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Maxwell Relation, Gibbs Free Energy, Thermal Expansion Coefficient

  1. Oct 26, 2017 #1
    1. The problem statement, all variables and given/known data
    By means of a Maxwell relation derived from the Gibbs free energy and making use of the third law of thermodynamics, prove that the thermal expansion coefficient β must be zero at T = 0. I tried but I got something funny.

    2. Relevant equations
    $$G=U-TS+PV$$
    $$dG=\mu dN-SdT+VdP$$
    $$S=Nk_B[\ln(\frac{V}{N}(\frac{4\pi mU}{3Nh^2})^{3/2})+\frac{5}{2}]$$
    $$PV=Nk_B T$$
    $$\beta = \frac{1}{V}\frac{\partial V}{\partial T} \Bigg| _{N,P}$$
    3. The attempt at a solution
    guestaccess.jpg
     
  2. jcsd
  3. Oct 26, 2017 #2

    TSny

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    I believe you have essentially shown that an ideal gas is not compatible with the third law.

    I think you should be able to finish the proof using your result upload_2017-10-26_11-51-36.png without assuming the ideal gas law.
     
    Last edited: Oct 26, 2017
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