Question

The frequency of $1^{st}$ line of Balmer series in $\text{H}$- atom is ${\nu }_0$. If the frequency of the same line emitted by singly ionized $\text{He}$ atom is $x{\nu }_0$, then $x$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is D $4$

The formula for the wavelength of emitted radiation in hydrogen spectral series is given by,

$\frac{1}{\lambda }=Z^{2}R(\frac{1}{n_f^2}-\frac{1}{n_i^2})$

Here, In both the cases, transition states are same,

$\Rightarrow \frac{1}{\lambda }\propto Z^2$

$\therefore \text{frequency}\nu \propto Z^2$

$\Rightarrow \frac{{\nu }_1}{{\nu }_2}={\left(\frac{Z_1}{Z_2}\right)}^2$

Let ${\nu }^{\prime }$ be the frequency of line emitted by ${\text{He}}^{+}$.

$\Rightarrow \frac{{\nu }_0}{{\nu }^{\prime }}={\left(\frac{1}{2}\right)}^2$

$\Rightarrow {\nu }^{\prime }=4{\nu }_0$