a hydrogen electron is elevated from level 1 to level 2. another electron is elevated from level 2 to level…

Explanation:

Step1: Recall energy - level formula

The energy of an electron in a hydrogen - like atom is given by $E =-\frac{13.6}{n^{2}}\text{ eV}$, where $n$ is the principal quantum number. The energy change $\Delta E$ for a transition from $n_i$ to $n_f$ is $\Delta E=E_f - E_i=- 13.6\left(\frac{1}{n_f^{2}}-\frac{1}{n_i^{2}}\right)\text{ eV}$.

Step2: Calculate energy change for $n = 1$ to $n = 2$ transition

For the transition from $n_i = 1$ to $n_f = 2$, $\Delta E_1=-13.6\left(\frac{1}{2^{2}}-\frac{1}{1^{2}}\right)=-13.6\left(\frac{1}{4} - 1\right)=-13.6\times\left(-\frac{3}{4}\right)=10.2\text{ eV}$.

Step3: Calculate energy change for $n = 2$ to $n = 4$ transition

For the transition from $n_i = 2$ to $n_f = 4$, $\Delta E_2=-13.6\left(\frac{1}{4^{2}}-\frac{1}{2^{2}}\right)=-13.6\left(\frac{1}{16}-\frac{1}{4}\right)=-13.6\times\left(-\frac{3}{16}\right)=2.55\text{ eV}$.

Step4: Compare the energy changes

Since $10.2\text{ eV}>2.55\text{ eV}$, the transition from level 1 to level 2 requires the greatest energy change.

Answer:

The transition from level 1 to level 2.