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Chapman–Enskog theory

A technique developed by Sydney Chapman and David Enskog in the early 1900s allows a more refined calculation of μ {\displaystyle \mu }. It is based on the Boltzmann equation, which provides a statistical description of a dilute gas in terms of intermolecular interactions. The technique allows accurate calculation of μ {\displaystyle \mu } for molecular models that are more realistic than rigid elastic spheres, such as those incorporating intermolecular attractions. Doing so is necessary to reproduce the correct temperature dependence of μ {\displaystyle \mu }, which experiments show increases more rapidly than the T 1 / 2 {\displaystyle T^{1/2}} trend predicted for rigid elastic spheres. Indeed, the Chapman–Enskog analysis shows that the predicted temperature dependence can be tuned by varying the parameters in various molecular models. A simple example is the Sutherland model, which describes rigid elastic spheres with weak mutual attraction. In such a case, the attractive force can be treated perturbatively, which leads to a simple expression for μ {\displaystyle \mu }:

where S {\displaystyle S} is independent of temperature, being determined only by the parameters of the intermolecular attraction. To connect with experiment, it is convenient to rewrite as

where μ 0 {\displaystyle \mu _{0}} is the viscosity at temperature T 0 {\displaystyle T_{0}}. This expression is usually named Sutherland's formula. If μ {\displaystyle \mu } is known from experiments at T = T 0 {\displaystyle T=T_{0}} and at least one other temperature, then S {\displaystyle S} can be calculated. Expressions for μ {\displaystyle \mu } obtained in this way are qualitatively accurate for a number of simple gases. Slightly more sophisticated models, such as the Lennard-Jones potential, or the more flexible Mie potential, may provide better agreement with experiments, but only at the cost of a more opaque dependence on temperature. A further advantage of these more complex interaction potentials is that they can be used to develop accurate models for a wide variety of properties using the same potential parameters. In situations where little experimental data is available, this makes it possible to obtain model parameters from fitting to properties such as pure-fluid vapour-liquid equilibria, before using the parameters thus obtained to predict the viscosities of interest with reasonable accuracy.

In some systems, the assumption of spherical symmetry must be abandoned, as is the case for vapors with highly polar molecules like H O. In these cases, the Chapman–Enskog analysis is significantly more complicated.