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Problem about ionisation

  1. Oct 15, 2016 #1
    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. jcsd
  3. Oct 16, 2016 #2

    haruspex

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    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?
     
  4. Oct 16, 2016 #3
    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.
     
  5. Oct 16, 2016 #4

    haruspex

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    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?
     
  6. Oct 16, 2016 #5
    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?
     
  7. Oct 16, 2016 #6

    haruspex

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    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.
     
  8. Oct 16, 2016 #7
    Yes. That is a good exam strategy!

    The correct answer given in the answer key is option (d).
     
  9. Oct 16, 2016 #8

    haruspex

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    That would have been my guess.
     
  10. Oct 17, 2016 #9
    I see :smile:
     
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