134-137] General Calculation of Pressure 119 



136. The equation of Van der Waals supplies the most convenient 

 basis for discussing the behaviour of a gas over those ranges of pressure, 

 volume and temperature for which the equation can be regarded as approxi- 

 mately true, i.e. over ranges in which the deviation from Boyle's Law is small. 

 We shall return to a discussion of the equation later. In the meantime we 

 shall investigate a general relation between pressure, volume and tempera- 

 ture, without the limitation that the deviations from Boyle's Law are to be 

 such as may be regarded as small. 



General Calculation of Pressure. 



137. We return now to the calculation started in 119, but shall make 

 no assumption as to the size of the molecules. 



The number of molecules which have velocities lying between u and 

 u + du, etc., and which will impinge on the element dS of the boundary 

 during the interval dt, is as before, equal to the number of molecules having 

 velocities satisfying these conditions, and lying within an element udSdt at 

 the beginning of this interval. This number, however, is no longer given 

 by expression (259). 



If the molecules possess spheres of molecular action satisfying the 

 conditions laid down in 82, the number required may be supposed to be 



Vif(u, v, w) dudvdwu dSdt (282), 



where v l is the "effective molecular density" (cf. 66) at the element in 

 question i.e. at a point distant \<r from the boundary. 



If the molecules do not possess spheres of molecular action, this expression 

 must be replaced by 



dudvdw udSdtll ...v l f(u, v, w, , ff 2 ...) df-idg^..., 



since v l is no longer a function solely of x, y, z. This case is too complex 

 for progress to be possible. 



If we revert to expression (282), and if, as in the previous calculation of 

 the pressure in a mixture of gases ( 123), we suppose the gas to consist of a 

 mixture of molecules of types a, ft ... , then we have in equation (258) 



\f a (u, v, w)u i dudvdw (283), 



A a. J 



where, as before, the integration extends over all values of u, v, and w for 

 which the molecules can collide with dS, and therefore over all possible 

 values of v and w, and over all positive values of u. 



Also, as before, the term 2w a w a ' in equation (258) is equal to the 



A 

 quantity on the right-hand of equation (283) except that the integration 



extends over all values of u, v, and w which are possible for molecules which 



