Question

A gas undergoes a process in which the pressure and volume are related by \( VP^n = \text{constant} \). The bulk modulus of the gas is

• A. nP
• B. \( P^{1/n} \)
• C. P/n
• D. \( P^n \)

Hint:

For a general process \( P^x V^y = C \), the bulk modulus is \( B = \frac{x}{y} P \). Here the equation is \( V^1 P^n = C \), so \( B = \frac{1}{n} P \). \

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Bulk modulus ($B$) is defined as the ratio of infinitesimal pressure increase to the resulting relative decrease of the volume. It measures a substance's resistance to uniform compression.
Step 2: Key Formula or Approach:
$B = -V \frac{dP}{dV}$.
Step 3: Detailed Explanation:
1. Given equation: $VP^n = C$ (Constant).
2. Differentiate both sides with respect to $V$: \[ \frac{d}{dV}(V P^n) = 0 \]
3. Use the product rule: \[ P^n \cdot (1) + V \cdot (n P^{n-1} \frac{dP}{dV}) = 0 \]
4. Rearrange to find $\frac{dP}{dV}$: \[ n V P^{n-1} \frac{dP}{dV} = -P^n \] \[ \frac{dP}{dV} = -\frac{P^n}{n V P^{n-1}} = -\frac{P}{nV} \]
5. Substitute this into the Bulk modulus formula: \[ B = -V \left( -\frac{P}{nV} \right) \] \[ B = \frac{P}{n} \]
Step 4: Final Answer
The bulk modulus of the gas is $P/n$.