A gas undergoes a process in which its pressure `P` and volume `V` are related as `VP^(n) =` constant. The bulk modulus of the gas in the process is:

Options

A. `nP`
B. `P^(1//n)`
C. `P//n`
D. `P^(n)`

Answer

To find the bulk modulus of the gas undergoing the process described by the equation \( VP^n = \text{constant} \), we can follow these steps:

### Step 1: Understand the Definition of Bulk Modulus
The bulk modulus \( B \) is defined as:
\[ B = -V \frac{dP}{dV} \]
This formula indicates how much the pressure \( P \) changes with respect to a change in volume \( V \).

### Step 2: Differentiate the Given Relation
We are given the relation \( VP^n = \text{constant} \). To find \( \frac{dP}{dV} \), we will differentiate this equation with respect to \( V \):
\[ \frac{d}{dV}(VP^n) = 0 \]
Using the product rule, we differentiate:
\[ P^n \frac{dV}{dV} + V \frac{d}{dV}(P^n) = 0 \]
This simplifies to:
\[ P^n + V \cdot n P^{n-1} \frac{dP}{dV} = 0 \]

### Step 3: Solve for \( \frac{dP}{dV} \)
Rearranging the equation gives:
\[ V n P^{n-1} \frac{dP}{dV} = -P^n \]
Now, we can isolate \( \frac{dP}{dV} \):
\[ \frac{dP}{dV} = -\frac{P^n}{V n P^{n-1}} = -\frac{P}{V n} \]

### Step 4: Substitute \( \frac{dP}{dV} \) into the Bulk Modulus Formula
Now we can substitute \( \frac{dP}{dV} \) back into the formula for bulk modulus:
\[ B = -V \left(-\frac{P}{V n}\right) \]
This simplifies to:
\[ B = \frac{P}{n} \]

### Conclusion
Thus, the bulk modulus of the gas in the process is:
\[ B = \frac{P}{n} \]