Question

The slopes of the isothermal and adiabatic $ p \text{-} v $ graphs of a gas are $ S_I $ and $ S_A $ respectively. If the heat capacity ratio of the gas is $ \gamma = \frac{3}{2} $, then $ \frac{S_I}{S_A} = $ ?

• A. \( \frac{3}{2} \)
• B. \( \frac{2}{3} \)
• C. \( \frac{1}{2} \)
• D. \( \frac{1}{3} \)

Hint:

Slope of adiabatic curve is \( \gamma \) times that of isothermal: \( \frac{S_I}{S_A} = \frac{1}{\gamma} \)

Answer 1

The Correct Option is B

Slopes: - Isothermal: \( S_I = -\frac{P}{V} \) - Adiabatic: \( S_A = -\gamma \cdot \frac{P}{V} \) So: \[ \frac{S_I}{S_A} = \frac{1}{\gamma} = \frac{1}{\frac{3}{2}} = \frac{2}{3} \]