Which of the plots shown in the figure represents speed ($$v$$) of the electron in a hydrogen atom as a function of the principal quantum number ($$n$$)?

The Bohr model gives two key relations for an electron revolving in the $$n^{\text{th}}$$ orbit of a hydrogen atom:

1. Angular‐momentum quantisation:$$m v_n r_n = n\hbar\qquad -(1)$$
2. Centripetal force equals Coulombic attraction:$$\frac{m v_n^{2}}{r_n} = \frac{e^{2}}{4\pi\epsilon_0 r_n^{2}}\qquad -(2)$$

From $$-(2)$$ we get $$m v_n^{2} r_n = \frac{e^{2}}{4\pi\epsilon_0}\qquad -(3)$$

Divide $$-(3)$$ by $$-(1)$$ to eliminate $$r_n$$:

$$\frac{m v_n^{2} r_n}{m v_n r_n} = \frac{\dfrac{e^{2}}{4\pi\epsilon_0}}{n\hbar}$$
$$\Rightarrow v_n = \frac{e^{2}}{4\pi\epsilon_0\hbar}\,\frac{1}{n}\qquad -(4)$$

Equation $$-(4)$$ shows that the speed $$v_n$$ is inversely proportional to the principal quantum number $$n$$:

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

Thus, when $$n$$ increases (electron jumps to higher orbits), its speed decreases hyperbolically. Among the four sketches given in the question, the only curve that falls as $$1/n$$ is the plot labelled “B”.

Therefore, the correct representation is:
Option A which is: $$B$$