18 PRESSURE OF GASES 35 



Each of these molecules would, after impact, rebound in 

 the opposite direction with unchanged speed if the piston 

 were at rest. But let the piston which compresses the gas 

 move forward with a speed a, in the direction opposite 

 that of the molecules which strike it with speed G. The 

 strength of the rebound is thereby increased in the ratio in 

 which the relative velocity G -f a exceeds the molecular 

 speed G. A rebounding molecule therefore experiences in 

 the impact an impulse, which is not ZmG as before, but 

 the greater one, 2m(Gr + a), which results from its losing 

 its initial speed G, and gaining the speed G + 2a in the 

 opposite direction ; its kinetic energy therefore increases 

 during the impact by 



Jm(G + 2a) 2 - JmG 2 = 2ma(G + a). 



This we may with sufficient exactness replace by 2waG, 

 since we have assumed that the compression goes on so 

 slowly that every disturbance of the molecular motion at 

 once subsides ; for we thereby also assume that the speed 

 a of the piston is negligible in respect of the speed G. 



Since each molecule gains this amount of energy at 

 every collision, the whole gain of energy by the gas in unit 

 time due to the impact of ^FNG molecules, as above, on 

 the piston is given by 



This product has a very simple meaning ; for the pressure 

 of the gas which the piston has to overcome is p = NmG\ 

 by 11, and the diminution which the volume V of the 

 gas experiences in unit time is 8V = Fa, as in this time 

 the piston moves through the length a in the cylinder, the 

 sectional area of which is F, and therefore the expression 

 found for the increment of energy is 



NmG 2 .Fa = p 8V. 



It is thus proved that the kinetic energy gained by the gas 

 during the compression is equal to p 8V, the work which 

 the piston must do to overcome the pressure of the gas. 



D 2 



