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An ideal gas at $127^{\circ} \mathrm{C}$ is compressed suddenly to $\frac{8}{27}$ of its initial volume. If $\gamma=\frac{5}{3}$ for an ideal gas, then rise in its temperature is
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$500 \mathrm{~K}$
The condition for adiabatic process is $\mathrm{TV}^{\gamma-1}=$ constant
Before compression, $\mathrm{T}_1 \mathrm{~V}_1^{\gamma-1}=$ const.
After compression, $\mathrm{T}_2 \mathrm{~V}_2^{\gamma-1}=$ const.
Dividing (ii) by (i):
$$
\begin{aligned}
& \mathrm{T}_2 \mathrm{~V}_2^{\gamma-1}=\mathrm{T}_1 \mathrm{~V}_1^{\gamma-1} \\
& \frac{\mathrm{T}_2}{\mathrm{~T}_1}=\left(\frac{\mathrm{V}_1}{\mathrm{~V}_2}\right)^{\gamma-1} \\
& \mathrm{~T}_2=\left(\frac{\mathrm{V}_1}{\mathrm{~V}_2}\right)^{\gamma-1}\left(\mathrm{~T}_1\right) \\
& \mathrm{T}_2=\left(\frac{\mathrm{V} \times 27}{8 \mathrm{~V}}\right)^{\frac{5}{3}-1}[400]=\frac{9}{4} \times 400 \\
& =900 \mathrm{~K} .
\end{aligned}
$$
Change in temperature,
$$
\begin{aligned}
& \Delta \mathrm{T}=\mathrm{T}_2-\mathrm{T}_1=900-400 \\
& \Delta \mathrm{T}=500 \mathrm{~K}
\end{aligned}
$$
Before compression, $\mathrm{T}_1 \mathrm{~V}_1^{\gamma-1}=$ const.
After compression, $\mathrm{T}_2 \mathrm{~V}_2^{\gamma-1}=$ const.
Dividing (ii) by (i):
$$
\begin{aligned}
& \mathrm{T}_2 \mathrm{~V}_2^{\gamma-1}=\mathrm{T}_1 \mathrm{~V}_1^{\gamma-1} \\
& \frac{\mathrm{T}_2}{\mathrm{~T}_1}=\left(\frac{\mathrm{V}_1}{\mathrm{~V}_2}\right)^{\gamma-1} \\
& \mathrm{~T}_2=\left(\frac{\mathrm{V}_1}{\mathrm{~V}_2}\right)^{\gamma-1}\left(\mathrm{~T}_1\right) \\
& \mathrm{T}_2=\left(\frac{\mathrm{V} \times 27}{8 \mathrm{~V}}\right)^{\frac{5}{3}-1}[400]=\frac{9}{4} \times 400 \\
& =900 \mathrm{~K} .
\end{aligned}
$$
Change in temperature,
$$
\begin{aligned}
& \Delta \mathrm{T}=\mathrm{T}_2-\mathrm{T}_1=900-400 \\
& \Delta \mathrm{T}=500 \mathrm{~K}
\end{aligned}
$$
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