What is the Relationship Between Gas Ionisation and Temperature?

[niranjan_learner]

Homework Statement

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 N_a , 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 )

The Attempt at a Solution

The Maxwell distribution is N_e=N_gexp(-E/k_BT). Where N_g and N_e 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, V_a enters the equation.

Please give me hints to solve this problem.

Problem Reference: TIFR GS 2012 Physical Sciences Question Paper.

 

[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?

 

[niranjan_learner]

Thank you for the reply, haruspex.

Can you rule out some?

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 N_iN_e decreases with increase in temperature, which is not physical.

Please tell me the other two.

 

[haruspex]

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 N_iN_e decreases with increase in temperature, which is not physical.

Please tell me the other two.

The N values are densities, i.e. Number per V_a. I find it easier to think about if I convert back to actual numbers by multiplying each equation by V_a². If we write M_x=N_xV_a, x=a,i,e, it gets rid of the V_a factor.
Now consider that the exp(-...) term is < 1. Does that seem inconsistent with what the Ms are saying?

 

[niranjan_learner]

Hey, I have objection on this statement:

The N values are densities, i.e. Number per Va.

N values are densities, number per unit volume. Now in my opinion, M_x=N_xV_a will indicate fraction of unit volume occupied by the particular species which has to be less than 1.

Did I miss something?

 

[haruspex]

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

Did I miss something?

You're right, I should have written (N_a+N_i+N_e)V_a=1.
So that blows that idea away.
However, I think we can rule out (b) on the basis that the N_as can be cancelled.

 

[niranjan_learner]

You're right, I should have written (N_a+N_i+N_e)V_a=1.
So that blows that idea away.
However, I think we can rule out (b) on the basis that the N_as can be cancelled.

Yes. That is a good exam strategy!

The correct answer given in the answer key is option (d).

 

[haruspex]

Yes. That is a good exam strategy!

The correct answer given in the answer key is option (d).

That would have been my guess.

 

[niranjan_learner]

That would have been my guess.

I see