Question

An electron in a hydrogen atom makes a transition from $n_{1}$ to $n_{2}$ where $n_{1}$ and $n_{2}$ are principal quantum numbers of the two states. Assume the Bohr model to be valid. The time period of the electron in the initial state is eight times that in the final state. Calculate the ratio of $n_{1}$ and $n_{2}$

Answer 1

SOLUTION : T $=\frac{2 \pi r}{v}$

we know that or, $T \propto \frac{r}{v}$

Now $r \propto n^{2}$ and $v \propto \frac{1}{n}$ for $H$-atom

$T \propto n^{3}$

$\frac{T_{1}}{T_{2}}=\frac{n_{1}^{3}}{n_{2}^{3}}$ Given $T_{1}=8 T_{2}$

$\frac{8 T_{2}}{T_{2}}=\frac{n_{1}^{3}}{n_{2}^{3}}$ or, $\frac{n_{1}^{3}}{n_{2}^{3}}=8$

or, $\frac{n_{1}}{n_{2}}=\frac{2}{1}$