Euler’s test for exactness

Clearly, dw is not an exact differential, since Af(Pext V) = 0 and V) = —P, and Euler s test for exactness is not satisfied. The integral f dw = — f Pgxt dVcannot be evaluated, unless the dependence of the external pressure upon the volume of the gas is known. There is, however, no necessary relationship between the external pressure and the volume of the gas therefore, an infinite number of paths exist taking the gas fromPi ViPi to P2V2T2. Each path is associated with a specific amount of work. 

Applying Euler s test for exactness, we obtain the equations... 

Clearly, Euler s test for exactness is not satisfied, and the differential du is not exact If we multiply Equation (6-53) by x, we obtain the equation... 

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