Properties of Gases and Vapours, $-c. 269 



If we write (v - b) for v, on the left hand side, this equation is 

 practically identical, for moderate pressures, with the modified form of 

 the equation of Van der Waals which was devised by Clausius* to 

 meet the objection that the term a/v in the equation of Van der Waals 

 did not satisfactorily represent the variation of the phenomena with 

 temperature. In adopting this equation of Rankine's to represent 

 their observations " On the Thermal Effects of Fluids in Motion,"! 

 Joule and Thomson substituted v = HB/p in the small term a/6v, 

 which thus became ap/R,@' 2 . For the purpose of their observations the 

 modification appeared to be unimportant, and they quote the equation 

 as Rankine's, but it really introduces a great simplification. If we 

 write the equation in the form, 



........................ (5), 



we observe that the small term is a function of the temperature only, 

 and is independent of the pressure or volume. The isothermals on 

 the p, v diagram are equilateral hyperbolas, identical in form with those 

 of an ideal gas. Or, if we plot the product pv against p, as is usual in 

 considering the deviations of a gas from Boyle's law, the isothermals 

 are straight lines inclined to the axis of p at various angles, which 

 diminish as the temperature rises. It is proved by the experiments of 

 Joule and Thomson, and more clearly by the subsequent observations 

 of Amagat and others, that the eqtiation, even in this simple form, 

 represents a very good first approximation to the deviations of actual 

 gases from Boyle's law at moderate pressures. The approximation 

 holds, for instance, in the case of COo, according to the observations of 

 Amagat, up to 50 or 100 atmospheres at temperatures between 100 

 and 200 C. The application of the equation to the case of vapours 

 may, however, be still further simplified, and rendered at the same time 

 more accurate, by two slight but important modifications. 



(1) It is practically certain that the equation of a perfect, or plu- 

 perfect, gas at high temperatures is not pv = R0, but p(v - b) = 

 R#, where b is the minimum volume or "co-volume" of Hirn and 

 Van der Waals. The co-volume b is variously regarded as being equal 

 to four times or 4 ^/2 times the absolute volume of the molecules. It 

 is relatively small at moderate pressures (about one-thousandth of v 

 at atmospheric pressure), and is often negligible, but may with great 

 probability be taken as equal to the volume of the liquid at tempera- 

 tures where the vapour pressure is small. 



(2) It is usual in the kinetic theory of gases, either tacitly or 

 explicitly, to make the fundamental assumption that the average total 

 kinetic energy of the molecules of a gas, including motions of vibration 

 and rotation, is directly proportional to the kinetic energy of transla- 



* ' Phil. Mag.,' June, 1880. 

 f ' Phil. Trans.,' 1862. 



