States of Matter: Gases and Liquids - Liquefaction of Gases

Liquefaction of gases is the process by which a gas is converted into a liquid state, typically achieved by lowering the temperature and increasing the pressure.

Critical Temperature ($T_c$): The temperature above which a gas cannot be liquefied, no matter how much pressure is applied. For $CO_2$, the $T_c$ is $31.1^\circ C$ ($304.25 K$).

Critical Pressure ($P_c$): The minimum pressure required to liquefy a gas at its critical temperature ($T_c$).

Critical Volume ($V_c$): The volume occupied by one mole of a gas at its critical temperature ($T_c$) and critical pressure ($P_c$).

Andrews Isotherms: Thomas Andrews studied the $P-V$ relationship of $CO_2$ at various temperatures. He found that below $T_c$, the isotherms show a horizontal portion where gas and liquid coexist in equilibrium.

Continuity of State: There is no sharp boundary between the gaseous and liquid states when a substance is transitioned along a path that avoids the liquid-vapor equilibrium region (passing above $T_c$ and $P_c$).

Van der Waals Constants and Critical Constants: The critical constants $T_c$, $P_c$, and $V_c$ can be expressed in terms of the van der Waals constants '$a$' (intermolecular forces) and '$b$' (molecular size).

Joule-Thomson Effect: When a gas under high pressure is allowed to expand into a region of low pressure through a porous plug or a fine orifice under adiabatic conditions, its temperature decreases (except for $H_2$ and $He$ at room temperature).