# Maximum number of atoms in the ground state

I was given the following question.

A sample contains $$100$$ identical and hypothetical $$H$$-like atoms ( $$Z$$ may be fractional). Out of the given atoms, some are in the ground state, while others are in a higher energy level ($$n=x$$). The $$P.E$$ of the electrons in the ground state is $$-192\ \mathrm{eV}$$.

The sample is subject to radiations of wavelength $$155\ \mathrm{nm}$$, and the electrons jump to a higher energy level ($$n=x+3$$). Upon back-transition, a total of $$10$$ different spectral lines are observed

If the final state the electrons reach after absorbing photons of $$\lambda=155\ \mathrm{nm}$$ is $$a$$ , and the maximum number of atoms which had electrons in the ground state initially is $$b$$, find $$\frac{b}{a}$$

By equating $${}^{x+3}C_2$$ to $$10$$, I found $$x$$ to be $$2$$. Then, $$a$$ must be $$x+3$$, which is $$5$$.

As to finding the value of $$b$$, I am stuck. Can we use the fact that $$\frac{|P.E|}{2} = |T.E|$$? The energy associated with each photon comes out to be $$8\ \mathrm{eV}$$, so the total energy absorbed is $$800\ \mathrm{eV}$$ (Since there are $$100$$ atoms). Maybe we can use this?

Any help would be greatly appreciated.

Edit:- The answers given are $$x+3=6$$ and $$96$$ , so I ended up getting the first part wrong as well.

• Look at the answers and comments to this question, chemistry.stackexchange.com/questions/111109/…. – porphyrin Apr 17 '19 at 19:10
• I have no idea what $P.E$ and $T.E$ are supposed to mean. Could you please use proper quantity symbols? – Faded Giant Apr 17 '19 at 20:03
• @Loong The potential and total energy – Poutnik Apr 18 '19 at 3:52
• @porphyrin The link doesn't answer my question at all :). I am stuck finding the value of $b$ , not $a$ – Aspirant Apr 18 '19 at 4:36

Of the $$100$$, initially some are in $$n=1$$, and some in $$n=2$$ and then some in $$n=5$$ after absorbing 155 nm then some in lots of other levels after photons emitted. Thus, eventually, none will be left in $$n = 5$$ but spread in all in the other levels including the ground state. However, all other possible transitions occur ($$n=2 \to 1$$ for example) so all end up in the ground state in total $$(100-b)$$. However, you do not know the initial amount $$b$$ so I'm not sure that this question can be answered except that the ratio is $$b/(100-b)$$.