A sample of monatomic hydrogen gas contains 100 atoms and all the atoms are in nth excited state. As the atoms come down to the ground state following various transitions, they release a total energy of $\frac{4800}{49}Rch$ (where 1 Rch = 13.6eV) . Find the value of N, where maximum possible number of photons emitted in the process is 7N x 100.

As all of atoms are in same nth state, they release equal energy (but number of transitions may be different for each).

$\text{ Released energy }=\frac{4800}{49}\times Rch$

$\text{ Energy released per atom }=\frac{48}{49}Rch=\frac{48}{49}\times 13.6\mathrm{eV}$

$=\left(\frac{49-1}{49}\right)\times 13.6=\left(\frac{1}{1^2}-\frac{1}{7^2}\right)\times 13.6\mathrm{eV}$

So, atoms are in 7^th excited state initially.
So, each atom can undergoes a maximum of

$\frac{n(n-1)}{2}=\frac{7\times 6}{2}=21\text{ transitions (per atom). }$

So, maximum possible number of photons emitted, by the sample of 100 atoms.

$\begin{array}{rcl} & =& 21\times 100=2100=7\times 3\times 100\\ \therefore N& =& 3\end{array}$