Polytropic Process MCQ - Practice Questions with Answers

A process $P{V}^N=C$  is called a polytropic process. So, any process in this world related to thermodynamics can be explained by a polytropic process. 

For example - 1. If N = 1, then the process becomes isothermal.

                       2. If N=0, then the process becomes isobaric.

                       3. If N = $\gamma$, then the process become adiabatic

Work done by the polytropic process -  

$$W_{1-2}=\int PdV$$

For a polytropic process,

$$\begin{array}{c}P{V}^N=P_1{V}_1^N=P_2{V}_2^N=C\\ P=\frac{C}{V^N}\end{array}$$

Substituting in Equation, we get,

$$\begin{array}{rl}\int PdV& =\int \frac{CdV}{V^N}=C\int V^{-N}dv\\ & ={\left[V^{1-N}\right]}_1^2=(V_2^{1-N}-V_1^{1-N})\\ W_{1-2}& =\frac{P_2{V}_2-P_1{V}_1}{1-N}\text{ or }\frac{P_1{V}_1-P_2{V}_2}{N-1}\dots \dots (\\ P_1{V}_1& =nR{T}_1\\ P_2{V}_2& =nR{T}_2\end{array}$$

So, equation (1) can be written as - 

$$W_{1-2}=\frac{nR(T_2-T_1)}{1-N}$$

And for one mole, $W_{1-2}=\frac{R(T_2-T_1)}{1-N}$
Specific heat for polytropic process -
We can write the equation of heat as - $Q=C\mathrm{\Delta }T$
Here $\mathrm{C}=$ Molar specific heat -
From the first law of thermodynamics

$$\begin{array}{c}Q=\mathrm{\Delta }U+W\\ \text{ or }C\mathrm{\Delta }T=C_v\mathrm{\Delta }T-\frac{R\mathrm{\Delta }T}{(N-1)}\\ \therefore C=C_v-\frac{R}{(N-1)}=\frac{R}{(\gamma -1)}-\frac{R}{(N-1)}\end{array}$$