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Doubt on one Maxwell relation

  1. Jan 8, 2009 #1


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    In thermodynamics, we always need to use exact differential relations to find the so called Maxwell relations. For a function of x and y, z=z(x,y)

    if dz = M(x,y)dx + N(x, y)dy

    the complete conditon for above equation to be hold is

    [tex]\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}[/tex]

    Now, let see one of the relation on Helmholtz free energy

    [tex]dF = -SdT - PdV[/tex]

    apply the condition mentioned above, we get

    [tex]\left(\frac{\partial P}{\partial T}\right)_V = \left(\frac{\partial S}{\partial V}\right)_T[/tex]

    Well, in many materials (including some textbooks), they like to write it

    [tex]\left ( {\partial T\over \partial p} \right )_{V,N} = \left ( {\partial V\over \partial S} \right )_{T,N}[/tex]

    I don't know why they like to inverse those relation! What interesting is only this relations I found to be in reverse order, the others Maxwells relations are just fine!
  2. jcsd
  3. Jan 9, 2009 #2


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    There's absolutely no difference; the reciprocals of Maxwell relations are also valid. We can also get to this one by looking at the potential [itex]H=E+PV[/itex] rather than [itex]F=E-TS[/itex]. It is odd that some textbooks would switch just one without comment. It would be a good teaching opportunity to explain that this is OK.
  4. Jan 13, 2009 #3
    this page: http://en.wikipedia.org/wiki/Exact_differential#Reciprocity_Relation
    might be enlightening. I know it helped me. At the school I go to, most of the engineers or physics majors (same math requirement more or less) I have talked to don't really know about these rules. I'm pretty sure that means we were never taught them. It's kind of annoying.
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