\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.    



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