Question

Two different adiabatic paths for the same gas intersect two isothermal curves as shown in the \(P-V\) diagram. The relation between the ratio \(\frac{V_a}{V_d}\) and the ratio \(\frac{V_b}{V_c}\) is:

• A. \(\frac{V_a}{V_d} = \left(\frac{V_b}{V_c}\right)^{-1}\)
• B. \(\frac{V_a}{V_d} \neq \frac{V_b}{V_c}\)
• C. \(\frac{V_a}{V_d} = \frac{V_b}{V_c}\)
• D. \(\frac{V_a}{V_d} = \left(\frac{V_b}{V_c}\right)^2\)

Answer 1

The Correct Option is C

To solve the problem, we need to understand the characteristics of adiabatic and isothermal processes depicted in the \( P-V \) diagram.

In the diagram, two adiabatic paths intersect with two isothermal curves. The points \(a, b, c,\) and \(d\) represent specific volume and pressure conditions. We are given four volume values: \(V_a, V_d, V_b,\) and \(V_c\).

Concept Explanation:

• Isothermal Process: A process where the temperature remains constant. For an ideal gas, the equation is \(PV = \text{constant}\).
• Adiabatic Process: A process where no heat is exchanged with the surroundings. The equation is \(PV^\gamma = \text{constant}\), where \(\gamma\) is the heat capacity ratio.

Reasoning:

• In the diagram, \(V_a\) and \(V_d\) lie on an isothermal curve, so the process \(a \to d\) is isothermal.
• Similarly, \(V_b\) and \(V_c\) lie on another isothermal curve, so the process \(b \to c\) is isothermal.
• Both processes \(a \to b\) and \(d \to c\) are adiabatic, passing through the same intermediate temperatures.

For adiabatic processes between the same isothermals, the relations are such that:

• \( \frac{V_a}{V_d} = \frac{V_b}{V_c} \)
• This is because the ratios depend on initial and final conditions between two points of intersecting isothermals.

Thus, the correct relation is: \(\frac{V_a}{V_d} = \frac{V_b}{V_c}\).

Answer 2

For an adiabatic process, the equation \(TV^{\gamma-1} = \text{constant}\) holds.

Between points \(a\) and \(d\):
\[ T_a \cdot V_a^{\gamma-1} = T_d \cdot V_d^{\gamma-1}. \]
\[ \frac{V_a}{V_d} = \frac{T_d}{T_a}. \]

Between points \(b\) and \(c\):
\[ T_b \cdot V_b^{\gamma-1} = T_c \cdot V_c^{\gamma-1}. \]
\[ \frac{V_b}{V_c} = \frac{T_c}{T_b}. \]

Given \(T_d = T_c\) and \(T_a = T_b\), we have:
\[ \frac{V_a}{V_d} = \frac{V_b}{V_c}. \]

Final Answer: \(\frac{V_a}{V_d} = \frac{V_b}{V_c}\).