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

Explanation

To find the isothermal bulk modulus of an ideal gas at pressure \( P \) and absolute temperature \( T \), we can follow these steps:

### Step 1: Understand the Concept of Bulk Modulus
The isothermal bulk modulus \( B \) is defined as the ratio of the change in pressure \( \Delta P \) to the fractional change in volume \( \frac{\Delta V}{V} \) at constant temperature. Mathematically, it can be expressed as:
\[ B = -\frac{\Delta P}{\frac{\Delta V}{V}} \]

### Step 2: Use the Ideal Gas Law
For an ideal gas, the relationship between pressure \( P \), volume \( V \), and temperature \( T \) is given by the ideal gas equation:
\[ PV = nRT \]
where \( n \) is the number of moles and \( R \) is the gas constant. In an isothermal process, \( n \), \( R \), and \( T \) are constant.

### Step 3: Differentiate the Ideal Gas Equation
Since \( PV \) is constant for an isothermal process, we can differentiate the equation:
\[ d(PV) = 0 \]
Using the product rule, we have:
\[ P dV + V dP = 0 \]

### Step 4: Rearrange the Differentiated Equation
From the equation \( P dV + V dP = 0 \), we can rearrange it to express \( dP \) in terms of \( dV \):
\[ P dV = -V dP \quad \Rightarrow \quad \frac{dP}{dV} = -\frac{P}{V} \]

### Step 5: Relate Changes to Bulk Modulus
Substituting \( \Delta P \) and \( \Delta V \) into the bulk modulus formula, we have:
\[ B = -\frac{\Delta P}{\frac{\Delta V}{V}} = -\frac{\Delta P \cdot V}{\Delta V} \]
Using \( \frac{dP}{dV} = -\frac{P}{V} \), we can express \( \Delta P \) in terms of \( \Delta V \):
\[ B = P \]

### Conclusion
Thus, the isothermal bulk modulus of the gas is:
\[ B = P \]