Liquefaction of Gases
Andrews' Isotherms and the Critical Constants.
A gas can be converted into a liquid if its molecules are brought close enough together that their intermolecular attractive forces overcome their thermal kinetic energy. This is achieved by increasing pressure (to bring molecules closer) and decreasing temperature (to slow them down).
1. Andrews' Isotherms for Carbon Dioxide
In 1869, Thomas Andrews measured the Volume of $CO_2$ at various Pressures, keeping the temperature constant (Isotherms). His famous graph completely changed our understanding of gases.
The Plateau: On the lowest curve ($T_1$), the horizontal line represents the volume shrinking rapidly without any increase in pressure. This is where condensation (liquefaction) is actively occurring. Both liquid and vapor coexist here.
2. The Critical Constants
Based on Andrews' observations, three vital constants are defined for every real gas:
1. Critical Temperature ($T_c$)
The temperature above which a gas cannot be liquefied, regardless of how much pressure is applied. Above $T_c$, the kinetic energy of the molecules is so high that intermolecular forces can never lock them into a liquid state.
2. Critical Pressure ($P_c$)
The minimum pressure required to liquefy the gas exactly at its critical temperature ($T_c$).
3. Critical Volume ($V_c$)
The volume occupied by exactly $1 \text{ mole}$ of the gas at its critical temperature and critical pressure.
3. Formulas (Derived from Van der Waals)
The critical constants can be mathematically derived from the Van der Waals constants '$a$' and '$b$'. These formulas are highly tested in competitive exams.
| Constant | Formula | Key Takeaway |
|---|---|---|
| $T_c$ | $\frac{8a}{27Rb}$ | Directly proportional to '$a$'. Gases with strong attractive forces have a high $T_c$. |
| $P_c$ | $\frac{a}{27b^2}$ | Depends on both '$a$' and the square of '$b$'. |
| $V_c$ | $3b$ | Critical volume is exactly three times the excluded volume per mole. |
$$Z_c = \frac{P_c V_c}{R T_c} = \frac{3}{8} = 0.375$$
For any gas following the Van der Waals equation, $Z$ at the critical point is exactly $0.375$, which proves it is not ideal ($Z=1$).
4. Ease of Liquefaction
How do we know which gas is easier to liquefy? Look at its Critical Temperature ($T_c$) and its Van der Waals constant '$a$'.
- Gases with high $T_c$ (like $H_2O, NH_3, SO_2$) are easy to liquefy because their $T_c$ is well above room temperature. You just need to apply pressure. They have a high '$a$' value (strong intermolecular forces).
- Gases with very low $T_c$ (like $H_2, He, N_2$) are extremely hard to liquefy. They must be aggressively cooled down to cryogenic temperatures before pressure will work. They have a low '$a$' value. Historically, these were called "permanent gases."
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