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Finding compressibility from given internal Energy function

  1. Jun 28, 2017 #1
    Hi everyone!

    1. The problem statement, all variables and given/known data


    Given is a function for the internal energy: ##U(T,V)=Vu(T)##
    Asked is to derive the entropy balance equation. In order to do so i need to find the "isothermal and adiabatic compressibility": $$\kappa_{T}=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}$$

    3. The attempt at a solution

    In order to calculate ##\kappa_{T}## I need to find a function ##V(P,T)## right? But how do I get this function from the given internal energy function?

    Thanks for your help!
     
  2. jcsd
  3. Jun 28, 2017 #2
    The isothermal and adiabatic compressibilities are second derivatives for U. So you may try taking derivatives of the internal energy equation using the technique of implicit differentiation.
     
  4. Jun 28, 2017 #3
    What is the general equation (not for this specific material) for dU in terms of dT and dV?
     
  5. Jun 28, 2017 #4
    I guess its $$dU(T,V)=\left(\frac{\partial U}{\partial T}\right)_{V} dT+\left(\frac{\partial U}{\partial V}\right)_{T} dV$$ Or one can write: $$dU(T,V)=C_{V}dT-PdV$$
    right?
     
  6. Jun 28, 2017 #5
    Wrong. $$dU(T,V)=C_{V}dT-\left[P-T\left(\frac{\partial P}{\partial T}\right)_V\right]dV$$
     
  7. Jun 28, 2017 #6
    Please state the entropy balance equation that you are supposed to derive?
     
  8. Jun 28, 2017 #7
    $$dS=\frac{1}{T}\left[(u+P)dV+V\frac{du}{dT}dT\right]$$
    I managed to derive the second term by simply using the definition of ##C_{V}## and taking the first derivative of ##U(T,V)## with respect to ##T## since: $$\left(\frac{\partial U}{\partial T}\right)_{V}=T \left(\frac{\partial S}{\partial T}\right)_{V}$$ This obviously leads to ##V\frac{du}{dT}##
    But I'm still struggling with the first term...
     
  9. Jun 28, 2017 #8
    This is much simpler than you think. Start out with:$$dS=\frac{dU}{T}+\frac{P}{T}dV$$From your equation for U(V,T), what is dU in terms of dV and dT?
     
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