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$(C)$ Free expansion of an ideal gas into a vacuum is an adiabatic process where $W = 0$ and $Q = 0$, so $\Delta U = 0$. For an ideal gas, $\Delta U =

Vedclass · Accepted Answer

$(A) \rightarrow p, (B) \rightarrow p \& s, (C) \rightarrow p \& s, (D) \rightarrow q \& p$

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$(C)$ Free expansion of an ideal gas into a vacuum is an adiabatic process where $W = 0$ and $Q = 0$, so $\Delta U = 0$. For an ideal gas, $\Delta U = nC_v\Delta T = 0$, implying $\Delta T = 0$. Thus, the temperature remains constant. This matches $(q)$.$(B)$ Polytropic process $PV^x = 	ext{constant}$ with $x = 2$. For a monoatomic gas, $\gamma = 5/3$. Since $x > \gamma$, the molar heat capacity $C = C_v + R/(1-x) = 3R/2 + R/(1-2) = 3R/2 - R = R/2 > 0$. Also, $T \propto PV \propto V^{-2} \cdot V = V^{-1}$, so as $V$ increases, $T$ decreases. Since $C > 0$ and $\Delta T $(C)$ Polytropic process with $x = 4/3$. Since $x  0$, meaning the gas gains heat. Matches $(p)$ and $(s)$.$(D)$ The graph shows $V$ increasing, if $P$ decreases as $V$ increases, the gas does work. If the process is such that $T$ increases, it gains heat. Based on standard interpretation of such problems, $(D)$ matches $(q)$ and $(s)$.

Column $I$	Column $II$
$(A)$ An insulated container has two chambers separated by a valve. Chamber $I$ contains an ideal gas and Chamber $II$ has a vacuum. The valve is opened.	$(p)$ The temperature of the gas decreases
$(B)$ An ideal monoatomic gas expands to twice its original volume such that its pressure $P \propto V^{-2}$	$(q)$ The temperature of the gas increases or remains constant
$(C)$ An ideal monoatomic gas expands to twice its original volume such that its pressure $P \propto V^{-4/3}$	$(r)$ The gas loses heat
$(D)$ An ideal monoatomic gas expands such that its pressure $P$ and volume $V$ follow the behavior shown in the graph	$(s)$ The gas gains heat

Column-$I$	Column-$II$
$(a)$ Heat engine	$(i)$ $\text{COP} = \frac{Q_2}{Q_1 - Q_2}$
$(b)$ Heat pump	$(ii)$ $\eta = \frac{Q_1 - Q_2}{Q_1}$
	$(iii)$ $\eta = \frac{T_1 - T_2}{T_1}$

Similar Questions

One mole of diatomic ideal gas undergoes a cyclic process $ABC$ as shown in the figure. The process $BC$ is adiabatic. The temperatures at $A, B$,and $C$ are $400\,K, 800\,K$,and $600\,K$ respectively. Choose the correct statement.

$A$ given mass of a gas is compressed isothermally until its pressure is doubled. It is then allowed to expand adiabatically until its original volume is restored and its pressure is then found to be $0.75$ of its initial pressure. The ratio of the specific heats of the gas is approximately:

Match the devices in Column-$I$ with their efficiency/coefficient of performance in Column-$II$:

Column-$I$ Column-$II$

$(a)$ Heat engine $(i)$ $\text{COP} = \frac{Q_2}{Q_1 - Q_2}$

$(b)$ Heat pump $(ii)$ $\eta = \frac{Q_1 - Q_2}{Q_1}$

$(iii)$ $\eta = \frac{T_1 - T_2}{T_1}$

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