Question

When an electron makes transition from $\left(n + 1\right)$ state to n state the wavelength of emitted radiations is related to $n (n >>> 1) $according to $\lambda \propto n^x$. What is the value of $x$ ?

Answer 1

$\frac{1}{\lambda} = RZ^{2} \left(\frac{1}{n^{2}} - \frac{1}{\left(n + 1\right)^{2}}\right)$
$\frac{1}{\lambda} = RZ^{2} \frac{\left(n+1\right)^{2} -n^{2}}{n^{2}\left(n+1\right)^{2}}$
$\frac{1}{\lambda} = RZ^{2} \frac{\left(n+1+n\right)\left(n +1 -n\right)}{n^{2} \left(n +1\right)}$
$\frac{1}{\lambda} = RZ^{2} \frac{\left(2n +1\right)}{n^{2}\left(n +1\right)^{2}}$
$\lambda\propto n^{3}$
$x =3$