ThermodynamicsHard
Question
One mole of an ideal gas with the adiabatic exponent ‘$\gamma$’ goes through a polytropic process as a result of which the absolute temperature of the gas increases $\tau$ -fold. The polytropic constant equals n. The entropy increment of the gas in this process is
Options
A.$\frac{(n - \gamma)R}{(n - 1)(\gamma - 1)}\ln\tau$
B.$\frac{(n - 1)(\gamma - 1)}{(n - \gamma)R}\ln\tau$
C.$\frac{(n - \gamma)R}{(\gamma - 1)}\ln\tau$
D.$\frac{(n - \gamma)R}{(n - 1)}\ln\tau$
Solution
$\Delta S = n.C_{V,m}.\ln\frac{T_{2}}{T_{1}} + nR.\ln\frac{V_{2}}{V_{1}}$
$= 1 \times \frac{R}{\gamma - 1} \times \ln\frac{T_{2}}{T_{1}} + 1 \times R \times {\ln\left( \frac{T_{1}}{T_{2}} \right)}^{1/n - 1}\left( \text{as }T.V^{n - 1} = \text{Constant} \right) $$$R.\ln\frac{T_{2}}{T_{1}}\left\lbrack \frac{1}{\gamma - 1} - \frac{1}{n - 1} \right\rbrack = \frac{(n - \gamma)R}{(n - 1)(\gamma - 1)}.\ln\tau$$
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