Question

Adiabatic bulk modulus of a gas at a pressure $ P $ is ($ \gamma $-ratio of specific heat capacity of the gas): Options:

• A. \( \gamma \)
• B. \( \gamma P \)
• C. \( P \)
• D. \( \frac{\gamma}{P} \)

Hint:

For an adiabatic process, the bulk modulus is directly proportional to the pressure and the ratio of specific heats (\( \gamma \)).

Answer 1

The Correct Option is B

We need to find the adiabatic bulk modulus of a gas at pressure \( P \), expressed in terms of \( \gamma \) and \( P \).
Step 1: Recall the definition of bulk modulus.
The bulk modulus \( K \) is: \[ K = -V \frac{\Delta P}{\Delta V} \] For an adiabatic process: \( P V^\gamma = \text{constant} \).
Step 2: Differentiate the adiabatic condition.
\[ P V^\gamma = C \] Differentiate with respect to \( V \): \[ V^\gamma \frac{dP}{dV} + P \cdot \gamma V^{\gamma - 1} = 0 \implies \frac{dP}{dV} = -\frac{P \gamma}{V} \] Step 3: Compute the bulk modulus. \[ K = -V \frac{dP}{dV} = -V \left( -\frac{P \gamma}{V} \right) = \gamma P \] Final Answer: \[ \boxed{\gamma P} \]