Question

In a mixture of H - $H{e}^{+}$ gas ($H{e}^{+}$ is singly ionized He atom). H atoms and $H{e}^{+}$ ions are excited to their respective first excited states. Subsequently, H atoms transfer their total excitation energy to $H{e}^{+}$ ions (by collisions ). Assume that the Bohr model of is atom exactly valid.

Answer 1

Energy released by H atom

$\Delta E_H=\frac{3}{4}\times 13.6eV$

Let $H{e}^{+}$ goes to nth state at the end

So energy required by $H{e}^{+}$ ions is

$\Delta E_{He}=13.6\times 4(\frac{1}{4}-\frac{1}{n^2})eV$

As $\Delta E_H={\Delta }_{He}$

$\frac{3}{4}\times 13.6=13.6\times 4(\frac{1}{4}-\frac{1}{n^2})$

$\frac{1}{4}-\frac{1}{n^2}=\frac{3}{16}$

$\frac{1}{n^2}=\frac{1}{4}-\frac{3}{16}$

$\frac{1}{n^2}=\frac{4-3}{16}$

$\frac{1}{n^2}=\frac{1}{16}$

$\frac{1}{n}=\frac{1}{4}$

$n=4$