Sutherland model

The Sutherland model describes the temperature dependence of the viscosity of gases . It was proposed by the Australian physicist William Sutherland in 1893 . The model is based on the kinetic gas theory , whereby a strongly idealized interaction potential is used to describe the intermolecular interactions . Despite the coarse approximations used, especially for the interaction potential used, the model reproduces the experimentally determined viscosities of many gases over a large temperature range with a systematic error of less than 10%.

Problem and model description

The viscosity of real gases is only inaccurately reproduced by the simplest kinetic model of gases, the model of the ideal gas . In particular, the observed temperature dependency is systematically stronger than the proportionality to the root of the temperature predicted by this model. The cause of this systematic deviation is the neglect of the long-range attractive van der Waals forces that act between all atoms and molecules.

The Sutherland model describes the repulsive intermolecular forces at small distances using a model of hard spheres , but also takes into account the attractive forces described above in the form of a simple spherically symmetrical potential outside the hard sphere

${\displaystyle V(r)\propto -r^{-n}}$

(r: intermolecular distance; n: whole number, mostly n = 6). This has the effect that in the Sutherland model the scattering cross section depends on the collision between two molecules on their relative speed: the lower the relative speed, the larger the scattering cross section. The reason for this is that the molecules remain at a short distance from one another for a longer period at a low relative speed and are therefore more exposed to the effect of their interaction potential. The resulting larger scattering cross-section in turn causes a decrease in the mean free path of the molecules and thus an increase in the viscosity of the gas.

Sutherland's formula

The viscosity of gases is described in the Sutherland model by the following formula:

${\displaystyle {\mu }={\mu }_{0}{\frac {T_{0}+C}{T+C}}\left({\frac {T}{T_{0}}}\right)^{3/2}}$

With:

• ${\displaystyle {\mu }}$ = Dynamic viscosity in (Pa · s) at temperature ${\displaystyle T}$
• ${\displaystyle {\mu }_{0}}$ = Reference viscosity in (Pa · s) at the reference temperature ${\displaystyle T_{0}}$
• ${\displaystyle T}$ = Temperature [K]
• ${\displaystyle T_{0}}$ = Reference temperature [K]
• ${\displaystyle C}$ = Sutherland's constant [K]

Sutherland constants and reference temperatures for some gases:

gas	${\displaystyle C}$ [K]	${\displaystyle T_{0}}$ [K]	${\displaystyle {\mu }_{0}}$ [10 −6  Pa s]
air	120	291.15	18.27
nitrogen	111	300.55	17.81
oxygen	127	292.25	20.18
Carbon dioxide	240	293.15	14.8
Carbon monoxide	118	288.15	17.2
hydrogen	72	293.85	8.76
Ammonia gas	370	293.15	9.82
Sulfur dioxide	416	293.65	12.54
helium	79.4	273	19th
argon	165.00	273.15	21.0
krypton	233.15	273.15	23.4

literature

• J. Hirschfelder, CF Curtiss, RB Bird: Molecular Theory of Gases and Liquids . Wiley, New York, 1954, ISBN 0-471-40065-3
• Karl Jousten: Wutz manual vacuum technology: theory and practice , Vieweg + Teubner, ISBN 3-8348-0133-X / ISBN 978-3-8348-0133-3 ; P. 40 ( as Google book )

Individual evidence

1. ↑ W. Sutherland, The viscosity of gas and molecular force , Philosophical Magazine 5 (1893), pp 507-531.
2. ↑ G. Turrell, Gas Dynamics , Wiley (1997), p. 52 ff.
3. ↑ FM White, Viscous Fluid Flow , 2nd ed., McGraw-Hill, (1991). Kim et al., Arxiv : physics / 0410237
4. ↑ Calculated from http://www.kayelaby.npl.co.uk/general_physics/2_2/2_2_3.html
5. ↑ Calculated from http://www.kayelaby.npl.co.uk/general_physics/2_2/2_2_3.html