Question:
For an ideal gas, the isothermal compressibility $k_T = -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T$ is _ _ _.
For an ideal gas, the isothermal compressibility $k_T = -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T$ is _ _ _.
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For ideal gas, isothermal compressibility always equals $\frac{1}{P}$, independent of temperature and moles
Updated On: Jun 1, 2026
- $\frac{1}{P}$
- $\frac{1}{T}$
- $\frac{nR}{P}$
- $nRT$
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The Correct Option is A
Solution and Explanation
Step 1: Start with ideal gas equation.
\[ PV = nRT \Rightarrow V = \frac{nRT}{P} \]
Step 2: Differentiate with respect to pressure.
\[ \left(\frac{\partial V}{\partial P}\right)_T = \frac{\partial}{\partial P}\left(\frac{nRT}{P}\right) = -\frac{nRT}{P^2} \]
Step 3: Substitute in compressibility formula.
\[ k_T = -\frac{1}{V}\left(-\frac{nRT}{P^2}\right) \]
Step 4: Simplify using $V = \frac{nRT}{P}$.
\[ k_T = \frac{1}{\frac{nRT}{P}} \cdot \frac{nRT}{P^2} \]
\[ k_T = \frac{P}{nRT} \cdot \frac{nRT}{P^2} = \frac{1}{P} \]
Step 5: Conclusion.
\[ \boxed{\frac{1}{P}} \]
\[ PV = nRT \Rightarrow V = \frac{nRT}{P} \]
Step 2: Differentiate with respect to pressure.
\[ \left(\frac{\partial V}{\partial P}\right)_T = \frac{\partial}{\partial P}\left(\frac{nRT}{P}\right) = -\frac{nRT}{P^2} \]
Step 3: Substitute in compressibility formula.
\[ k_T = -\frac{1}{V}\left(-\frac{nRT}{P^2}\right) \]
Step 4: Simplify using $V = \frac{nRT}{P}$.
\[ k_T = \frac{1}{\frac{nRT}{P}} \cdot \frac{nRT}{P^2} \]
\[ k_T = \frac{P}{nRT} \cdot \frac{nRT}{P^2} = \frac{1}{P} \]
Step 5: Conclusion.
\[ \boxed{\frac{1}{P}} \]
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