$\left(\frac{\partial S}{\partial U}\right)_{V,N}$, $\left(\frac{\partial S}{\partial V}\right)_{U,N}$, $\left(\frac{\partial S}{\partial N}\right)_{U,V}$ (3)
2. A box of light can be treated as an ideal gas which obeys the equations of state
$\frac{U}{V} = aT^4$
$\frac{U}{V} = 3P$, (4)
where $a$ is a constant, $T$ is the absolute temperature, and $P$ is the pressure of the gas.
(a) Show that this \"photon gas\" behaves adiabatically, such that $PV^\gamma = $ constant,
and determine the adiabatic constant, $\gamma$.
(b) The entropy of this photon gas depends on the internal energy, $U$, and the volume, $V$. Suppose that we write
$S = AU^xV^y$,
where $A$ is a constant, and $x$ and $y$ are numbers. Using Eqs. (3) and (4) above,
determine $A$, $x$, and $y$.
(c) What is the chemical potential, $\mu$, for this box of light?
