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\sect{State equations for ideal Fermi gas}

${N}$ fermions with ${\frac{1}{2}}$ spin and mass ${\mathsf {m}}$
are in a tank with volume ${V}$. The gas is in thermic equilibrium
in temperature ${T}$.

Assume it's possible to relate to the temperature as a low one, and
find explicit expressions, up to second order in temperature, for
the state equations
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\[ \mu=\mu \left(T;\frac{N}{V}\right)\]
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\[ E=E\left(T,V;N \right)\]
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\[ {P}={P}\left(T,V;N \right)\]
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Define what is a low temperature.
Use  only  ${N,\mathsf{m},V,T}$.
Write expressions also for the heat capacity ${C_{v}}$ and the
compressibility ${K_{T}}$.
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\[ K_{T}\equiv \frac{1}{V}\left(\frac{\partial V}{\partial {P}}\right)_{T}\]
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Guideline: Write an expression for ${N=N\left(\beta \mu\right)}$ and find
${\mu\left(\beta,N/V\right)}$ while keeping terms up to
${O\left(T^{2}\right)}$

Similar to the calculation of ${N\left(\beta\mu\right)}$ it is
possible to calculate ${E\left(\beta\mu\right)}$ up to second order
in temperature.

Now there's to place the expression for
${\mu\left(T;\frac{N}{V}\right)}$ you found earlier, and write the
result as a development of ${T}$ while keeping terms op to second
order only! This is the "trickiest" phase..., You'll have to use the
development
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\[\left(1+\chi\right)^{\alpha}=
1+\alpha\chi+\frac{\alpha\left(\alpha-1\right)}{2}\chi^{2}
+\theta\left(\chi^{3}\right)\]
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several times and to make sure not to losing the first and the
second order terms during the algebra process.




