text: Since its original descriptions, the Urey–Bigeleisen–Mayer equation has taken many forms. Given an isotopic exchange reaction , such that  designates a molecule containing an isotope of interest, the equation can be expressed by relating the equilibrium constant, , to the product of partition function ratios, namely the translational, rotational, vibrational, and sometimes electronic partition functions. Thus the equation can be written as:  where  and  is each respective partition function of molecule or atom . 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  or . 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. Some common, significant modifications to the equation include accounting for pressure effects, 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.