JEE Main 2023 — Structure of Atom Question with Solution

Paschen series are spectral lines coming to $n=3$

$1^{\mathrm{st}}$ line of Paschen series makes a transition from, $4\to 3$

So, $2^{\mathrm{nd}}$ line of Paschen series makes a transition from, $5\to 3$

We know that,

$\frac{1}{\mathrm{\lambda }}=\mathrm{R}\left[\frac{1}{{\mathrm{n}}_1^2}-\frac{1}{{\mathrm{n}}_2^2}\right]{\mathrm{Z}}^2$

For $1^{\mathrm{st}}$ line, $\frac{1}{\lambda }=R\left[\frac{1}{(3{)}^2}-\frac{1}{(4{)}^2}\right]Z^2$

and for $2^{\mathrm{nd}}$ line,

$\frac{1}{{\lambda }^{\text{'}}}=R\left[\frac{1}{(3{)}^2}-\frac{1}{(5{)}^2}\right]Z^2$

or $\frac{\left({\displaystyle \frac{1}{\lambda }}\right)}{\left({\displaystyle \frac{1}{{\lambda }^{\text{'}}}}\right)}=\frac{R\left[\frac{7}{9\times 16}\right]Z^2}{R\left[\frac{16}{9\times 25}\right]Z^2}$

Given, $\lambda =720 \mathrm{nm}$

or $\frac{\left({\displaystyle \frac{1}{720}}\right)}{\left({\displaystyle \frac{1}{\lambda \text{'}}}\right)}=\frac{175}{256}$

or  $\frac{{\lambda }^{\text{'}}}{720}=\frac{175}{256}$

or ${\mathrm{\lambda }}^{\text{'}} \approx 492 \mathrm{nm}$