w 



WO I 



198 Messrs. M. P. Applebey and D. L. Chapman on 



mean kinetic energy of translation of a molecule was propor- 

 tional to the temperature. Strictly this is not true. We are 

 only justified in concluding that 



(p + £){*-l>) = *»»'?«, .... (1) 



here the letters have their usual significance. In other 

 ds, the equation relates the pressure and volume of the 

 gas with the total kinetic energy of translation of the mole- 

 cules, but not with the temperature. If any relation is to be 

 established between the pressure, volume, and thermodynamic 

 temperature of a gas, some assumption will have to be made 

 in order to replace that which is no longer strictly valid. 



We assume that in a closed space which contains a large 

 number of like molecules the ratio of the number of molecules 

 per unit volume whose potential energy is A to the number of 

 molecules per unit volume whose potential energy is zero is given 



A 



by the expression e kt , k being the gas constant for a single 

 molecule, and t the thermodynamic temperature. 



It will be observed that the distinction between this 

 assumption and the corresponding proposition deduced from 

 Maxwell's laws of the partition and distribution of energy, 

 is that t is substituted in the assumption for § of the 

 mean kinetic energy of translation in the corresponding 

 proposition. 



Starting from this assumption, a modified form of Van der 

 Waals' equation can be deduced in the following way. 

 Consider a column of gas in a field of force — let us say 

 gravity. Take the axis of x in the direction of the force. 

 The density of the gas will increase in the direction of 

 increasing x. The increasing density of the gas will set up 

 a field of cohesive force acting in the same direction as 



gravity and equal in magnitude to <*—, where p is the 



ax 



density of the gas and a a constant. Let m be the mass 

 of a molecule of the gas. The sum of the forces due to 

 gravity and cohesion, acting in the direction of x, on 



this molecule will be mlg + a-V g being the acceleration 



of gravity. From this must be subtracted the upthrust 

 due to the fluid displaced by the molecule. Let the 

 volume of fluid displaced be represented by b' . Then 

 the total downward force acting on the molecule be- 

 comes : (m — b'p) I g -r ex. -j- ). If the molecule is displaced 



