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I Can internal energy be calculated from equation of state?

  1. Oct 15, 2016 #1
    We know,
    $$dU=TdS-PdV$$
    ##\int PdV## can be calculated if the equation of state is given.
    I tried to express ##S## as a function of ##P ,V## or ##T## (any two of those).
    $$dS=\left(\frac{\partial S}{\partial V}\right)_T dV+\left(\frac{\partial S}{\partial T}\right)_V dT$$
    $$=\left(\frac{\partial P}{\partial T}\right)_V dV+\left(\frac{\partial S}{\partial P}\right)_V \left(\frac{\partial P}{\partial T}\right)_V dT~~~ [Using ~~Maxwell's~~ relation]$$
    $$=\left(\frac{\partial P}{\partial T}\right)_V dV-\left(\frac{\partial V}{\partial T}\right)_S \left(\frac{\partial P}{\partial T}\right)_V dT~~~[Using ~~Maxwell's~ ~relation]$$
    All the terms except ##\left(\frac{\partial V}{\partial T}\right)_S## can be calculated using the equation of state.
    Any suggestion will be appreciated.
     
  2. jcsd
  3. Oct 15, 2016 #2
    You can't do it solely in terms of the equation of state. You need to use the heat capacity as well.
     
  4. Oct 15, 2016 #3
    Thanks. That's exactly what I wanted to know.
     
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