Under isothermal condition ,the pressure of a gas is given by `P=aV_(-3), where a is a constant and V is the volume of the gas. The bulk modulus at constant temperature is equal to

Options

A. `P`
B. `P/2`
C. `3P`
D. `2P`

Answer

To find the bulk modulus of the gas under isothermal conditions, we start with the given relationship between pressure and volume:

\[ P = a V^{-3} \]

where \( a \) is a constant and \( V \) is the volume of the gas.

### Step-by-step Solution:

1. **Understand the Definition of Bulk Modulus**:
The bulk modulus \( B \) is defined as:
\[ B = -V \frac{dP}{dV} \]
This represents the material's resistance to uniform compression.

2. **Differentiate the Pressure with Respect to Volume**:
We need to find \( \frac{dP}{dV} \). Start by differentiating the given pressure equation:
\[ P = a V^{-3} \]
Using the power rule for differentiation:
\[ \frac{dP}{dV} = a \cdot (-3) V^{-4} = -\frac{3a}{V^4} \]

3. **Substitute \( \frac{dP}{dV} \) into the Bulk Modulus Formula**:
Now, substitute \( \frac{dP}{dV} \) into the bulk modulus formula:
\[ B = -V \left(-\frac{3a}{V^4}\right) \]
Simplifying this gives:
\[ B = \frac{3a}{V^3} \]

4. **Final Expression for Bulk Modulus**:
Thus, the bulk modulus at constant temperature is:
\[ B = 3P \]
Since \( P = a V^{-3} \), we can express \( B \) in terms of pressure:
\[ B = 3 \cdot a V^{-3} \]

### Conclusion:
The bulk modulus at constant temperature is equal to \( 3P \) where \( P = a V^{-3} \).