# Derivation of the equation of the work done in a polytropic process

I am completing a question that is requiring calculation of the work done in a polytropic process. In the worked solutions the work equation seems to be reversed for the answer, as can be seen in the screenshot- the top row is p1v1-p2v2 insted of p2v2-p1v1. Any clue why?

Stay away from running things by the rule book, you could have done the calculation yourself.

$W=\int_{1}^{2} pdV$

$pV^n=const=C$

$W=\int_{1}^{2} \frac{C}{V^n}dV$

$W= \left[\frac{C \cdot V^{1-n}}{1-n}\right]_1^2$

$W= \frac{C \cdot V_2^{1-n}}{1-n}-\frac{C \cdot V_1^{1-n}}{1-n}$

multiply with $\frac{-1}{-1}$

$W= \frac{- C \cdot V_2^{1-n}}{n-1}+\frac{C \cdot V_1^{1-n}}{n-1}$

$W= \frac{C \cdot V_1^{1-n}}{n-1}-\frac{C \cdot V_2^{1-n}}{n-1}$

replace with $CV_i^{-n}=p_i$

$W= \frac{p_1 \cdot V_1- p_2 \cdot V_2}{n-1}$

I went the longer way instead of commenting on the switched denominator to add some value for other readers to this question.

• Thanks @idkfa makes a lot more sense with the equation explained. Commented Mar 28, 2016 at 19:56

This is probably late and not needed, but the equation with p2v2-p1v1 is just the negative of this one. The denominator for p2v2-p1v1 is 1-n, but for this one is n-1. so they are essentially the same thing.