Polytropic Process MCQ - Practice Questions & Answers

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An ideal gas undergoes a quasi-static, reversible process in which its molar heat capacity C remains constant. If during this process the relation of pressure P and volume V is given by PVⁿ=constant, then n is given by (Here C_P and C_V are molar specific heat at constant pressure and constant volume, respectively)

$n=\frac{C_{p}}{C_{v}}$

$n=\frac{C-C_{p}}{C-C_{v}}$

$n=\frac{C_{p}-C}{C-C_{v}}$

$n=\frac{C-C_{v}}{C-C_{p}}$

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Concepts Covered - 1

Polytropic Process

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

For example - 1 . If $\mathrm{N}=1$, then the process become isothermal.
2. If $\mathrm{N}=0$, then the process become isobaric.
3. If $\mathrm{N}=\gamma$, then the process become adiabatic

Work done by 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}
$

Subsiting 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 equation of heat as - $Q=C \Delta T$
Here C = Molar specific heat -
From the first law of thermodynamics

$
\begin{aligned}
& \quad 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{aligned}
$