Question

1 mole of $H{e}^{+}$ ion is excited. Spectral analysis showed the existence of 50% ions in $3^{rd}$ level, 25% in $2^{nd}$ level, and remaining 25% in ground state. The total energy evolved when all the ions return to ground state is:

• A. $232.88\times {10}^{4}J$
• B. $331.13\times {10}^{4}J$
• C. $58.22\times {10}^{4}J$
• D. $660\times {10}^{4}J$

Answer 1

The correct option is B $331.13\times {10}^{4}J$

$\Delta E=\left(IP{)}_Z\right(\frac{1}{n_1^2}-\frac{1}{n_2^2})$

$IP$ of $H{e}^{+}=+13.6\times 4=+54.4eV$

$\therefore \Delta E_{(3\to 1)}=\left(54.4\right)\frac{N_A}{2}(\frac{1}{1^2}-\frac{1}{3^2})$, where $N_A=6.023\times {10}^{23}$

$\Delta E_{(2\to 1)}=\left(54.4\right)\frac{N_A}{4}(\frac{1}{1^2}-\frac{1}{2^2})$

Hence, total energy $=\left(54.4\right)N_A(\frac{4}{9}+\frac{3}{16})eV=54.4\times 6.02\times {10}^{23}\times \frac{91}{144}eV$

$=54.4\times 6.02\times {10}^{23}\times \frac{91}{144}\times 1.6\times {10}^{-19}=331.13\times {10}^{4}J$