Answer
The proof is below.
Work Step by Step
We know that equation 18.3 states:
$W=-\int_{V_1}^{V_2}PdV$
For an adiabatic process, we know the following equation is true:
$PV^{\gamma}=c$
Where c is a constant.
Substituting this equation into equation 18.3 and taking the integral, it follows:
$W = P(\frac{V_2-V_1}{\gamma-1})$
$W = (\frac{PV_2-PV_1}{\gamma-1})$
This is equation 18.12.
