Chapter 29, Problem 1OQ

To determine

The number of electrons can be placed in the $n=3$ energy level in a hydrogen atom.

Answer to Problem 1OQ

The number of electrons can be placed in the $n=3$ energy level in a hydrogen atom is $\underset{\_}{18}$ .

Explanation of Solution

In this case, the principal quantum number, $n=3$ .

Write the expression for the orbital quantum number,

    $l=n-1$        (I)

Here, $l$ is the orbital quantum number and $n$ is the principal quantum number.

Write the expression for the orbital magnetic quantum number,

    $m_l=2l+1$        (II)

Here, $m_l$ is the orbital magnetic quantum number.

In this case, there are $2$ is the possible spin orientations for each value of orbital magnetic quantum number.

    $m_s=\pm \frac{1}{2}$        (III)

Here, $m_s$ is the spin orientation for each value of orbital magnetic quantum number.

Write the expression for the total possible values for each value of the orbital quantum number,

    $T=2\left(2l+1\right)$        (IV)

Here, $T$ is the grand total possible value for each value of the orbital quantum number.

Write the expression for the grand total of $n=3$ states,

    $T=T_1+T_2+T_3$        (V)

$T_1,T_2\&T_3$ are the possible states.

Conclusion:

Substitute $3$ for $n$ in (I),

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

So, the possible values of orbital magnetic quantum number, $l=0,1,2.$

Substitute $3$ for $n$ and $0$ for $l$ in (IV),

    $\begin{array}{c}{T}_1=2\left[2\left(0\right)+1\right]\\ =2\end{array}$

Substitute $3$ for $n$ and $1$ for $l$ in (IV),

    $\begin{array}{c}{T}_2=2\left[2\left(1\right)+1\right]\\ =2\left[3\right]\\ =6\end{array}$

Substitute $3$ for $n$ and $2$ for $l$ in (IV),

    $\begin{array}{c}{T}_1=2\left[2\left(2\right)+1\right]\\ =2\left[5\right]\\ =10\end{array}$

Substitute $2$ for $T_1$ , $6$ for $T_2$ and $10$ for $T_3$ in (V),

    $\begin{array}{c}T=2+6+10\\ =18\end{array}$

Thus, the number of states are $3s,3p\&3d$ .

Therefore, the number of electrons can be placed in the $n=3$ energy level in a hydrogen atom is $\underset{\_}{18}$.