\sect{Baruch's  A53.}

The partition functions of a diatomic molecules AB or A$_2$ (within an ideal gas)
has the form
\[f_{AB}=g_{AB}(T)(m_Am_B)^{3/2} \qquad \mbox{or} \qquad
f_{A_2}=\half g_{A_2}(T)m_A^3\] where $m_A,\,m_B$ are atomic
masses and B is an isotope of A; $g_{AB}$ and $g_{A_2}$ are
independent of the isotope masses.
\begin{itemize}
\item[(a)] a)  Explain the origin of the factor $\half$.
\item[(b)]  In the reaction H$_2$+Cl$_2 \leftrightarrows$2HCl the
Cl atom has two isotopes Cl$^{35}$ and Cl$^{37}$.
Write the relevant four reactions and their laws of mass action.
\item[(c)]   Show that the relative abundance of Cl$^{35}$ and Cl$^{37}$
in Cl$_2$ is
the same as in HCl, i.e. the various densities $n$ satisfy
\[
\frac{2n_{Cl_2^{37}}+n_{Cl^{35}Cl^{37}}}{2n_{Cl_2^{35}}+n_{Cl^{35}Cl^{37}}}
=\frac{n_{HCl^{37}}}{n_{HCl^{35}}} \]\\
\end{itemize}

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