Question

For an ideal gas, the isothermal compressibility $k_T = -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T$ is _ _ _.

• A. $\frac{1}{P}$
• B. $\frac{1}{T}$
• C. $\frac{nR}{P}$
• D. $nRT$

Hint:

For ideal gas, isothermal compressibility always equals $\frac{1}{P}$, independent of temperature and moles

Answer 1

The Correct Option is A

Step 1: Start from the gas law.
\[ V = \frac{nRT}{P} \]

Step 2: Differentiate at constant T.
\[ \left(\frac{\partial V}{\partial P}\right)_T = -\frac{nRT}{P^2} \]

Step 3: Put into the definition.
\[ k_T = -\frac{1}{V}\left(-\frac{nRT}{P^2}\right) = \frac{P}{nRT}\cdot\frac{nRT}{P^2} = \frac{1}{P} \]

Step 4: Answer.
\[ \boxed{\frac{1}{P}} \]