Question

In the hydrogen atom spectrum, let $E_1$ and $E_2$ are energies for the transition $n=2\to n=1$ and $n=3\to$ $n=2$ respectively. The ratio $E2/E1$ is

• A. $\frac{2}{3}$
• B. $\frac{3}{2}$
• C. $\frac{2}{9}$
• D. $\frac{5}{27}$

Answer 1

Answer: (D)

Energy value of nth state for $H$-electron is
$E_n=\frac{E_1}{n^2}$
Now, given transitions are

Energy radiated in transition $n=3$ to $n=2$ is
$E_2=\frac{E_1}{9}-\frac{E_1}{4}=\frac{-5E_1}{36}$
Also, energy radiated in transition $n=2$ to $n=1$ is
$E_1=\frac{E_1}{4}-\frac{E_1}{1}=\frac{-3E_1}{4}$
Hence, ratio of $\frac{E_2}{E_1}$ is $\frac{5}{27}$.