Statistical Mechanics of Mixtures of Isotopes

The excess free energy of a mixture of isotopes is expanded in a Taylor series in powers of the relative mass differences $\frac{(m_0-m_j)}{m_j}$, where $m_j$ and $m_0$ are the masses of the particles of the $j\mathrm{th}$ component and of a reference isotope, respectively. This expansion is only useful if it assumed that all the particles in the mixture obey classical or Boltzmann statistics. When this assumption is made it is found that the linear term in the expansion vanishes identically. The second-order term has the form $\Sigma i^s\Sigma j^s{x}_i{x}_j{({\lambda }_i-{\lambda }_j)}^2·Q$, where $Q$ is a universal function of $m_0$, $T$, and $V$, ${\lambda }_j$ equals $\frac{(m_0-m_j)}{m_j}$, and $x_j$ is the mole fraction of the $j\mathrm{th}$ component. This expression gives the explicit dependence of the excess free energy on the mole fractions and on the relative mass differences ${\lambda }_j$. From this result it can be shown, among other things, that a phase separation of the isotopes in a mixture should take place at a sufficiently low temperature. It can also be shown that there is an approximate law of corresponding states between different mixtures of isotopes. The theory is directly applicable to all solid mixtures and to fluid mixtures of the hydrogen isotopes. Unfortunately owing to the lack of experimental data it is impossible to test the theory rigorously. Finally it is shown how the theory can be used to interpret the behavior of ${\mathrm{He}}^3$-${\mathrm{He}}^4$ solutions. When sufficient experimental data becomes available the theory should throw considerable light on the influence of quantum statistics on the properties of these solutions.