Since its original descriptions, the Urey–Bigeleisen–Mayer equation has taken many forms. Given an isotopic exchange reaction A+B^*=A^*+B, such that ^* denotes a molecule containing an isotope of interest, the equation can be expressed by relating the
equilibrium constant, K_, to the product of
partition function ratios, namely the
translational,
rotational,
vibrational, and sometimes electronic partition functions. Thus the equation can be written as: K_{eq} = \frac{[A^*][B]}{[A][B^*]} where [A]=\prod^n Q_{n,A} and Q_n is each respective partition function of molecule or atom A. It is typical to approximate the rotational partition function ratio as quantized
rotational energies in a
rigid rotor system. The Urey model also treats
molecular vibrations as simplified
harmonic oscillators and follows the
Born–Oppenheimer approximation. Isotope partitioning behavior is often reported as a
reduced partition function ratio, a simplified form of the Bigeleisen–Mayer equation notated mathematically as \frac{s}{s'}f or (\frac{Q^*}{Q})_r. The reduced partition function ratio can be derived from
power series expansion of the function and allows the partition functions to be expressed in terms of frequency. It can be used to relate molecular vibrations and intermolecular forces to equilibrium isotope effects. As the model is an approximation, many applications append corrections for improved accuracy. nuclear geometry, and corrections for
anharmonicity and quantum mechanical effects. For example,
hydrogen isotope exchange reactions have been shown to disagree with the requisite assumptions for the model but correction techniques using
path integral methods have been suggested. == History of discovery ==