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Differentiating thermodynamic coefficients

  1. Jan 26, 2014 #1
    1. The problem statement, all variables and given/known data
    In oppgave 1 a) I am supposed to show that the given equality is true (namely that the isoterm compressibility coefficient partial-differentiated with regards to temperature = isobar coefficient differentiated with regards to pressure multiplied by minus one).

    http://web.phys.ntnu.no/~stovneng/TFY4165_2014/oving2.pdf

    3. The attempt at a solution


    I can naturally see that the way to attack this problem is to simply carry out the differentiations, but how does it make sense to partially differentiate ##\alpha_V## with regards to ##p## when ##\alpha_V## assumes constant pressure? And besides, ##\alpha_V## measures the change of relative volume as a function of ##T##, yet constant temperature is assumed for ##\frac{\partial \alpha_V}{\partial p}##? It's a similar problem with the differentiating the isoterm coefficient with regards to temperature while keeping pressure constant..
     
    Last edited: Jan 26, 2014
  2. jcsd
  3. Jan 26, 2014 #2

    BvU

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    Not strong in Norwegian. alpha_v may be defined at constant pressure, that doesn't mean its value is independent of the pressure!
     
  4. Jan 26, 2014 #3
    Okay, but alpha_v is the change of relative volume with regards to temperature, yet ##\frac{\partial \alpha_V}{\partial p}## assumes constant temperature? How does it make sense to calculate how ##\alpha_V## (which is a measure of how the volume varies with regards to temperature) varies while the temperature is constant?
     
  5. Jan 26, 2014 #4
    ##\alpha_V## is a function of both temperature and pressure. Unfortunately, many, if not most, texts on thermodynamics do a very bad job at explaining functional dependencies among all the things they operate with. The unconventional notation ##\left({\partial X \over \partial Y}\right)_{Z}## is my personal anti-favorite, because in my experience very few students understand what it really signifies. Just for the record, it means that ##X## is considered to be a function of ##Y## and ##Z##, and is differentiated with respect to ##Y##. The usually added bit that ##Z## is held constant is redundant because that follows from the definition of partial differentiation. I suspect it is not only redundant, but is severely confusing, because the obvious redundancy makes the inexperienced reader wonder what really is going on.
     
  6. Jan 26, 2014 #5
    Please write down your defining equations for αV and κT. Once you get a look at these, you will see immediately how to prove what you are trying to prove.
     
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