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 $$480049Rch$$ (where$$ $$1$$ Rch $$=$$ $$13.6eV) . Find the value of N, where maximum possible number of photons emitted in the process is $$7N$$ x $$100$$.

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 $$480049Rch$$ (where$$ $$1$$ Rch $$=$$ $$13.6eV) . Find the value of N, where maximum possible number of photons emitted in the process is $$7N$$ x $$100$$.

Infinity Learn · Accepted Answer

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

$Released energy = \frac{4800}{49} \times R c h$

$Energy released per atom = \frac{48}{49} R c h = \frac{48}{49} \times 13.6 eV$

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

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

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

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

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

Hydrogen spectrum

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