In ideal gas of $27^{\circ} \mathrm{C}$ is compressed adiabatically to $(8 / 27)$ of its original volume. If…

In ideal gas of $27^{\circ} \mathrm{C}$ is compressed adiabatically to $(8 / 27)$ of its original volume. If $\gamma=\frac{5}{3}$, the rise in temperature of a gas is
  1. 300 K
  2. 375 K
  3. 400 K
  4. 450 K

Solution

$\begin{aligned} \therefore \quad \frac{\mathrm{T}_2}{\mathrm{~T}_1} & =\left(\frac{\mathrm{V}_1}{\mathrm{~V}_2}\right)^{\gamma-1} \\ & =\left(\frac{27}{8}\right)^{\frac{5}{3}-1} \quad \ldots\left(\text { given, } \mathrm{V}_2=\frac{8}{27} \mathrm{~V}_1\right) \\ & =\left(\frac{27}{8}\right)^{\frac{2}{3}}=\frac{9}{4} \\ \therefore \quad \mathrm{~T}_2 & =\frac{9}{4} \times \mathrm{T}_1=\frac{9}{4} \times(27+273)=675 \mathrm{~K} \\ \therefore \quad \mathrm{~T}_2 & -\mathrm{T}_1=675-(27+273)=375 \mathrm{~K}\end{aligned}$

Asked in: MHT CET 2024 (16 May Shift 1)

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