Question 2.4: Obtain an expression for the work done during a polytropic p......

For a polytropic process, the relation is pV^n = c where c is a constant. For a system change from state 1 to state 2, the work done is given by W_{12} = \int\limits_{1}^{2}{pdV}.

W_{12} = \int\limits_{1}^{2}{\frac{c}{V^n}dν } = c\frac{V^{-n+1}}{-n+1} ]^2_1 = \frac{c}{-n+1} \left[V^{-n+1}_2 – V^{-n+1}_1\right] = \frac{1}{-n+1} \left[p_2V^{-n+1}_2 – p_1V^{-n+1}_1\right] = \frac{p_1V_1- p_2V_2}{n-1}

Note: The formula for adiabatic process is given by replacing n by γ. However, the formula does not hold good for isothermal process when n becomes 1. The formula needs to be derived by integrating p as c/V.