Question

The frequency of first line of balmer series in hydrogen atom is V1 the frequency of corresponding line emitted by singly ionized helium atom is ​

Answer 1

from Rydberg's equation,

$\frac{1}{\lambda}=RZ^2\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$

or, $\nu=RcZ^2\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$

for first line of Balmer's series

$n_1=2,n_2=3$

case 1: $\nu=\nu_1,Z=1$

so, $\nu_1=Rc(1)^2\left(\frac{1}{2^2}-\frac{1}{3^2}\right)$.....(1)

case 2: $\nu=x,Z=2$

then, $x=Rc(2)^2\left(\frac{1}{2^2}-\frac{1}{3^2}\right)$....(2)

from equations (1) and (2),

$\frac{\nu_1}{x}=\frac{1}{2^2}$

or, $x=4\nu_1$

hence, answer is $4\nu_1$