Question

The electron in a hydrogen atom makes a transition $n_1\to n_2$, where $n_1$ and $n_2$ are the principal quantum numbers of the two energy states. Assume Bohr's model to be valid. The time period of the electron in the initial state is eight times that in the final state. What are the possible values of $n_1$ and $n_2$?

Answer 1

Here, $Z=1$. We can derive the expression for the time period of electron in $n^{th}$ orbit as
$T_n=\frac{1}{f_n}\propto n^3$
Hence
$\frac{T_1}{T_2}=\frac{n_1^3}{n_2^3}$. As $T_1=8T_2$, therefore we get
$8={\left(\frac{n_1}{n_2}\right)}^3\Rightarrow n_1=2n_2$
Thus, the possible values of $n_1$ and $n_2$ are $n_1=2,n_2=1;n_1=4,n_2=2;n_1=6,n_2=3$ and so on.