To determine the energy change for an electron transitioning from the [tex]\( n=2 \)[/tex] level to the [tex]\( n=5 \)[/tex] level in a hydrogen atom, we can use the energy levels of the hydrogen atom, which are given by the formula:
[tex]\[
E_n = -\frac{13.6 \text{ eV}}{n^2}
\][/tex]
However, in this situation, all the specific values and constants are given directly in joules.
The energy of an electron in a particular quantum level [tex]\( n \)[/tex] in hydrogen is:
[tex]\[
E_n = -2.18 \times 10^{-19} \text{ J} / n^2
\][/tex]
First, we will calculate the initial energy level when the electron is at [tex]\( n=2 \)[/tex]:
[tex]\[
E_{\text{initial}} = \frac{-2.18 \times 10^{-19} \text{ J}}{2^2}
\][/tex]
[tex]\[
E_{\text{initial}} = \frac{-2.18 \times 10^{-19} \text{ J}}{4}
\][/tex]
[tex]\[
E_{\text{initial}} = -5.45 \times 10^{-20} \text{ J}
\][/tex]
Next, we calculate the final energy level when the electron is at [tex]\( n=5 \)[/tex]:
[tex]\[
E_{\text{final}} = \frac{-2.18 \times 10^{-19} \text{ J}}{5^2}
\][/tex]
[tex]\[
E_{\text{final}} = \frac{-2.18 \times 10^{-19} \text{ J}}{25}
\][/tex]
[tex]\[
E_{\text{final}} = -8.72 \times 10^{-21} \text{ J}
\][/tex]
The energy change [tex]\( \Delta E \)[/tex] associated with the transition from [tex]\( n=2 \)[/tex] to [tex]\( n=5 \)[/tex] is the difference between the final energy and the initial energy:
[tex]\[
\Delta E = E_{\text{final}} - E_{\text{initial}}
\][/tex]
[tex]\[
\Delta E = -8.72 \times 10^{-21} \text{ J} - (-5.45 \times 10^{-20} \text{ J})
\][/tex]
[tex]\[
\Delta E = -8.72 \times 10^{-21} \text{ J} + 5.45 \times 10^{-20} \text{ J}
\][/tex]
[tex]\[
\Delta E = 4.578 \times 10^{-20} \text{ J}
\][/tex]
So, the energy change associated with the transition from [tex]\( n=2 \)[/tex] to [tex]\( n=5 \)[/tex] in the hydrogen atom is:
[tex]\[
4.58 \times 10^{-20} \text{ J}
\][/tex]
Therefore, the correct answer is:
[tex]\[ 4.58 \times 10^{-20} \text{ J} \][/tex]
