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.
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.
