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Three moles of an ideal gas being initially at a temperature $T_{f} = 273 K$ were isothermally expanded 5 times its initial volume and then isochorically heated so that the pressure in the final state becomes equal to that in the initial state. The total heat supplied in the process in $80 k J$ . Find $γ (= \frac{C_{p}}{C_{V}})$ of the gas.

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answer is 1.4.

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Detailed Solution

First process

$T = constant$

$p \propto \frac{1}{V}$

$∴ p \propto \frac{1}{V}$

$V$ is made 5 times. Therefore, $p$ will become $\frac{1}{5}$ th.

Second process

$\Rightarrow V = constant$

Pressure again becomes five times. So, temperature will also become 5 times.

$Q_{1} + Q_{2} = 80 \times 10^{3}$

$∴ n R T \ln (\frac{V_{f}}{V_{i}}) + n C_{V} Δ T = 80 \times 10^{3}$

$∴ (3) (8.31) (273) \ln (5) + (3) (\frac{8.31}{γ - 1}) (5 \times 273 - 273)$

$= 80 \times 10^{3}$

Solving we get,

$γ = 1.4$

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