Chapter 7, Problem 7.72P

(a)

Interpretation Introduction

Interpretation:

The correct $l$ value for the combination $n=3$;$l=0$;$m_l=-1$ is to be determined.

Concept introduction:

Quantum numbers are a set of four numbers that gives complete information about an electron within an atom. Out of the four, the three quantum numbers that define the shape, size, and orientation of an orbital are the principal quantum number, the orbital angular momentum quantum number, and the magnetic quantum number.

The formula to calculate the highest possible value of angular momentum quantum number $l$ for a given value of a principal quantum number $n$ is as follows:

$\text{Highest value for }l=n-1$        (1)

Answer to Problem 7.72P

The $l$ value must be either 1 or 2 in order to obtain an allowable combination of quantum numbers.

Explanation of Solution

The principal quantum number is denoted by $n$. The number 3 represents the principal quantum number.

For a given value of $n$, the only possible values for $l\left(\text{angular momentum quantum number}\right)$ can vary from 0 to $n-1$.

Substitute 3 for $n$ in equation (1) to calculate the highest value for $l$.

$\begin{array}{c}\text{Highest value for }l=3-1\\ =2\end{array}$

Therefore, the only possible values of $l$ for $n=3$ are 0, 1 and 2.

For a given value of $l$, the only possible values for $m_l$ extend from $-l$ to $+l$.

Therefore for $l=0$, the only possible value for $m_l$ is 0. Hence, the combination $n=3$;$l=0$;$m_l=-1$ is not allowed.

For $l=1$, the possible $m_l$ values are $-1,0,+1$. The $m_l$ value of $-1$ is thus possible for $l=1$.

For $l=2$, the possible $m_l$ values are $-2,-1,0,+1,+2$. The $m_l$ value of $-1$ is thus possible for $l=2$.

The value $m_l=-1$ is only possible if $l$ is either 1 or 2.

Conclusion

The $l$ value must be either 1 or 2 in order to obtain an allowable combination of quantum numbers.

(b)

Interpretation Introduction

Interpretation:

The correct $l$ value for the combination $n=3$;$l=3$;$m_l=+1$ is to be determined.

Concept introduction:

Quantum numbers are a set of four numbers that gives complete information about an electron within an atom. Out of the four, the three quantum numbers that define the shape, size, and orientation of an orbital are the principal quantum number, the orbital angular momentum quantum number, and the magnetic quantum number.

The formula to determine the total number of orbitals is,

$\text{Number of orbitals}=2l+1$        (1)

Answer to Problem 7.72P

The $l$ value must be either 1 or 2 in order to obtain an allowable combination of quantum numbers.

Explanation of Solution

The principal quantum number is denoted by $n$. The number 3 represents the principal quantum number.

For a given value of $n$, the only possible values for $l\left(\text{angular momentum quantum number}\right)$ can vary from 0 to $n-1$.

Substitute 3 for $n$ in equation (1) to calculate the highest value for $l$.

$\begin{array}{c}\text{Highest value for }l=3-1\\ =2\end{array}$

Therefore, the only possible values of $l$ for $n=3$ are 0, 1 and 2.

For a given value of $l$, the only possible values for $m_l$ extend from $-l$ to $+l$.

Therefore for $l=0$, the only possible value for $m_l$ is 0. Hence, a value of $l=0$ will not make an allowed combination.

For $l=1$, the possible $m_l$ values are $-1,0,+1$. The $m_l$ value of $+1$ is thus possible for $l=1$.

For $l=2$, the possible $m_l$ values are $-2,-1,0,+1,+2$. The $m_l$ value of $+1$ is thus possible for $l=2$.

The value $m_l=+1$ is only possible if $l$ is either 1 or 2.

Conclusion

The $l$ value must be either 1 or 2 in order to obtain an allowable combination of quantum numbers.

(c)

Interpretation Introduction

Interpretation:

The correct $l$ value for the combination $n=7$;$l=2$;$m_l=+3$ is to be determined.

Concept introduction:

Quantum numbers are a set of four numbers that gives complete information about an electron within an atom. Out of the four, the three quantum numbers that define the shape, size, and orientation of an orbital are the principal quantum number, the orbital angular momentum quantum number, and the magnetic quantum number.

