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 (K) of an ideal gas given its pressure (P) and absolute temperature (T), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Ideal Gas Law**:
The ideal gas law is given by the equation:
\[ PV = nRT \]
where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the universal gas constant, and \(T\) is the absolute temperature.

2. **Identify the Isothermal Process**:
In an isothermal process, the temperature (T) remains constant. Therefore, for an ideal gas undergoing an isothermal process, the product \(PV\) remains constant.

3. **Differentiate the Ideal Gas Equation**:
Since \(PV = \text{constant}\), we can differentiate this equation:
\[ d(PV) = 0 \]
Applying the product rule gives:
\[ PdV + VdP = 0 \]

4. **Rearranging the Equation**:
From the differentiated equation, we can express the relationship between changes in pressure and volume:
\[ PdV = -VdP \]
Dividing both sides by \(VdV\) gives:
\[ \frac{dP}{dV} = -\frac{P}{V} \]

5. **Define the Isothermal Bulk Modulus**:
The isothermal bulk modulus \(K\) is defined as:
\[ K = -\frac{dP}{dV} \cdot \frac{V}{P} \]
Substituting the expression we found for \(\frac{dP}{dV}\):
\[ K = -\left(-\frac{P}{V}\right) \cdot \frac{V}{P} \]
Simplifying this gives:
\[ K = \frac{P}{V} \cdot V = P \]

6. **Conclusion**:
Therefore, the isothermal bulk modulus of the gas is equal to the pressure \(P\).

### Final Answer:
The isothermal bulk modulus of the gas is \(K = P\).

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