WORK OF THE EXPANSION OF AIR. 243 



The work of expansion between the pressures p v and p z 

 is represented by the area of the space MPjPcjN (Anal. 

 Mechs., Art. 222). To find an algebraic expression for this 

 work, let p and v be the corresponding pressure and volume 

 at any intermediate point P in the expansion. Then the 

 work done on the piston during the expansion from v to 

 v + dv ispdv, and the whole work clone during the expan- 

 sion from v 1 to v g , represented by the area MP 1 P 2 N, 



= / *pdv =p l v j j ^[from (1)] . 



/t>, "Vi V 



>g = Pi" i lg* r> (2) 



\ 



which is the work of expanding a given mass of air 

 from a higher pressure p l to a lower pressure p t . 



COR. In order to compress a given mass of air whose 

 volume is v 9 and whose pressure is p z , into a volume v^ of 

 the pressure p lf the work to be done 



which is the work of compressing a given mass of 

 air from a lower pressure p z to a higher pressure p l . 



Sen. The expressions in (2) and (3) for the work done 

 during the expansion and compression of air, are correct 

 only when the temperature of the air remains constant 

 while the change of volume or density is taking place ; but 

 the temperature of the air remains constant only when the 

 change of volume takes place so slowly that the heat in the 

 confined air has sufficient time to communicate any excess 

 to the walls of the vessel and to the exterior air. If the 

 change of density occurs so quickly that it is accompanied 

 by a change of temperature, when the air is expanded the 



* Hyp. log. 



