The
kinetic theory of gases allows accurate calculation of the temperature-variation of gaseous viscosity. The theoretical basis of the kinetic theory is given by the
Boltzmann equation and
Chapman–Enskog theory, which allow accurate statistical modeling of molecular trajectories. In particular, given a model for intermolecular interactions, one can calculate with high precision the viscosity of monatomic and other simple gases (for more complex gases, such as those composed of
polar molecules, additional assumptions must be introduced which reduce the accuracy of the theory). The viscosity predictions for four molecular models are discussed below. The predictions of the first three models (hard-sphere, power-law, and Sutherland) can be simply expressed in terms of elementary functions. The
Lennard–Jones model predicts a more complicated T-dependence, but is more accurate than the other three models and is widely used in engineering practice.
Hard-sphere kinetic theory If one models gas molecules as
elastic hard spheres (with mass m and diameter \sigma), then elementary kinetic theory predicts that viscosity increases with the square root of absolute temperature T: : \mu = 1.016 \cdot \frac{5}{16 \sigma^2} \left(\frac{k_{\rm B} m T}{\pi}\right)^{1/2} where k_\text{B} is the
Boltzmann constant. While correctly predicting the increase of gaseous viscosity with temperature, the T^{1/2} trend is not accurate; the viscosity of real gases increases more rapidly than this. Capturing the actual T dependence requires more realistic models of molecular interactions, in particular the inclusion of attractive interactions which are present in all real gases.
Power-law force A modest improvement over the hard-sphere model is a repulsive inverse power-law force, where the force between two molecules separated by distance r is proportional to 1/r^{\nu}, where \nu is an empirical parameter. This is not a realistic model for real-world gases (except possibly at high temperature), but provides a simple illustration of how changing intermolecular interactions affects the predicted temperature dependence of viscosity. In this case, kinetic theory predicts an increase in temperature as T^s, where s = (1/2) + 2 / (\nu - 1). More precisely, if \mu' is the known viscosity at temperature T', then : \mu = \mu' (T / T')^s Taking \nu \rightarrow \infty recovers the hard-sphere result, s = 1/2. For finite \nu, corresponding to softer repulsion, s is greater than 1/2, which results in faster increase of viscosity compared with the hard-sphere model. Fitting to experimental data for hydrogen and helium gives predictions for s and \nu shown in the table. The model is modestly accurate for these two gases, but inaccurate for other gases.
Sutherland model Another simple model for gaseous viscosity is the Sutherland model, which adds weak intermolecular attractions to the hard-sphere model. If the attractions are small, they can be treated
perturbatively, which leads to \mu = \frac{5}{16 \sigma^2} \left(\frac{k_{\text{B}} m T}{\pi}\right)^{1/2} \cdot \left(1 + \frac{S}{T} \right)^{-1} where S, called the Sutherland constant, can be expressed in terms of the parameters of the intermolecular attractive force. Equivalently, if \mu' is a known viscosity at temperature T', then \mu = \mu' \left(\frac{T}{T'} \right)^{3/2} \frac{T' + S}{T + S} Values of S obtained from fitting to experimental data are shown in the table below for several gases. The model is modestly accurate for a number of gases (
nitrogen,
oxygen,
argon,
air, and others), but inaccurate for other gases like
hydrogen and
helium. In general, it has been argued that the Sutherland model is actually a poor model of intermolecular interactions and useful only as a simple interpolation formula for a restricted set of gases over a restricted range of temperatures.
Lennard-Jones Under fairly general conditions on the molecular model, the kinetic theory prediction for \mu can be written in the form \mu = \frac{5}{16 \sqrt{\pi}} \frac{\sqrt{m k_{\text{B}} T}}{\sigma^2 \Omega(T)} where \Omega is called the
collision integral and is a function of temperature as well as the parameters of the intermolecular interaction. It is completely determined by the kinetic theory, being expressed in terms of integrals over collisional trajectories of pairs of molecules. In general, \Omega is a complicated function of both temperature and the molecular parameters; the power-law and Sutherland models are unusual in that \Omega can be expressed in terms of elementary functions. The Lennard–Jones model assumes an intermolecular pair potential of the form V(r) = 4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12} - \left(\frac{\sigma}{r} \right)^6 \right] where \epsilon and \sigma are parameters and r is the distance separating the
centers of mass of the molecules. As such, the model is designed for spherically symmetric molecules. Nevertheless, it is frequently used for non-spherically-symmetric molecules, provided these do not possess a large
dipole moment. The collisional integral \Omega for the Lennard-Jones model cannot be expressed exactly in terms of elementary functions. Nevertheless, it can be calculated numerically, and the agreement with experiment is good – not only for spherically symmetric molecules such as the
noble gases, but also for many polyatomic gases as well. \Omega(T) = 1.16145 \left(T^* \right)^{-0.14874} + 0.52487 e^{-0.77320 T^*} + 2.16178 e^{-2.43787 T^*} where T^* \equiv k_{\text{B}} T / \epsilon. This equation has an average deviation of only 0.064 percent in the range 0.3 . Values of \sigma and \epsilon estimated from experimental data are shown in the table below for several common gases. == Liquids ==