=2$ is $5$.

In each orbital, the maximum number of electrons that can be present is $2$.

Therefore maximum number of electron for which the value of quantum number is $n=3,l=2$ is $5\times 2=\underset{\_}{10}$.

Conclusion

The possible number of electrons that can have the value of quantum number $n=3,l=2$ is $\underset{\_}{10}$.

Interpretation Introduction

Interpretation:

The maximum number of electrons for the given quantum numbers to be predicted.

Concept Introduction:

In an electronic configuration, each subshell can only have a definite number of orbitals. In $s$ subshell, the number of orbitals is 1; in subshell $p$ the number of orbitals is $3$, in $d$ subshell the number of orbitals is $5$, in the $f$ subshell the number orbitals is 7 and in subshell $g$ the number of orbitals is $9$. The maximum number of electrons present in each orbital is 2.

To determine: The maximum number of electrons present in an atom having quantum number $n=2\text{and}l=1$.

Answer to Problem 81E

Answer

The maximum number of electrons present in an atom with quantum number $n=2\text{and}l=1$ is $\underset{\_}{6}$.

Explanation of Solution

The number of subshell having the same quantum number $n=2\text{and}l=1$,

Principal quantum number $\left(n\right)$	Orbital quantum number $\left(l=0\text{to}\left(n-1\right)\right)$	Magnetic quantum number $\left(m_l=-l\text{to}+l\right)$
2	0	0
2	1	$-1,0+1$
2	2	$-2,-1,0,+1,+2$

The total number of orbital for quantum number $n=2,l=1$ is $3$.

In each orbital, the maximum number of electrons that can be present is $2$.

Therefore, maximum number of electrons for which the value of quantum number is $n=2\text{and}l=1$ is $3\times 2=\underset{\_}{6}$.

Conclusion

The possible number of electrons that can have the value of quantum number $n=2\text{and}l=1$ is $\underset{\_}{6}$.