Question

Adiabatic bulk modulus of a substance is defined as:

• A. $-\dfrac{1}{v}\left(\dfrac{\partial v}{\partial P}\right)_T$
• B. $-v\left(\dfrac{\partial P}{\partial v}\right)_T$
• C. $-\dfrac{1}{v}\left(\dfrac{\partial v}{\partial P}\right)_S$
• D. $-v\left(\dfrac{\partial P}{\partial v}\right)_S$

Hint:

Bulk modulus can be isothermal or adiabatic depending on the process. Always check whether derivative is at $T$ or $S$.

Answer 1

The Correct Option is D

Step 1: Recall definition.
Bulk modulus $K$ is defined as the negative ratio of pressure change to relative volume change: \[ K = -V \left( \frac{\partial P}{\partial V} \right)_{\text{condition}} \]
Step 2: Adiabatic case.
For adiabatic bulk modulus, the condition is at constant entropy $S$: \[ K_s = -v \left(\frac{\partial P}{\partial v}\right)_S \] Hence, the correct expression is option (D). Final Answer: \[ \boxed{-v \left(\dfrac{\partial P}{\partial v}\right)_S} \]