A quasi-static cycle of a monoatomic ideal gas contains an isothermal process $(\boldsymbol{a} \boldsymbol{b})$, followed by an isochoric process $(\boldsymbol{b} \boldsymbol{c})$ and an adiabatic process $(\boldsymbol{c} \boldsymbol{a})$ as shown in the figure. The volumes of the gas are $V_{1}$ and $V_{2}$ at $\boldsymbol{a}$ and $\boldsymbol{b}$, respectively. If the cycle has heat input $Q_{\text {in }}$ and output $Q_{\text {out }}$, then the efficiency of the cycle is defined as $\eta=\frac{Q_{\text {in }}-Q_{\text {out }}}{Q_{\text {in }}}$. The correct statement(s) is/are: [Given $: \ln 2 \approx 0.7$ ]

(A) If $V_{2} / V_{1}=8$, the heat released in the process $\boldsymbol{b} \boldsymbol{c}$ is smaller than the heat absorbed in the process $\boldsymbol{a} \boldsymbol{b}$.; (B) For a given value of $V_{2} / V_{1}, \eta$ does not depend on the temperature of the isothermal process; (C) If $V_{2} / V_{1}=8$, then the temperature of the gas at $\boldsymbol{a}$ is 4 times the temperature of the gas at $\boldsymbol{c}$.

(D) If $V_{2} / V_{1}=8$, then the pressure of the gas at $\boldsymbol{a}$ is 4 times the pressure of the gas at $\boldsymbol{b}$.

JEE Advanced · 2026 · 17 May 2026 · Paper 1 · Heat & Thermodynamics

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A quasi-static cycle of a monoatomic ideal gas contains an isothermal process

(a b)

, followed by an isochoric process

(b c)

and an adiabatic process

(c a)

as shown in the figure. The volumes of the gas are

V_{1}

and

V_{2}

a

and

b

, respectively. If the cycle has heat input

Q_{in}

and output

Q_{out}

, then the efficiency of the cycle is defined as

η = \frac{Q _{in} - Q _{out}}{Q _{in}}

. The correct statement(s) is/are:
[Given

: ln 2 \approx 0.7

]

(A)If $V_{2} / V_{1} = 8$ , the heat released in the process $b c$ is smaller than the heat absorbed in the process $a b$ .✓ Correct
(B)For a given value of $V_{2} / V_{1}, η$ does not depend on the temperature of the isothermal process✓ Correct
(C)If $V_{2} / V_{1} = 8$ , then the temperature of the gas at $a$ is 4 times the temperature of the gas at $c$ .✓ Correct
(D)If $V_{2} / V_{1} = 8$ , then the pressure of the gas at $a$ is 4 times the pressure of the gas at $b$ .

Answer: (A), (B), (C)

Solution

For monoatomic gas:

C_{v} = \frac{3 R}{2}

γ = \frac{5}{3}

.

(A) For a clockwise P-V cycle:

Q_{ab} + Q_{bc} + Q_{ca} > 0

. Since

Q_{ca} = 0

(adiabatic),

Q_{ab} > - Q_{bc}

, meaning heat absorbed in ab exceeds heat released in bc ✓

(B)

η = 1 - \frac{C _{v}}{R} \cdot (\frac{1 - T _{c} / T _{a}}{l n ( V _{2} / V _{1} )})

. Using adiabatic relation

T_{a} V_{1}^{γ - 1} = T_{c} V_{2}^{γ - 1}

, the ratio

T_{c} / T_{a}

depends only on

V_{2} / V_{1}

, making

η

independent of

T_{a}

✓

(C)

\frac{T _{a}}{T _{c}} = (\frac{V _{2}}{V _{1}})^{γ - 1} = 8^{2/3} = 4

✓

(D) For isothermal process ab:

\frac{P _{a}}{P _{b}} = \frac{V _{2}}{V _{1}} = 8 \neq = 4

✗

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