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 -  

$$
W_{1-2}=\int P d V
$$

For a polytropic process,

$$
\begin{gathered}
P V^N=P_1 V_1^N=P_2 V_2^N=C \\
P=\frac{C}{V^N}
\end{gathered}
$$

Substituting in Equation, we get,

$$
\begin{aligned}
\int P d V & =\int \frac{C d V}{V^N}=C \int V^{-N} d v \\
& =\left[V^{1-N}\right]_1^2=\left(V_2^{1-N}-V_1^{1-N}\right) \\
W_{1-2} & =\frac{P_2 V_2-P_1 V_1}{1-N} \text { or } \frac{P_1 V_1-P_2 V_2}{N-1} \ldots \ldots( \\
P_1 V_1 & =n R T_1 \\
P_2 V_2 & =n R T_2
\end{aligned}
$$

So, equation (1) can be written as - 

$$
W_{1-2}=\frac{n R\left(T_2-T_1\right)}{1-N}
$$

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

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