In a continuous$-$flow process, a gas is throttled to half its density. The throttling is well described as a constant$-$enthalpy process. The constant$-$volume heat capacity of the gas is approximately $(5/2)$kB and its equation of state is P $=$ k$\_$B T$/$v $-$ a$/$v$^2,$ where a is a positive constant. Find an expression for the final temperature T$2$ in terms of the initial temperature T$1$ and molar volume v$1.$ Does the gas heat up or cool down due to throttling?

Answer and tip:

$9\mathrm{.}18,T_2=T_1-\frac{2a}{7k_B}{}_1$

Consider a $2-$step process of expansion at constant temperature followed by

heating$/$cooling at constant volume: $dh=(\frac{\text{d}\text{e}\text{l}\text{h}}{\text{d}\text{e}\text{l}\text{v}}{)}_{T}dv+(\frac{\text{d}\text{e}\text{l}\text{h}}{\text{d}\text{e}\text{l}\text{T}}{)}_{v}dT=0.$ Relate the partial

differentials to measurable properties involving $P,v,$ and $T$ using $dh=Tds+vdP.$

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