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1. Oct 15, 2016

### niranjan_learner

1. The problem statement, all variables and given/known data
When a gas is enclosed in an impermeable box and heated to a high temperature T, some of the neutral atoms lose an electron and become ions. If the number density of neutral atoms, ions and electrons is Na , N i and N e , respectively, these can be related to the average volume V a occupied by an atom/ion and the ionisation energy E by the relation

2. Options
(a) N e (N a + N i ) = (N a /V a ) exp (−E/k B T )
(b) N a (N e + N i ) = (N a /V a ) exp (−E/k B T )
(c) N e N i = (N a /V a ) exp (+E/k B T )
(d) N e N i = (N a /V a ) exp (−E/k B T )

3. The attempt at a solution
The Maxwell distribution is Ne=Ngexp(-E/kBT). Where Ng and Ne are number of atoms in ground state and exited state respectively, E is the energy difference between the states. I don't know how to apply this concept here and how the average volume occupied by atom, Va enters the equation.

Please give me hints to solve this problem.

Problem Reference: TIFR GS 2012 Physical Sciences Question Paper.

2. Oct 16, 2016

### haruspex

I have no idea how to derive any such equations, but I believe can see reasons for ruling out three of them.
Can you rule out some?

3. Oct 16, 2016

### niranjan_learner

Thank you for the reply, haruspex.

I can rule out one of them. Namely option (c). N e N i = (N a /V a ) exp (+E/k B T )
Because then the product NiNe decreases with increase in temperature, which is not physical.

Please tell me the other two.

4. Oct 16, 2016

### haruspex

The N values are densities, i.e. Number per Va. I find it easier to think about if I convert back to actual numbers by multiplying each equation by Va2. If we write Mx=NxVa, x=a,i,e, it gets rid of the Va factor.
Now consider that the exp(-...) term is < 1. Does that seem inconsistent with what the Ms are saying?

5. Oct 16, 2016

### niranjan_learner

Hey, I have objection on this statement:
N values are densities, number per unit volume. Now in my opinion, Mx=NxVa will indicate fraction of unit volume occupied by the particular species which has to be less than 1.

Did I miss something?

6. Oct 16, 2016

### haruspex

You're right, I should have written (Na+Ni+Ne)Va=1.
So that blows that idea away.
However, I think we can rule out (b) on the basis that the Nas can be cancelled.

7. Oct 16, 2016

### niranjan_learner

Yes. That is a good exam strategy!

8. Oct 16, 2016

### haruspex

That would have been my guess.

9. Oct 17, 2016

I see