\sect{The law of mass action for diatomic molecules}
Consider a diatomic AB molecule, where $A$ and $B$ are different spin~0 atoms,
each having a 1-unit atomic mass $\mass_0$.
The length of the molecule is~$a$,
the binding energy is $-\varepsilon_0$,
and the vibration frequency of the bond is~$\omega_0$.
The vibration amplitude is much smaller compared with~$a$.
The temperatures are not low, namely ${ T \gg 1/(\mass_0 a^2) }$,
such that the rotation-spectrum can be treated as a continuum.
For higher temperatures (${T \gg \omega_0 }$) also the vibration-spectrum
can be treated using a classical approximation. \\
In item (3) below we consider Hydrogen $H_{2}$, Deuterium $D_{2}$, and $HD$ molecules.
The respective masses of the atoms are ${\mass_H, \mass_D}$.
Note that the Deuterium nucleus has spin~1.
Assume that neither the energy nor the ``spring constant" of the binding
are affected by the ${H \mapsto D}$ replacement.
\Dn
(1) Find the one molecule partition function $Z^{AB}$ for an AB molecule
that is held in a container that has volume~$L^3$.
Assume that the temperature is not low, but not necessarily high.
\Dn
(2) Write the law of mass action for the reaction ${A+B\leftrightarrow AB}$.
Find an explicit expression for the equilibrium constant $K(T)$
in the high temperature regime.
\Dn
(3) Write the law of mass action for the reaction ${H_{2}+D_{2} \leftrightarrow 2 HD}$.
Express the equilibrium constant $K(T)$
in terms of one-particle partition functions $Z^{C}$,
were $C$ stands for $H_{2}$, and $D_{2}$, and $HD$.
\Dn
(4) Find expressions for the ratio $Z^{C}/Z^{AB}$ in the high temperature regime,
where $A$ and $B$ are distinct spinless atoms
that have the same masses as that of the $C$ constituents.
Explain why the high temperature assumpation is essential in order to get a simple result.
\Dn
(5) What is the explicit result for ${K(T)}$ of item (3) in the high temperature regime?
\Dn
{\bf Tip:} The Hamiltonian of a diatomic molecule consist of center of mass degrees of freedom,
and of a relative motion degrees of freedom. The latter involves the reduced mass ${\mass_A\mass_B/(\mass_A+\mass_B)}$. For intermediate calculations you can use the notation $\alpha$ for spring constant.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%