To find the total number of photons emitted during the transition of electrons from the [tex]\(n_2\)[/tex] shell to the [tex]\(n_1\)[/tex] shell in a [tex]\( He^+ \)[/tex] ion, we need to solve the given system of equations and then use the information to calculate the number of photons.
The given system of equations is:
[tex]\[
\begin{cases}
2n_2 + 3n_1 = 18 \\
2n_2 - 3n_1 = 6
\end{cases}
\][/tex]
Step 1: Solve the system of equations for [tex]\(n_2\)[/tex] and [tex]\(n_1\)[/tex]
We start by adding the two equations to eliminate [tex]\( n_1 \)[/tex]:
[tex]\[
(2n_2 + 3n_1) + (2n_2 - 3n_1) = 18 + 6
\][/tex]
This simplifies to:
[tex]\[
4n_2 = 24
\][/tex]
Divide both sides by 4:
[tex]\[
n_2 = 6
\][/tex]
Next, substitute [tex]\( n_2 = 6 \)[/tex] back into one of the original equations to solve for [tex]\( n_1 \)[/tex]:
Using [tex]\( 2n_2 - 3n_1 = 6 \)[/tex]:
[tex]\[
2(6) - 3n_1 = 6
\][/tex]
This simplifies to:
[tex]\[
12 - 3n_1 = 6
\][/tex]
Subtract 12 from both sides:
[tex]\[
-3n_1 = -6
\][/tex]
Divide both sides by -3:
[tex]\[
n_1 = 2
\][/tex]
So, we have [tex]\( n_2 = 6 \)[/tex] and [tex]\( n_1 = 2 \)[/tex].
Step 2: Calculate the total number of photons emitted
Transitions to the [tex]\( n_1 \)[/tex] level from higher levels will involve the following steps and numbers of photons:
- From [tex]\( n_2 = 6 \)[/tex] to [tex]\( n_1 = 2 \)[/tex]
- Emitting photons for every transition from [tex]\( n_2 \)[/tex] down to [tex]\( n_1 \)[/tex]
The formula to calculate the total number of transitions (photons) is:
[tex]\[
\text{Total photons} = \frac{n_{2}(n_{2} + 1)}{2} - \frac{n_{1}(n_{1} - 1)}{2}
\][/tex]
Substitute the values [tex]\( n_2 = 6 \)[/tex] and [tex]\( n_1 = 2 \)[/tex] into this formula:
[tex]\[
\text{Total photons} = \frac{6(6 + 1)}{2} - \frac{2(2 - 1)}{2}
\][/tex]
This simplifies to:
[tex]\[
\text{Total photons} = \frac{6 \times 7}{2} - \frac{2 \times 1}{2}
\][/tex]
[tex]\[
\text{Total photons} = \frac{42}{2} - \frac{2}{2}
\][/tex]
[tex]\[
\text{Total photons} = 21 - 1
\][/tex]
[tex]\[
\text{Total photons} = 20
\][/tex]
Therefore, the total number of photons emitted when electrons transition to the [tex]\( n_1 = 2 \)[/tex] shell is 20.
The correct answer is:
(C) 20
