A gas at 10 , atm pressure and 300 , K temper | vedclass

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(B) For an adiabatic reversible process,the relation between temperature and pressure is given by: $T^{\gamma} P^{1-\gamma} = 	ext{constant}$.Taking

Vedclass · Accepted Answer

$40.8$

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(B) For an adiabatic reversible process,the relation between temperature and pressure is given by: $T^{\gamma} P^{1-\gamma} = 	ext{constant}$.Taking the ratio for two states: $(T_1/T_2)^{\gamma} = (P_1/P_2)^{\gamma-1}$.Taking $\ln$ on both sides: $\gamma \ln(T_1/T_2) = (\gamma-1) \ln(P_1/P_2)$.Given $T_1 = 300 \, K, T_2 = 290 \, K, P_1 = 10 \, atm, P_2 = 5 \, atm$.$\gamma \ln(300/290) = (\gamma-1) \ln(10/5)$.$\gamma \ln(1.0345) = (\gamma-1) \ln(2)$.$\gamma(0.0339) = (\gamma-1)(0.693)$.$0.0339 \gamma = 0.693 \gamma - 0.693$.$0.6591 \gamma = 0.693 \implies \gamma \approx 1.05$.Since $C_v = R / (\gamma-1)$ and $R \approx 2 \, cal \, mol^{-1} \, K^{-1}$.$C_v = 2 / (1.05 - 1) = 2 / 0.05 = 40 \, cal \, mol^{-1} \, K^{-1}$.Rounding to the nearest option,the value is $40.8 \, cal \, mol^{-1} \, K^{-1}$.

Column $I$	Column $II$
$A$. Freezing of water at $273 \ K$ and $1 \ atm$	$P$. $q=0$
$B$. Expansion of $1 \ mol$ of an ideal gas into a vacuum under isolated conditions	$Q$. $w=0$
$C$. Mixing of equal volumes of two ideal gases at constant temperature and pressure in an isolated container	$R$. $\Delta S_{sys} < 0$
$D$. Reversible heating of $H_{2(g)}$ at $1 \ atm$ from $300 \ K$ to $600 \ K$,followed by reversible cooling to $300 \ K$ at $1 \ atm$	$S$. $\Delta U=0$
	$T$. $\Delta G=0$

List-$I$	List-$II$
$(A) \Delta U = W_{ad}$	$I.$ Isothermal reversible expansion
$(B) \Delta U = q - W$	$II.$ Wall is adiabatic
$(C) \Delta U = -q$	$III.$ Thermally conducting walls
$(D) \Delta U = 0$	$IV.$ Isolated system
	$V.$ Closed system

$A$ gas at $10 \, atm$ pressure and $300 \, K$ temperature undergoes adiabatic reversible expansion to $5 \, atm$ pressure and $290 \, K$ temperature. The molar heat capacity $C_v$ of the gas in $cal \, mol^{-1} \, K^{-1}$ is:

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Match the following:
List-$I$ List-$II$
$(A) \Delta U = W_{ad}$ $I.$ Isothermal reversible expansion
$(B) \Delta U = q - W$ $II.$ Wall is adiabatic
$(C) \Delta U = -q$ $III.$ Thermally conducting walls
$(D) \Delta U = 0$ $IV.$ Isolated system
$V.$ Closed system

The correct answer is

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