The formula to determine the total number of orbitals is,

$\text{Number of orbitals}=2l+1$        (1)

Answer to Problem 7.72P

The $l$ value must be 3, 4, 5 or 6 in order to obtain an allowable combination of quantum numbers.

Explanation of Solution

The principal quantum number is denoted by $n$. The number 7 represents the principal quantum number.

For a given value of $n$, the only possible values for $l\left(\text{angular momentum quantum number}\right)$ can vary from 0 to $n-1$.

Substitute 7 for $n$ in equation (1) to calculate the highest value for $l$.

$\begin{array}{c}\text{Highest value for }l=7-1\\ =6\end{array}$

Therefore, the only possible values of $l$ for $n=7$ are 0, 1, 2, 3, 4, 5 and 6.

For a given value of $l$, the only possible values for $m_l$ extend from $-l$ to $+l$.

Therefore for $l=0$, the only possible value for $m_l$ is 0. Hence, a value of $l=0$ will not make an allowed combination.

For $l=1$, the possible $m_l$ values are $-1,0,+1$. Hence, a value of $l=1$ will not make an allowed combination.

For $l=2$, the possible $m_l$ values are $-2,-1,0,+1,+2$. Hence, a value of $l=2$ will not make an allowed combination.

For $l=3$, the possible $m_l$ values are $-3\text{, }-2,-1,0,+1,+2,\text{ +3}$. The $m_l$ value of $+3$ is thus possible for $l=3$.

For $l=4$, the possible $m_l$ values are $-4,-3,-2,-1,0,+1,+2,\text{ +3, }+4$. The $m_l$ value of $+3$ is thus possible for $l=4$.

For $l=5$, the possible $m_l$ values are $-5,-4,-3,-2,-1,0,+1,+2,\text{ +3, }+4,+5$. The $m_l$ value of $+3$ is thus possible for $l=5$.

For $l=6$, the possible $m_l$ values are $-6,-5,-4,-3,-2,-1,0,+1,+2,\text{ +3, }+4,+5,\text{ +6}$. The $m_l$ value of $+3$ is thus possible for $l=6$.

The value $m_l=+3$ is only possible if $l$ is 3, 4, 5 or 6.

Conclusion

The $l$ value must be 3, 4, 5 or 6 in order to obtain an allowable combination of quantum numbers.

(d)

Interpretation Introduction

Interpretation:

The correct $l$ value for the combination $n=4$;$l=1$;$m_l=-2$ is to be determined.

Concept introduction:

Quantum numbers are a set of four numbers that gives complete information about an electron within an atom. Out of the four, the three quantum numbers that define the shape, size, and orientation of an orbital are the principal quantum number, the orbital angular momentum quantum number, and the magnetic quantum number.

The formula to determine the total number of orbitals is,

$\text{Number of orbitals}=2l+1$        (1)

Answer to Problem 7.72P

The $l$ value must be 2or 3 in order to obtain an allowable combination of quantum numbers.

Explanation of Solution

The principal quantum number is denoted by $n$. The number 4 represents the principal quantum number.

For a given value of $n$, the only possible values for $l\left(\text{angular momentum quantum number}\right)$ can vary from 0 to $n-1$.

Substitute 4 for $n$ in equation (1) to calculate the highest value for $l$.

$\begin{array}{c}\text{Highest value for }l=4-1\\ =3\end{array}$

Therefore, the only possible values of $l$ for $n=4$ are 0, 1, 2 and 3.

For a given value of $l$, the only possible values for $m_l$ extend from $-l$ to $+l$.

For $l=1$, the possible $m_l$ values are $-1,0,+1$. Hence, a value of $l=1$ will not make an allowed combination.

For $l=2$, the possible $m_l$ values are $-2,-1,0,+1,+2$. The $m_l$ value of $-2$ is thus possible for $l=2$.

For $l=3$, the possible $m_l$ values are $-3\text{, }-2,-1,0,+1,+2,\text{ +3}$. The $m_l$ value of $-2$ is thus possible for $l=3$.

The value $m_l=-2$ is only possible if $l$ is 2 or 3.

Conclusion

The $l$ value must be 2or 3 in order to obtain an allowable combination of quantum numbers.