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 Question:
We need to find the Bulk Modulus \((B)\) for a gas following the polytropic-like process \( V P^n = C \).

Step 2: Key Formula or Approach:
The Bulk Modulus is defined as: \[ B = -V \frac{dP}{dV} \]

Step 3: Detailed Explanation:
Given the relation: \( V P^n = k \)
Taking the natural logarithm on both sides: \[ \ln V + n \ln P = \ln k \] Differentiating with respect to volume \( V \): \[ \frac{1}{V} + n \left( \frac{1}{P} \right) \frac{dP}{dV} = 0 \] Rearranging for \( \frac{dP}{dV} \): \[ \frac{n}{P} \frac{dP}{dV} = -\frac{1}{V} \Rightarrow \frac{dP}{dV} = -\frac{P}{nV} \] Now, substitute this into the expression for Bulk Modulus: \[ B = -V \left( -\frac{P}{nV} \right) = \frac{P}{n} \]

Step 4: Final Answer:
The bulk modulus of the gas is \( \frac{P}{n} \).