Use Bohr's model of the hydrogen atom

For an electron in the $n$th orbit of hydrogen, Bohr's quantization condition is: $mvr = n\frac{h}{2\pi}$

Use centripetal force balance

The electrostatic force provides the centripetal force: $\frac{mv^2}{r} = \frac{1}{4\pi\varepsilon_0}\frac{e^2}{r^2}$

Recall standard Bohr result for velocity

Solving the above equations gives the speed of electron in the $n$th orbit as: $v_n = \frac{1}{4\pi\varepsilon_0}\frac{e^2}{\hbar}\cdot \frac{1}{n}$

Hence, $v_n \propto \frac{1}{n}$

Check the options

• A: $v_n \propto \dfrac{1}{n}$ ✅ Correct
• B: $v_n \propto n^2$ ❌ Incorrect
• C: $v_n \propto n$ ❌ Incorrect
• D: $v_n \propto \dfrac{1}{n^2}$ ❌ Incorrect

Final answer

The correct option is: $\boxed{A}$

Comparison with stored correct answer

Stored correct answer: A

My derived answer is also A, so they agree.