In the P-V diagram shown,there are two adiaba | vedclass

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(B) For an adiabatic process,the relation between temperature and volume is $T V^{\gamma-1} = 	ext{constant}$.For the adiabatic path $BC$,the points

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$\left(\frac{V_c}{V_d}\right)$

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(B) For an adiabatic process,the relation between temperature and volume is $T V^{\gamma-1} = 	ext{constant}$.For the adiabatic path $BC$,the points $B$ and $C$ lie on the adiabatic curve connecting the isothermals $T_1$ and $T_2$. Thus:$T_1 V_b^{\gamma-1} = T_2 V_c^{\gamma-1}$$\Rightarrow \left(\frac{V_b}{V_c}ight)^{\gamma-1} = \frac{T_2}{T_1} \quad ---(1)$For the adiabatic path $AD$,the points $A$ and $D$ lie on the adiabatic curve connecting the isothermals $T_1$ and $T_2$. Thus:$T_1 V_a^{\gamma-1} = T_2 V_d^{\gamma-1}$$\Rightarrow \left(\frac{V_a}{V_d}ight)^{\gamma-1} = \frac{T_2}{T_1} \quad ---(2)$Equating $(1)$ and $(2)$:$\left(\frac{V_b}{V_c}ight)^{\gamma-1} = \left(\frac{V_a}{V_d}ight)^{\gamma-1}$$\Rightarrow \frac{V_b}{V_c} = \frac{V_a}{V_d}$Rearranging the terms,we get:$\frac{V_b}{V_a} = \frac{V_c}{V_d}$Therefore,option $(B)$ is correct.

In the $P-V$ diagram shown,there are two adiabatic parts of the same gas intersecting two isothermals at $T_1$ and $T_2$. The ratio $\left(\frac{V_b}{V_a}\right)$ is equal to

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