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Thermodynmaics- Energy equation

  1. Aug 31, 2011 #1
    1. The problem statement, all variables and given/known data

    The equation of state PV= U/3 applies to a photon gas at pressure P, volume V and internal energy U. By using the 'energy equation', show that the internal energy of a phont gas is proportional to the fourth power of temperature (i.e U [itex]\propto[/itex]T[itex]^{4}[/itex]). You can assume that U/V is only a function of temperature.

    2. Relevant equations

    Energy equation: (dU/dvV)[itex]_{T}[/itex] = T (dP/dT)[itex]_{V}[/itex] - P


    3. The attempt at a solution

    U = 3PV
    (dU/dV)[itex]_{T}[/itex] = 3P as (dP/dV)[itex]_{T}[/itex] = 0 as it is only a function of T

    From here.. I'm not sure what to do. Help?
     
  2. jcsd
  3. Sep 3, 2011 #2

    dynamicsolo

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    Homework Helper

    (typoes corrected)

    There is nothing in what you are told that allows you to state that [itex](\frac{dP}{dV})_{T} = 0 [/itex] . This would lead you to the result 3P = -P .

    What you know is that [itex]P = \frac{U}{3V} [/itex] ; we also know that U must depend on temperature T , particularly since we're being asked to find a proportionality for it. You need to differentiate P implicitly with respect to T , with V held constant; this will give you a relation between [itex]\frac{\partial P}{\partial T}[/itex] and [itex]\frac{\partial U}{\partial T}[/itex] . Use this to replace [itex]\frac{\partial P}{\partial T}[/itex] in your energy relation, bring all the P's to one side, and use the given relation U = 3PV to replace P . You will have a separable differential equation that will lead you to the proportionality you seek.
     
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