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## Homework Statement

Prove that the work done by an ideal gas with constant heat capacities during a quasi-static adiabatic expansion is equal to

W= (PfVf)/(Y-1)[1 - (Pi/Pf)^((Y-1)/Y)]

where Y = gamma, which is heat capacity at constant pressure over heat capacity at constant volume

## Homework Equations

## The Attempt at a Solution

Alright so this is my attempt and im not sure where to go from here...

In an adiabatic quasi-static process we can write the formula

PV^Y = constant

constant = K for simplification

Since its adiabatic no heat change so Q=0

Using the first law of thermo

Q= ΔU -W

We know that W = -PdV

and P= K/V^Y

so...

W = ΔU

W = -PdV

W = -(K/V^Y)*dV

W = -K∫(1/V^Y)*dV

W = -K[V^(1-Y)/(1-Y)]*∫dV

W = -(K/(1-Y))[Vf^(1-Y) - Vi^(1-Y)]

W = -(K/(1-Y))[Vf^(-Y)*Vf - Vi^(-Y)*Vi]

W = -(1/(1-Y))[((Vf*K)/(Vf^Y)) - ((Vi*K)/(Vi^Y))]

since Pi = K/Vi^Y and Pf = K/Vf^Y sub those in

W = -(1/(1-Y))(Vf*Pf - Vi*Pi)

Times this by (-1/-1)

and we get

W = (PfVf - PiVi)/(Y-1)

This is where I get to not sure where to go from here to make this into

W= (PfVf)/(Y-1)[1 - (Pi/Pf)^((Y-1)/Y)]

Any suggestions and help would be greatly appreciative.