Chapter 21, Problem 21.58AP

In a cylinder, a sample of an ideal gas with number of moles n undergoes an adiabatic process. (a) Starting with the expression $W=-{\displaystyle \int PdV}$ and using the condition PV^γ = constant, show that the work done on the gas is

$W=\left(\frac{1}{\gamma -1}\right)(P_f{V}_f-P_i{V}_i)$

(b) Starting with the first law of thermodynamics, show that the work done on the gas is equal to nC_V(T_f − T_i). (c) Are these two results consistent with each other? Explain.