Question

The smallest ratio of time periods of an electron revolving in the second orbit of a hydrogen atom to another electron revolving in the third orbit of $H{e}^{+}$ is $x:y$. Find the value of $x+y$

Answer 1

We know that time period of an electron in the $n^{th}$ orbit of Hydrogen like atom is:
$T_n\propto \frac{n^3}{Z^2}$
$\frac{T_{(n=2,Z=1)}}{T_{(n=3,Z=2)}}=\frac{\left(2^3/1^2\right)}{\left(3^3/2^2\right)}=\frac{32}{27}$
$x+y=32+27=59$