Question
Show how the polytropic exponent $n$ can be evaluated if you know the end state properties, $\left(P_{1}, V_{i}\right)$ and $\left(P_{2}, V_{2}\right)$.
Step 1
Step 1: The polytropic process is defined by the equation $P V^n = C$, where $P$ is the pressure, $V$ is the volume, $n$ is the polytropic exponent, and $C$ is a constant. Show more…
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If $\alpha$ is the polytropic index then $p V^{\alpha}=$ constant, $T V^{\alpha-1}=$ constant. Now $\quad \frac{v^{\prime}}{v}=\frac{n^{\prime}}{n} \frac{\left\langle v^{\prime}\right\rangle}{\langle\nu>}=\frac{V}{V^{\prime}} \sqrt{\frac{T^{\prime}}{T}}=\frac{V T^{-1 / 2}}{V^{\prime} T^{-1 / 2}}=1$ Hence $\quad \frac{1}{\alpha-1}=-\frac{1}{2} \quad$ or $\quad \alpha=-1$ Then $C=\frac{i R}{2}+\frac{R}{2}=3 R$
Thermodynamics And Molecular Physics
Kinetic theory of Gases. Boltzmann's Law and Maxwell's Distribution
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