Which of the following relations defines the coefficient of isothermal compressibility ($C_g$) for a gas?
Here, $p$, $T$, and $v$ represent the pressure, temperature and volume of the gas, respectively.
Show Hint
- $C_g = -\dfrac{1}{v}\left(\dfrac{\partial v}{\partial p}\right)_T$
- $C_g = -\dfrac{1}{v}\left(\dfrac{\partial p}{\partial v}\right)_T$
- $C_g = -\dfrac{1}{p}\left(\dfrac{\partial v}{\partial p}\right)_T$
- $C_g = -\dfrac{1}{p}\left(\dfrac{\partial p}{\partial v}\right)_T$
The Correct Option is A
Solution and Explanation
- Mathematically, it is given by:
\[ C_g = -\frac{1}{v}\left(\frac{\partial v}{\partial p}\right)_T \]
- Here, $v$ is the molar volume (or volume), $p$ is the pressure, and the negative sign ensures that $C_g$ is a positive quantity, since an increase in pressure generally decreases the volume of the gas.
- Option (A) exactly matches this definition.
- Option (B) involves $\dfrac{\partial p}{\partial v}$, which is not the standard definition.
- Option (C) incorrectly takes $\dfrac{1}{p}$ instead of $\dfrac{1}{v}$.
- Option (D) again uses $\dfrac{\partial p}{\partial v}$ with $\dfrac{1}{p}$, which is incorrect.
Therefore, the correct definition of isothermal compressibility is given by option (A).
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Identify the option that has the most appropriate sequence such that a coherent paragraph is formed:
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