D

To find the minimum velocity \( u \) required for one hydrogen atom to gain excitation energy, we need to ensure that the kinetic energy of the moving hydrogen atom is equal to the ionization energy.
Step 1: Calculate the minimum energy required.
The ionization energy given is 13.6 eV. We need to convert this to joules.
1 eV = 1.6 × 10^-19 J, hence
\( E = 13.6 \text{ eV} \times 1.6 \times 10^{-19} \text{ J/eV} = 2.176 \times 10^{-18} \text{ J} \).
Step 2: Relate this energy to kinetic energy.
The kinetic energy (KE) of a hydrogen atom is given by:
\( KE = \frac{1}{2} m u^2 \).
where \( m = 1.0078 \times 1.66 \times 10^{-27} \text{ kg} \approx 1.6735 \times 10^{-27} \text{ kg} \).
Step 3: Set the kinetic energy equal to the excitation energy.
So, we have:
\( \frac{1}{2} (1.6735 \times 10^{-27}) u^2 = 2.176 \times 10^{-18} \).
Step 4: Solve for u.
\( u^2 = \frac{2 \times 2.176 \times 10^{-18}}{1.6735 \times 10^{-27}} \)
\( u^2 = \frac{4.352 \times 10^{-18}}{1.6735 \times 10^{-27}} \)
\( u^2 \approx 2.60 \times 10^{9} \)
Taking the square root:
\( u \approx \sqrt{2.60 \times 10^{9}} \approx 50988 \text{ m/s} \)
Therefore, the answer is D.