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Partial differentiation - Constants

  1. Nov 25, 2012 #1
    1. The problem statement, all variables and given/known data

    Consider the following equality:

    ([itex]\frac{∂S}{∂V}[/itex])T = ([itex]\frac{∂P}{∂T}[/itex])V

    If I rearrange the equality so that I write:

    ([itex]\frac{∂S}{∂P}[/itex])? = ([itex]\frac{∂V}{∂T}[/itex])?

    What variables will be constant in each side?
    I'm having some trouble in a few thermodynamics problems because of this...

    2. Relevant equations



    3. The attempt at a solution

    I don't know how to do this. Normally I just ignore and assume that every variable except the variable of differentiation is a constant.
    If anyone could confirm or correct me I'd be thankful.

    Thanks.
     
  2. jcsd
  3. Nov 26, 2012 #2

    SammyS

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    The following may help you.

    For the case in which three variables can be related by a function of the form f(x, y, z) = 0, then the following relations hold.

    [itex]\displaystyle \left(\frac{\partial x}{\partial y}\right)_z\left(\frac{\partial y}{\partial z}\right)_x\left(\frac{\partial z}{\partial x}\right)_y = -1[/itex]

    [itex]\displaystyle \left(\frac{\partial x}{\partial y}\right)_z = \frac{1}{\left(\frac{\partial y}{\partial x}\right)_y}[/itex]

    See the following two Wikipedia entries:

    http://en.wikipedia.org/wiki/Triple_product_rule

    http://en.wikipedia.org/wiki/Exact_differential#Cyclic_relation
     
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