1. Energy of the absorbed photon

Given wavelength, $\lambda = 980\,\text{\AA}$

Using $E = \frac{hc}{\lambda}$ with $hc = 12500\,\text{eV\AA}$ we get $E = \frac{12500}{980}\,\text{eV}$ $E \approx 12.76\,\text{eV}$

2. Energy needed to excite hydrogen from ground state

For hydrogen atom, $E_n = -\frac{13.6}{n^2}\,\text{eV}$

Initially the atom is in ground state, so $E_1 = -13.6\,\text{eV}$

If it absorbs a photon and reaches level $n$, then $E_{\text{photon}} = E_n - E_1$

So, $12.76 = -\frac{13.6}{n^2} - (-13.6)$ $12.76 = 13.6\left(1 - \frac{1}{n^2}\right)$

Divide by $13.6$: $\frac{12.76}{13.6} = 1 - \frac{1}{n^2}$ $0.938 \approx 1 - \frac{1}{n^2}$

Thus, $\frac{1}{n^2} \approx 0.062 = \frac{1}{16}$

Hence, $n^2 = 16 \Rightarrow n=4$

3. Radius of hydrogen atom in nth orbit

Bohr radius formula: $r_n = n^2 a_0$

For $n=4$, $r_4 = 4^2 a_0 = 16a_0$

4. Option check

• A: $4a_0$ ❌
• B: $9a_0$ ❌
• C: $25a_0$ ❌
• D: $16a_0$ ✅

Therefore, the correct answer is D.