An ideal gas at 127^{circ} C is compressed su | vedclass

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(B) sudden compression is an adiabatic process. The relation between temperature and volume for an adiabatic process is $T_1 V_1^{\gamma-1} = T_2 V_2^

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$500$

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(B) sudden compression is an adiabatic process. The relation between temperature and volume for an adiabatic process is $T_1 V_1^{\gamma-1} = T_2 V_2^{\gamma-1}$.Initial temperature $T_1 = 127 + 273 = 400 \ K$.Given $V_2 = \frac{8}{27} V_1$, so $\frac{V_1}{V_2} = \frac{27}{8}$.Using the adiabatic relation:$T_2 = T_1 \left( \frac{V_1}{V_2} ight)^{\gamma-1}$$T_2 = 400 	imes \left( \frac{27}{8} ight)^{\frac{5}{3}-1} = 400 	imes \left( \frac{27}{8} ight)^{\frac{2}{3}}$$T_2 = 400 	imes \left( \left( \frac{3}{2} ight)^3 ight)^{\frac{2}{3}} = 400 	imes \left( \frac{3}{2} ight)^2 = 400 	imes \frac{9}{4} = 900 \ K$.The rise in temperature is $\Delta T = T_2 - T_1 = 900 \ K - 400 \ K = 500 \ K$.

An ideal gas at $127^{\circ} C$ is compressed suddenly to $\frac{8}{27}$ of its initial volume. If $\gamma=\frac{5}{3}$ for the ideal gas, then the rise in its temperature is: (in $K$)

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