A given quantity of a ideal gas is at pressure P and absolute temperature T. The isothermal bulk modulus of the gas is

Options

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

Explanation

To find the isothermal bulk modulus of an ideal gas, we can follow these steps:

### Step 1: Understand the Definition of Isothermal Bulk Modulus
The isothermal bulk modulus (K) is defined as the negative ratio of the change in pressure (dp) to the fractional change in volume (dV/V) at constant temperature. Mathematically, it is expressed as:
\[ K = -\frac{dp}{dV} \cdot \frac{V}{V} \]

### Step 2: Use the Ideal Gas Law
The ideal gas law states that:
\[ PV = nRT \]
Where:
- P = pressure
- V = volume
- n = number of moles of gas
- R = universal gas constant
- T = absolute temperature

Since we are considering isothermal conditions, T is constant.

### Step 3: Differentiate the Ideal Gas Law
To find the relationship between pressure and volume, we can differentiate the ideal gas law:
\[ P \cdot V = nRT \]
Differentiating both sides gives:
\[ P \, dV + V \, dP = 0 \]
Rearranging this equation gives:
\[ P \, dV = -V \, dP \]
From this, we can express \( dP \) in terms of \( dV \):
\[ dP = -\frac{P}{V} \, dV \]

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

### Conclusion
The isothermal bulk modulus of the ideal gas is equal to the pressure of the gas:
\[ K = P \]