Question

An electron in a hydrogen atom makes a transition $n_1\to n_2$ where $n_1$ and $n_2$ are principal quantum numbers to be valid. The time period of the electron in initial state is eight times that in the final state. Then the ratio of $n_1$ and $n_2$ will be

• A. 8 : 1
• B. 4 : 1
• C. 2 : 1
• D. 1 : 2

Answer 1

Answer: (C)

Let T be the periodic time and r the radius of circular orbit, then $T^2\propto r^3$ So, ${\left(\frac{T_1}{T_2}\right)}^2\propto {\left(\frac{r_1}{r_2}\right)}^3$ or $\frac{r_1}{r_2}={\left(\frac{T_1}{T_2}\right)}^{2/3}$ $={(8)}^{2/3}=4$ If n is principal quantum number, then $r=\frac{{\epsilon }_0{h}^2{n}^2}{\pi m{e}^2}\propto n^2$ So, $\frac{r_1}{r_2}={\left(\frac{n_1}{n_2}\right)}^2$ or $\frac{n_1}{n_2}=\sqrt{\frac{r_1}{r_2}}$ $=\sqrt{\frac{4}{1}}=2:1$