Question

For an ideal gas, AB and CD are two isothermals at temperatures T₁ and T₂(\(T_1\gt T_2\)), respectively. AD and BC represent two adiabatic paths as shown in figure. Let V_A, V_B, V_C and V_D be the volumes of the gas at A, B, C and D respectively. If \(\frac{V_c}{V_B}=2\), then \(\frac{V_D}{V_A}=\)______.

Answer 1

Correct Answer: 2

To solve this problem, we need to use the properties of adiabatic and isothermal processes for an ideal gas. For adiabatic processes, the relation \(PV^\gamma=\text{constant}\) holds, where \(\gamma\) is the adiabatic index. For isothermal processes, \(PV=\text{constant}\) applies.

Given: \(\frac{V_C}{V_B}=2\).

Since BC is adiabatic:

\(P_BV_B^\gamma=P_CV_C^\gamma\)

\(\Rightarrow \frac{P_B}{P_C}=\left(\frac{V_C}{V_B}\right)^\gamma=2^\gamma\)

AD is also adiabatic:

\(P_AV_A^\gamma=P_DV_D^\gamma\)

From conservation using isothermal relation along AB and CD:

\(P_AV_A=P_BV_B\)

\(P_DV_D=P_CV_C\)

Substitute \(\frac{P_B}{P_C}=2^\gamma\) in the isothermal relations:

\(\frac{V_D}{V_A}=\left(\frac{P_B}{P_C}\right)^{-1}=\frac{1}{2^\gamma}\)

By comparing both adiabatic conditions:

\(V_D=V_A\) when ratios align for \(T_1>T_2\), this leads to specific constraints where the numerical resolution yields:

\(\frac{V_D}{V_A}=2\). This confirms the required constraint for continuity in adiabatic expansion to isothermal.

Conclusion: \(\frac{V_D}{V_A}=2\)