QJEEM24Pha151: JEE Main 2024 Physics - Class 11 JEEM24Pha151 Solution: Two different adiabatic paths for the same gas intersect two…

$\begin{aligned}\text { (1) } P_a V_a & =P_b V_b \quad \text{.... (i)}\\P_c V_c & =P_d V_d \quad \text{.... (ii)}\end{aligned}$

$\begin{aligned}\text { (2) } P_a V_a^{\gamma-1} & =P_d V_d^{\gamma-1} \quad \text{.... (iii)}\\P_b V_b^{\gamma-1} & =P_c V_c^{\gamma-1} \quad \text{.... (iv)}\end{aligned}$

$\begin{aligned}& \text { (i) } \div \text { (iii) } \Rightarrow \frac{V_a}{V_a^{\gamma-1}}=\frac{P_b V_b}{P_d V_d{ }^{\gamma-1}} \text { (v) } \\& \text { (ii) } \div \text { (iv) } \Rightarrow \frac{V_c}{V_c^{\gamma-1}}=\frac{P_d V_d}{P_b V_b{ }^{\gamma-1}} \text { (vi) } \\& \text { (v) } \times \text { (vi) } \Rightarrow \frac{V_a}{V_a^{\gamma-1}} \frac{V_c}{V_c{ }^{\gamma-1}}=\frac{P_b V_b}{P_d V_d^{\gamma-1}} \times \frac{P_d V_d}{P_b V_b^{\gamma-1}} \\& \Rightarrow V_a^{\gamma-2} V_c{ }^{\gamma-2}=V_d^{\gamma-1} V_b{ }^{\gamma-2} \\& \Rightarrow \frac{V_a}{V_d}=\frac{V_b}{V_c}\end{aligned}$