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Groupwork 8 Maxwell Relations

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    Work in groups on these problems. You should try to answer the questions without referring to your textbook. If you get stuck, try asking another group for help.

    Two different ways to cast the first law of thermodynamics are

    \(\Delta U=q+w\)     and      \(du=\sigma q+\sigma w\) 

    If the process under discussion is performed reversibly, then

    \(dU=\sigma q_{rev}+\sigma w_{rev} =TdS-PdV\)

     

    For a process carried out at constant volume and temperature, then \(-PdV=0\) and \(dU=TdS\).  We rearrange to get \(dU-Tds=0\).  Again, assuming constant volume and temperature, we realize that \(dU-TdS=0\) implies \(d(U-TS)=0\). 

     

    From this we define the Helmholtz energy, \(A=U-TS\) (also sometimes called the Helmholtz free energy).

     

    Then if we consider the case where we do not hold temperature and volume constant,\(dA=d(U-TS)=dU-TdS-SdT

    substituting \(TdS-PdV\) in for dU, we get

     

    \(dA=TdS-PdV-TdS-SdT=-PdV-SdT\)

     

    We can also express the total differential of A by differentiating with respect to temperature and volume, 

    \(dA=(\frac{\partial A}{\partial T})_VdT+(\frac{\partial A}{\partial V})_TdV\)

    Then by comparing with \(dA=-PdV-SdT\), we find

    \((\frac{\partial A}{\partial T})_V=-P\) and \((\frac{\partial A}{\partial V})_T=-S\)

                                                                                                        

    We can take this one step further.  If we take the derivative of each expression with respect to the opposite variable, we get

    \(\frac{\partial}{\partial V}(\frac{\partial A}{\partial T})=\frac{\partial}{\partial V}(-P)=-(\frac{\partial P}{\partial V})_T\)  and  \(\frac{partial}{partial T}(\frac{\partial A}{\partial V})=\frac{\partial}{\partial T}(-S)=-(\frac{\partial S}{\partial T})_V

    so

    \(-(\frac{\partial P}{\partial V})_T=-(\frac{\partial S}{\partial T})_V\) or \((\frac{\partial P}{\partial V})_T=(\frac{\partial S}{\partial T})_V\)

     

    This one of four Maxwell Relations that relate the four variables, P, V, T, and S.

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    Define the Gibbs energy (also known as Gibbs free energy) \(G=H-TS\)

     

    Using this definition of G, what is the total differential dG?

     

    Simplify your expression for dG so that it only has two terms.  (Hint: consider the method above for dA and \(dH=dU+PdV\)

     

    Consider that

    \(dG=(\frac{\partial G}{\partial T})_P+(\frac{\partial G}{\partial P})_TdP\)

    Compare your expression for dG above to get values for

    \((\frac{\partial G}{\partial T})_P=\)

    \((\frac{\partial G}{\partial P})_T=\)

     

    What is the simplified expression for \((\frac{\partial}{\partial P}(\frac{\partial G}{\partial T})_p)_T=\)

     

    What is the simplified expression for \((\frac{\partial}{\partial T}(\frac{\partial G}{\partial P})_T)_P=\)

     

    What is the Maxwell relation arising from these double partial derivatives? 

     

    Bonus – if you have time, draw a plot of G vs. T and G vs. P.  Consider the phases of the substances.