Question

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, 1Rch = 13.6 eV). Find the value of N, where maximum possible number of photons emitted in the process is 7N $\times$ 100. ___

Answer 1

As all of atoms are in same $n^{th}$ state, they release equal energy.
(but number of transitions may be different for each)
Released energy $=\frac{4000}{49}\times Rch$
Energy released per atom
$=\frac{48}{49}Rch=\frac{48}{49}\times 13.6eV$
$=\left(\frac{49-1}{49}\right)\times 13.6$
$=(\frac{1}{1^2}-\frac{1}{7^2})\times 13.6eV$
So, atoms are in $7^{th}$ excited state initially.
So, each atom can undergoes a maximum of
$\frac{n(n-1)}{2}=\frac{7\times 6}{2}=21$ transitions (per atom).
So, maximum possible number of photons emitted, by the sample of 100 atoms.
$=21\times 100=2100$
$=7\times 3\times 100$
$\therefore N=3$