Question

When an electron makes a transtion from its $n+1$ state to the $nth$ state, the frequency of the emitted radiations is related to $n$ according to (where $n>>1)$:

• A. $v=\frac{2cR{Z}^2}{n^3}$
• B. $v=\frac{cR{Z}^2}{n^4}$
• C. $v=\frac{cR{Z}^2}{n^2}$
• D. $v=\frac{2cR{Z}^2}{n^2}$

Answer 1

The correct option is A $v=\frac{2cR{Z}^2}{n^3}$

Wave number $\stackrel{-}{v}$ for a transition $(n+1)\to n$
$\stackrel{-}{v}=\frac{1}{\lambda }=R[\frac{1}{n^2}-\frac{1}{(n+1{)}^2}]Z^2$

$\Rightarrow \frac{1}{\lambda }=\frac{(2n+1)}{n^2(n+1{)}^2}R{Z}^2$

$2n+1\approx 2n(\because n>>1)$
and $(n+1{)}^2\approx n^2$

$\therefore \frac{1}{\lambda }=\frac{\left(2n\right)}{(n^2.n^2)}R{Z}^2$

$\Rightarrow \frac{v}{c}=\frac{1}{\lambda }=\frac{2R{Z}^2}{n^3}$

$\Rightarrow v=\frac{2cR{Z}^2}{n^3}$