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Thermodynamics, manipulating partial derivatives

  1. Sep 21, 2014 #1
    Hello PF! It's been a while since I last posted here. I have come across a problem in my textbook, which asks me to find expressions for V as a function of T and P, starting from the coefficients of thermal expansion and compressibility.
    [tex]\alpha = \frac{1}{V} \left(\frac{\partial V}{\partial T} \right)_P[/tex]
    [tex]\beta = -\frac{1}{V} \left(\frac{\partial V}{\partial P} \right)_T[/tex]
    I've already solved the problem, I separated the differentials and integrated, then cleared for V. Now, here's where I have trouble. This is how I manipulated the differentials:
    [tex]\alpha \ dT= \frac{1}{V} \ dV[/tex]
    [tex]\beta \ dP= -\frac{1}{V} \ dV[/tex]
    Both my professor, and Castellan's PChem text say this is correct, however, neither my professor nor the book explain why the separation of a partial derivative works in this case. Math professors have always said we can't "break" a partial derivative the same way we do for a regular derivative. If anyone could offer some insight about this case, it would be very helpful. Thanks!
     
  2. jcsd
  3. Sep 21, 2014 #2
    [tex]\alpha dT=\frac{dV}{V}[/tex] under the condition p=const or dp=0
    [tex]\beta dP=\frac{-dV}{V}[/tex] under the condition T=const or dT=0
    You need these conditions.
     
  4. Sep 21, 2014 #3
    If V is a function V(P,T) of P and T, then
    [tex]dV=\left(\frac{\partial V}{\partial P}\right)_TdP+\left(\frac{\partial V}{\partial T}\right)_PdT[/tex]

    Chet
     
  5. Sep 22, 2014 #4
    Thank you both! I understand now, I thought they were just magically turning the partial derivative into a regular one. I wasn't aware of how both coefficients relate to the differential of V. We haven't covered partial derivatives or differentials of multivariable functions in Calc III, my professor is going a bit slowly. But we have started to use them in Thermodynamics I, so everything I know about the subject right now is from what I have read on my own.

    Oh, and I'm sorry you had to move my thread, I thought it could be posted in the physics section since I wasn't asking for help on solving the problem itself, just a doubt with the math involved, but I'll be more careful next time.
     
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