\sect{Baruch's A25.}
Adiabatic cooling:
Consider an ideal gas whose $N$ atoms have mass $m$, spin $\half$ and a magnetic moment $\mu$; the energy levels of each particle are $\frac{p^2}{2m}\pm\mu B$ in a magnetic field $B$ where $p$ is the momentum.
\begin{itemize}
\item[(a)] Evaluate the entropy as $S_{kin}+S_{spin}$ due to kinetic and spin terms,
respectively, and show that by reducing $B$ to zero adiabatically the initial $T_i$ and final $T_f$ temperatures satisfy
\[ \ln \frac{T_f}{T_i}=\frac{2}{3Nk_B}[S_{spin}(T_i, B)-S_{spin}(T_f, 0)]\]
\item[(b)] Find the lower limit for $\frac{T_f}{T_i}$ by allowing the large $B$ limit.
\item[(c)] Extend (b) to the case of space dimensionality $d$ and general spin $S$.
\end{itemize}
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