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

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