Question

Isothermal compressibility of an ideal gas \[ -\frac{1}{V}\left(\frac{dV}{dP}\right)_T \] is equal to

• A. \(\dfrac{nR}{P}\)
• B. \(nR\)
• C. \(R\)
• D. \(\dfrac{1}{P}\)

Hint:

For an ideal gas, always remember: Isothermal compressibility \( \kappa_T = \dfrac{1}{P} \). It depends only on pressure and not on temperature.

Answer 1

The Correct Option is D

Step 1: Write the ideal gas equation.
For an ideal gas, \[ PV = nRT \] At constant temperature \(T\), \[ V = \frac{nRT}{P} \]
Step 2: Differentiate with respect to pressure.
\[ \frac{dV}{dP} = -\frac{nRT}{P^2} \]
Step 3: Substitute in compressibility expression.
Isothermal compressibility is defined as: \[ \kappa_T = -\frac{1}{V}\left(\frac{dV}{dP}\right)_T \] Substituting values: \[ \kappa_T = -\frac{1}{\frac{nRT}{P}} \left(-\frac{nRT}{P^2}\right) \] \[ \kappa_T = \frac{P}{nRT} \cdot \frac{nRT}{P^2} \] \[ \kappa_T = \frac{1}{P} \]
Step 4: Conclusion.
Hence, the isothermal compressibility of an ideal gas is \(\dfrac{1}{P}\).