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Relationship Between V and T in Adiabatic Expansion

  1. Sep 22, 2015 #1
    1. The problem statement, all variables and given/known data
    The energy and pressure of black body radiation depend on T and V as Eq(1) & Eq(2). Suppose that the temperature and volume of a box of radiation change adiabatically. Find the relation between dE and dT in this process. Next, using Eq(1), show that T ∝V^-1/3

    2. Relevant equations
    Eq(1): E = σVT^4;
    Eq(2): p = 1/3σT^4;
    ΔE = Q - W;

    Since Q = 0;
    ΔE = -W

    3. The attempt at a solution

    To begin with we've (a few people working together) have tried what appears to be an overly simple method.
    E = σ V T^4
    dE/dT = 4 σ V T^3
    dE = 4 σ V T^3 dT
    V = dE/(4 σ T^3 dT)

    ∴ V∝T^-1/3

    But this seems overly simplistic, especially since volume is changing too. Any formulae/approaches we're missing?
     
  2. jcsd
  3. Sep 22, 2015 #2

    haruspex

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    Do you realise that in your final answer V and T are swapped compared with what was given to be proved?
    Certainly the method is not valid.
    You do not seem to have used Eq 2 at all. I would think you need to use that and some relationship between V, E and p.
     
  4. Sep 22, 2015 #3
    That is correct. I apologize. It was a mistake of plain anticlimactic proportions.

    And yes, I did end up using Eq(2), and we got the answer after some rather petty algebra and a step of differential equations. Problem solved.
     
  5. Sep 24, 2015 #4

    haruspex

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    Was that petty or pretty?
     
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