Viscosity

All ordinary fluids exhibit viscosity, which is a type of internal friction. A continuous application of force is needed to keep a fluid flowing, just as a continuous force is needed to keep a solid body moving in the presence of friction. Consider the case of a fluid slowly flowing through a long capillary tube. A pressure difference of Δp must be maintained across the ends to keep the fluid flowing, and the resulting flow rate is proportional to Δp. The rate is inversely proportional to the viscosity (η) since the friction that opposes the flow increases as η increases. It also depends on the geometry of the tube, but this effect will not be considered here. The SI units of η are N · s/m² or Pa · s. An older unit of the centimetre-gram-second version of the metric system that is still often used is the poise (1 Pa · s = 10 poise). At 20° C the viscosity of water is 1.0 × 10^-3 Pa · s and that of air is 1.8 × 10^-5 Pa · s. To a rough approximation, liquids are about 100 times more viscous than gases.

There are three important properties of the viscosity of dilute gases that seem to defy common sense. All can be explained, however, by the kinetic theory (see below Kinetic theory of gases). The first property is the lack of a dependence on pressure or density. Intuition suggests that gas viscosity should increase with increasing density, inasmuch as liquids are much more viscous than gases, but gas viscosity is actually independent of density. This result can be illustrated by a pendulum swinging on a solid support. It eventually slows down owing to the viscous friction of the air. If a bell jar is placed over the pendulum and half the air is pumped out, the air remaining in the jar damps the pendulum just as fast as a full jar of air would have done. Robert Boyle noted this peculiar phenomenon in 1660, but his results were largely either ignored or forgotten. The Scottish chemist Thomas Graham studied the flow of gases through long capillaries, which he called transpiration, in 1846 and 1849, but it was not until 1877 that the German physicist O.E. Meyer pointed out that Graham’s measurements had shown the independence of viscosity on density. Prior to Meyer’s investigations, the kinetic theory had suggested the result, so he was looking for experimental proof to support the prediction. When James Clerk Maxwell discovered (in 1865) that his kinetic theory suggested this result, he found it difficult to believe and attempted to check it experimentally. He designed an oscillating disk apparatus (which is still much copied) to verify the prediction.

The second unusual property of viscosity is its relationship with temperature. One might expect the viscosity of a fluid to increase as the temperature is lowered, as suggested by the phrase “as slow as molasses in January.” The viscosity of a dilute gas behaves in exactly the opposite way: the viscosity increases as the temperature is raised. The rate of increase varies approximately as T^s, where s is between 1/2 and 1, and depends on the particular gas. This behaviour was clearly established in 1849 by Graham.

The third property pertains to the viscosity of mixtures. A viscous syrup, for example, can be made less so by the addition of a liquid with a lower viscosity, such as water. By analogy, one would expect that a mixture of carbon dioxide, which is fairly viscous, with a gas like hydrogen, which is much less viscous, would have a viscosity intermediate to that of carbon dioxide and hydrogen. Surprisingly, the viscosity of the mixture is even greater than that of carbon dioxide. This phenomenon was also observed by Graham in 1849.

Finally, there is no obvious correlation of gas viscosity with molecular weight. Heavy gases are often more viscous than light gases, but there are many exceptions, and no simple pattern is apparent.

Heat conduction

If a temperature difference is maintained across a fluid, a flow of energy through the fluid will result. The energy flow is proportional to the temperature difference according to Fourier’s law, where the constant of proportionality (aside from the geometric factors of the apparatus) is called the heat conductivity or thermal conductivity of the fluid, λ. Mechanisms other than conduction can transport energy, in particular convection and radiation; here it is assumed that these can be eliminated or adjusted for. The SI units for λ are J/m · s · K or watt per metre degree (W/m · K), but sometimes calories are used for the energy term instead of joules (one calorie = 4.184 J). At 20° C the thermal conductivity of water is 0.60 W/m · K, and that of many organic liquids is roughly only one-third as large. The thermal conductivity of air at 20° C is only about 2.5 × 10^-2 W/m · K. To a rough approximation, liquids conduct heat about 10 times better than do gases.

The properties of the thermal conductivity of dilute gases parallel those of viscosity in some respects. The most striking is the lack of dependence on pressure or density. Based on this fact, there seems to be no advantage to pumping out the inner chambers of thermos bottles. As far as conduction is concerned, it does not provide any benefits until practically all the air has been removed and free-molecule conduction is occurring. Convection, however, does depend on density, so some degree of insulation is provided by pumping out only some of the air.

The thermal conductivity of a dilute gas increases with increasing temperature, much like its viscosity. In this case, such behaviour does not seem particularly odd, probably because most people do not have a preconceived idea of how thermal conductivity should behave, unlike the situation with viscosity.

There are some differences in the behaviour of thermal conductivity and viscosity; one of the most notable has to do with mixtures. At first glance the thermal conductivity of a gaseous mixture seems to be as expected, since it falls between the conductivities of its components, but a closer look reveals an odd regularity. The conductivity of the mixture is always less than an average based on the number of moles (or molecules) of each component in the mixture. This appears to be related to the different effect that molecular weight has on thermal conductivity and viscosity. Light gases are usually better conductors than are heavy gases, whereas heavy gases are often (but not always) more viscous than are light gases. There also seems to be some correlation between molar heat capacity and thermal conductivity. The foregoing properties of thermal conductivity pose more puzzles that the kinetic theory of gases must address.