36 MOLECULAR MOTION AND ITS ENERGY 19 



19. Cooling by Expansion 



A.U exactly the same manner the reverse phenomenon 



janay be explained by the kinetic theory, viz. that a gas 



V must cool when it does work by expanding, and that it 



thereby loses an amount of molecular energy equal to the 



work done. 



If by its pressure the gas pushes back the piston with a 

 speed which, as before, we will denote by a, the molecular 

 speed of a molecule which impinges on the piston diminishes 

 from G to G %a, and there passes therefore from the 

 molecule to the piston at each impact the energy %maG. 

 Thus the molecules that strike the piston in unit time, 

 in all, lose the total energy 



.Fa=p SF, 



which is the work done by the gas in expanding through 

 the volume SF against the pressure^. 



The rise of temperature that accompanies the compres- 

 sion of a gas and the fall that results from its expansion can 

 from this be easily calculated if the value of the specific 

 heat of the gas at constant volume is known. We have 

 only to apply to this problem the general theorem of 

 thermodynamics that heat and energy are equivalent to 

 each other. If we represent by A the heat which is equi- 

 valent to a unit of work, Ap SF is the heat which is added 

 to the gas during the compression of its volume from F to 

 F SV or which leaves it during the expansion from F to 

 F 4- SV. We can otherwise express this heat in terms of 

 &S, the change produced in the temperature $. If c is the 

 specific heat at constant volume, pV the mass of the gas in 

 the cylinder, and therefore pVc its heat-capacity, the rela- 

 tion between the heat produced by compression and the 

 corresponding rise of temperature is 



= -ApSV, 



and this holds good too for the case of the gas cooling by 

 expansion. The negative sign has to be introduced into the 

 formula to indicate that an increase of volume corresponds 



