# 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.