Question

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.

• A. $C_g = -\dfrac{1}{v}\left(\dfrac{\partial v}{\partial p}\right)_T$
• B. $C_g = -\dfrac{1}{v}\left(\dfrac{\partial p}{\partial v}\right)_T$
• C. $C_g = -\dfrac{1}{p}\left(\dfrac{\partial v}{\partial p}\right)_T$
• D. $C_g = -\dfrac{1}{p}\left(\dfrac{\partial p}{\partial v}\right)_T$

Hint:

Always remember: Isothermal compressibility $C_g$ is defined with respect to {volume change per pressure change at constant temperature}, given by $C_g = -\frac{1}{v}\left(\frac{\partial v}{\partial p}\right)_T$.

Answer 1

The Correct Option is A

- The isothermal compressibility of a gas is defined as the fractional decrease in volume per unit increase in pressure at constant temperature.
- 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).