Thermal Physics - Adiabatic

The adiabatic expansion of an ideal gas is described by the equation PV ^γ = C, where γ and C are constants. The work done by the gas is expanding adiabatically from the state (V_i, P_i) to (V_f, P_f) is equal to

A. P_fV_f
B. ½(P_i + P_f)(V_f  − V_i)
C. (P_fV_f   − P_fV_i)/(1 − γ)
D. P_i(V_f ^1+ γ − V_i ^1+ γ )/(1 + γ)
E. P_f (V_f ^1- γ − V_i ^1- γ)/(1 + γ)

(GR9677 #73)

Solution:

PV ^γ = C
P = CV ^−γ

W = ∫ P dV
= C _Vi∫^V_f  V ^−γ dV
= [C/(−γ + 1)] V ^−γ+1 _Vi|^V_f
= [C/(1 − γ)] (V_f ^1−γ− V_i ^1−γ)

= (V_fCV_f ^−γ− V_i CV_i ^1−γ)/(1 − γ)

= (V_fP_f  − V_i P_i )/(1 − γ)

= (P_fV_f  − P_iV_i )/(1 − γ)

Answer: C