CONDUCTION IN GASES AND LIQUIDS 36. The theory of conduction of heat by diffusion in gases has a particular interest, since it is possible to predict the value of the conductivity on certain assumptions, if the viscosity is known. On the kinetic theory the molecules of a gas are relatively far apart and there is nothing exactly analogous to friction between two adjacent layers A and B in relative motion. There is, however, a continual interchange of molecules between A and B, which pro duces the same effect as viscosity in a liquid. Faster-moving par ticles diffusing from A to B carry their momentum with them, and tend to accelerate B ; an equal number of slower particles diffusing from B to A act as a drag on A. This action and reaction between layers in relative motion is equivalent to a frictional stress tending to equalize the velocities of adjacent layers. The magnitude of the stress per unit area parallel to the direction of flow is evidently proportional to the velocity gradient, or the rate of change of velocity per cm. in passing from one layer to the next. It must also depend on the rate of interchange of molecules, that is to say, (I) on the number passing through each square centimetre per second in either direction, (2) on the average distance to which each can travel before collision (i.e. on the "mean free path"), and (3) on the average velocity of translation of the molecules, which varies as the square root of the temperature. Similarly if A is hotter than B, or if there is a gradient of temperature be tween adjacent layers, the diffusion of molecules from A to B tends to equalize the temperatures, or to conduct heat through the gas at a rate proportional to the temperature gradient, and de pending also on the rate of interchange of molecules in the same way as the . viscosity effect.
Conductivity and viscosity in a gas should vary in a similar manner since each depends on diffusion in a similar way. The mechanism is the same, but in one case we have diffusion of momentum, in the other case diffusion of heat. Viscosity in a gas was first studied theoretically from this point of view by J. Clerk Maxwell, who predicted that the effect should be independent of the density within wide limits. This, at first sight, paradoxical result is explained by the fact that the mean free path of each molecule increases in the same proportion as the density is dimin ished, so that as the number of molecules crossing each square centimetre decreases, the distance to which each carries its mo mentum increases, and the total transfer of momentum is un affected by variation of density. Maxwell himself verified this prediction experimentally for viscosity over a wide range of pres sure. By similar reasoning the thermal conductivity of a gas should be independent of the density. Maxwell predicted a value 0.000055 C.G.S. for the conductivity of air, and a value seven times greater for hydrogen on account of the greater velocity and range of its molecules. A. Kundt and E. Warburg (Jour. Phys., v. 118) found that the rate of cooling of a thermometer in air be tween 150 mm. and Imm. pressure remained constant as the pressure was varied. At higher pressures the effect of conduction was masked by convection currents.
The question of the variation of conductivity with temperature is more difficult. If the effects depended merely on the velocity of translation of the molecules, both conductivity and viscosity should increase directly as the square root of the absolute temperature; but the mean free path also varies in a manner which cannot be predicted by theory and which appears to be different for different gases (Rayleigh, Proc. R.S., January 1896). Experi ments by the capillary tube method have shown that the viscosity varies more nearly as 8:, but indicate that the rate of increase diminishes at high temperatures. The conductivity probably changes with temperature in the same way, being proportional to the product of the viscosity and the specific heat ; but the experi mental investigation presents difficulties on account of the neces sity of eliminating the effects of radiation and convection, and the results of different observers of ten differ considerably from theory and from each other. The values found for the conduc tivity of air at o° C range from .000048 to .000057, and the temperature-coefficient from •0015 to .0028.
Experimental determinations of the thermal conductivities of gases are still somewhat scarce and discordant owing to the great practical difficulties, but are of special interest for the elucidation of the law of action between molecules. The hot-wire method of T. Andrews (Phil. Trans., 1840) offers special facilities for rela tive measurements, such as the comparison of conductivities of different gases, or of the same gas at different temperatures, and has frequently been applied with this object in recent years. It has also been improved by introducing the usual compensation for end-effects, and employing more accurate methods of electrical measurement, but it remains liable to the difficulties depending on the small dimensions of the wire and the elimination of the corrections for radiation. The determination of the thermal con ductivities of gases gives a means of testing the value of the numerical coefficient f in the relation, k= between the con ductivity k, the viscosity n, and the specific heat s at constant volume. According to the theoretical investigations of S. Chap man (Phil. Trans., A, 21I, P. 433, 1911) the value of the coefficient f should be 2.5 for a gas constituted of spherically symmetrical molecules, which agrees with Maxwell's theory based on the in verse fifth-power law of force, and also with experiment for monatomic molecules. Unfortunately the variation of viscosity with temperature does not satisfy the fifth-power law, which re quires that the viscosity should be directly proportional to T. The conclusion is that monatomic gases may have spherically symmetrical molecules, but that the law of force is different. Theory gives no clear indication with regard to the appropriate value of f for other types of molecules. Experiment gives approx imately a linear relation, f=2.816-y-2.2, between f and the ratio y of the specific heats. This gives f=7/4 for diatomic gases, which show fair agreement with each other. The experi mental values for polyatomic gases are much less certain.
The thermal conductivity of liquids shows in one respect a remarkable contrast to that of gases, in that it has little or no relation to the viscosity. Excluding liquid metals, different liquids, such as water and glycerine, may vary widely in viscosity and yet differ little in conductivity: Most liquids show a very rapid diminution of viscosity with rise of temperature, without any cor responding change of similar magnitude in conductivity. But the experimental evidence is very discordant, as in the case of gases. The conductivity of liquids has been investigated by sim ilar methods, generally variations of the thin plate or guard-ring method. A critical account of the subject is contained in a paper by C. Chree (Phil. Mag., July 1887). Many of the experiments were made by comparative methods, taking a standard liquid such as water for reference. A determination of the conductivity of water by S. R. Milner and A. P. Chattock, employing an elec trical method, deserves mention on account of the careful elimina tion of various errors (Phil. Mag., July 1899). Their final result was k==.001433 at 20° C, which may be compared with the results of other observers, G. Lundquist (1869), .001S5 at 40° C; A. Winkelmann (1874), •00104 at 15° C; H. F. Weber (corrected by H. Lorberg), •00138 at 4° C, and .00152 at 23.6° C; C. H. Lees (Phil. Trans., 1898), •00136 at 25° C, and •00120 at 47° C; C. Chree, .00124 at 18° C, and .00136 at 19.5° C. The variations of these results illustrate the experimental difficulties. It appears probable that the conductivity of a liquid increases with rise of temperature, although the contrary would appear from the work of Lees.