Heat and Thermodynamics 4 Question 17
19. 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 $p V^{n}=$ constant, then $n$ is given by (Here, $C _p$ and $C _V$ are molar specific heat at constant pressure and constant volume, respectively)
(2016 Main)
(a) $n=\frac{C _p}{C _V}$
(b) $n=\frac{C-C _p}{C-C _V}$
(c) $n=\frac{C _p-C}{C-C _V}$
(d) $n=\frac{C-C _V}{C-C _p}$
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Solution:
- $\Delta Q=\Delta U+\Delta W$
In the process $p V^{n}=$ constant, molar heat capacity is given by
$$ \begin{array}{r} C=\frac{R}{\gamma-1}+\frac{R}{1-n}=C _V+\frac{R}{1-n} \\ C-C _V=\frac{R}{1-n} \Rightarrow 1-n=\frac{C _p-C _V}{C-C _V} \\ \Rightarrow \quad n=1-\left(\frac{C _p-C _V}{C-C _V}\right) \\ =\frac{\left(C-C _V\right)-\left(C _p-C _V\right)}{C-C _V}=\frac{C-C _p}{C-C _V} \end{array} $$
