An ideal gas whose adiabatic exponent equals to $γ = \frac{7}{5}$ is expanded according to the law $P = 2 V$ . The initial volume of the gas is equal to $V_{0} = 1 unit$ . As a result of expansion volume increases 4 times. (Take $R = \frac{25}{3}$ units )

$Column I$ $Column II$

$(A)$ Work done by the gas $(P)$ 25 units

$(B)$ Increment in internal energy of the gas $(Q)$ 90 units

$(C)$ Heat supplied to the gas $(R)$ 75 units

$(D)$ Molar heat capacity of the gas in the process $(S)$ 15 units

$(D)$ 55 units

(A) \to (S); (B) \to (R); (C) \to (Q); (D) \to (P)

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(A) \to (P); (B) \to (Q); (C) \to (R); (D) \to (T)

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(A) \to (T); (B) \to (S); (C) \to (P); (D) \to (S)

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(A) \to (S); (B) \to (R); (C) \to (P); (D) \to (Q)

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Solution

The correct option is A $(A) \to (S); (B) \to (R); (C) \to (Q); (D) \to (P)$
Given,
$γ = \frac{7}{5}, P = 2 V, V_{o} = 1$ unit
and $V_{f} = 4 V_{0}$

Work done in an adiabatic process $W = \int_{V_{i}}^{V_{f}} P d V$
$W = \int_{V_{0}}^{4 V_{0}} 2 V d V = [V^{2}]_{V_{0}}^{4 V_{0}} = 15 V_{0}^{2} = 15 units$
Hence, $(A) \to (S)$ .

Increment in internal energy
Ideal gas equation : $PV = n R T,$
where, $P = 2 V$
so, $2 V^{2} = n R T 2 (V_{f}^{2} - V_{0}^{2}) = n R (Δ T) 2 (4^{2} - 1^{2}) V_{0}^{2} = n R (Δ T) \Rightarrow n R Δ T = 30 V_{0}^{2} . . . . (1)$

Internal energy in an adiabatic process is given by
$Δ U = n C_{v} Δ T = \frac{n R}{γ - 1} Δ T$
using eq. (1),
$\frac{30 V_{0}^{2}}{γ - 1} = \frac{30 (1)^{2}}{\frac{7}{5} - 1} = \frac{30}{2} (5) = 75 units$
Hence, $(B) \to (R)$ .

Calculation of heat supply,
$Q = W + Δ U$
As we have calculated early,
$W = 15$ unit
and $Δ U = 75$ unit
$Q = 15 + 75 = 90 units$
Hence, Hence, $(C) \to (Q)$ .

Molar heat capacity for $P V^{x}$ process is given as:
$C = C_{v} + \frac{R}{1 - x} = \frac{5}{2} R + \frac{R}{1 - (- 1)} = \frac{5}{2} R + \frac{R}{2} = 3 R = 3 \times \frac{25}{3} = 25 units$
Hence, Hence, $(D) \to (P)$ .

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$Column I$	$Column II$
$(A)$ Work done by the gas	$(P)$ 25 units
$(B)$ Increment in internal energy of the gas	$(Q)$ 90 units
$(C)$ Heat supplied to the gas	$(R)$ 75 units
$(D)$ Molar heat capacity of the gas in the process	$(S)$ 15 units