UAm(Yr^ 



106 PROCEEDINGS OF THE AMERICAN ACADEMY. 



to consider consists of a single molecule. The molecule consists of 

 a heavy particle of mass m, with two weightless springs projecting 

 on either side of length /q, giving a total diameter of the molecule of 

 2^0. (Figure 113.) This molecule travels back and forth in a hori- 

 zontal line between the two opposite vertical walls of the enclosure. 



The distance apart of the walls is the 

 volume of the enclosure, the force 

 exerted by the springs on the walls 

 during an encounter gives the pres- 

 sure exerted by the substance, and 

 the kinetic energy of the particle at 

 its maximum velocity represents 

 Figure 113. Model of a sub- temperature, 

 stance consisting of one molecule. ^, . , , . . i ,i , .i 



The model is to show that at high This model is so simple that the 



compressions the pressure is not entire discussion may be carried 

 given by the change of momentum through with rigorous mathematical 

 oi the molecules strikmg the walls i • mi i 



of the vessel in unit time. analysis. Ihe pressure exerted on 



the walls is evidently to be found 

 by taking the time average (over a long interval of time) of the force 

 exerted on the walls by the springs during the encounters. Now in 

 the solution of this problem there are three different cases. 



I. The first case is when the distance apart of the walls of the 

 vessel is greater than 21q, the diameter of the molecule. Under these 

 conditions we have collision without interference, which are the only 

 conditions to which the usual analysis applies. In this case it may be 

 proved by a detailed mathematical solution which will not be given 

 here that the time average of the force is exactly equal to the change 

 of momentum in unit time with respect to one of the walls, as it 

 should be. The results of the mathematical analysis for this case 

 may be given in the form of a distinctive equation of the substance, 



P[{v — b) + a P] = 2T. 



Here P is pressure, v = volume (distance apart of walls), h = diameter 



|2 



of molecule (2/o), o. = tt^j, where k is elasticity of the springs, and 



T = absolute temperature (kinetic energy of particle). The equation 

 bears a resemblance to van der Waal 's equation without the attrac- 

 tive forces. In van der Waal's equation a = 0, or k = oo , which 

 means that the time of collision of the particles with the wall may be 

 neglected in comparison with the time of free flight. 



II. The distance apart of the walls is less than 2/o, the diameter of 



