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Math Help - Maxwell Relations

  1. #1
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    Maxwell Relations

    Hey I'm having a little problem with this question
    The enthalpy of a gas is defined by H=U+pV where U satisfies  dU=TdS-pdV Determine a relationship between the differentials of H, S and p. Hence show that

     \left ( \frac{\partial V}{\partial S} \right )_p= \left ( \frac{\partial T}{\partial p} \right )_S

    By regarding U as a function of p and V show that

     (\frac{\partial S}{\partial V})_p (\frac{\partial T}{\partial p})_V  - (\frac{\partial S}{\partial p})_V (\frac{\partial T}{\partial V})_p = 1<br />
    I can do the first one absolutely fine, just take the total derivative of H and substitute in the expression in for dU then use the fact that mixed partials commute.

    The second part I'm having more trouble with, any chance of a hint?

    Cheers
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  2. #2
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    Quote Originally Posted by thelostchild View Post
    Hey I'm having a little problem with this question


    I can do the first one absolutely fine, just take the total derivative of H and substitute in the expression in for dU then use the fact that mixed partials commute.

    The second part I'm having more trouble with, any chance of a hint?

    Cheers
    Here's an idea. Write

     \frac{\partial V}{ \partial S} = \frac{\partial T}{ \partial p}

    as (with Jacobian's)

     \frac{\partial(V,p)}{\partial(S,p)} = \frac{\partial(S,T)}{\partial(S,p)}

    then

     \frac{\partial(S,T)}{\partial(V,p)}= 1

    expanding gives your result.
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