1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Maxwell relations with heat capacity.

  1. Feb 28, 2008 #1
    1. The problem statement, all variables and given/known data
    Use the Maxwell relations and the Euler chain relation to express (ds/dt)p in terms of the heat capacity Cv = (du/dt)v. The expansion coefficient alpha = 1/v (dv/dt)p, and the isothermal compressibility Kt = -1/v (dV/dp)T. Hint. Assume that S= S(p,V)

    2. Relevant equations
    dQ(rev) = Tds
    The maxwell relations
    Euler Chain relation

    3. The attempt at a solution

    Alright, my attempts at this involved trying find common partial derivatives from the information already given. I couldn't find anything. But then looking at the hint I thought that there might be a way to express the change in entropy with respect to pressure and volume. I get this ds = (dU + PdV)/T assuming constant pressure. I am really not sure what I am suppose to do. I especially don't get what the expansion coefficient and thermal compressibility has to do with anything, but that might be because I can't see the big picture with this problem.

    A step by step explanation would be greatly appreciated.
  2. jcsd
  3. Mar 1, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    I'd rather consider S to be a function of T and V. Then I could differentiate S as


    Then I'd differentiate with respect to T at constant p:


    You should be able to figure out the rest. This is a useful trick for when you want to compare derivatives taken under different conditions.
  4. Mar 3, 2008 #3
    I am very thankful for your reply. However, I managed to solve the problem several hours after my post. Your method though is something that I didn't think of, so I do appreciate it.
  5. Nov 15, 2009 #4
    I'm not sure about the sentence "all variables and given/known data".

    In fact I succeed in order to obtain a relation between:
    [tex] \left(\frac{\partial s}{\partial T}\right)_p[/tex]
    and [tex] c_v, \alpha, k_T, T, v[/tex] just following the mapes's hint to consider S as a function of T and V. Where minuscle letter for extensive quantity means: "this quantity is molar", and all transformation are intended to involve a costant number of molecules.

    In fact:

    [tex] \left(\frac{\partial s}{\partial T}\right)_p = \left(\frac{\partial s}{\partial T}\right)_v + \left(\frac{\partial s}{\partial v}\right)_T \left(\frac{\partial v}{\partial T}\right)_T [/tex]

    The Maxwell's relation following from [tex] d(-p dv - sdT)=0[/tex] tell us:
    [tex] \left(\frac{\partial s}{\partial v}\right)_p= \left(\frac{\partial p}{\partial T}\right)_v[/tex]
    Now the Euler's chain rule give us the link between the first derivative in second addend, the compressibility and the thermal expansion coefficient. In fact:

    [tex] \left(\frac{\partial p}{\partial T}\right)_v \left(\frac{\partial T}{\partial v}\right)_p\left(\frac{\partial v}{\partial p}\right)_T = -1[/tex]


    [tex]\left(\frac{\partial T}{\partial v}\right)_p = 1/\left(\frac{\partial v}{\partial T}\right)_p [/tex]

    so that:

    [tex] \left(\frac{\partial p}{\partial T}\right)_v = \frac{\alpha}{k_T}[/tex]

    In order to complete the derivation we need to use the given alternative definition of specific heat:

    [tex] c_v = T \left(\frac{\partial s}{\partial T}\right)_v = \left(\frac{\partial u}{\partial T}\right)_v [/tex]

    Which follow from:

    [tex]T dS = dU + p dV [/tex]

    So obtaining:

    [tex] \left(\frac{\partial s}{\partial T}\right)_p = \frac{c_v}{T} + \frac{\alpha^2 v}{k_T}[/tex]

    is this what was required?

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook