Lecture 23

Phase Equilibrium

Liquid-Vapor Equilibrium

Liquids, like gases, have molecules with a range of kinetic energies. At any given temperature, some molecules in a liquid have enough energy to escape the surface and enter the vapor phase. This process is known as vaporization or evaporation.

• Vaporization/Evaporation: The process by which molecules escape from the liquid surface into the vapor phase.

• In a closed system, molecules that escape into the vapor phase exert a pressure called vapor pressure.

• As more molecules enter the vapor phase, some will return to the liquid (a process called condensation), becoming trapped again by intermolecular forces.

• Eventually, the rates of vaporization and condensation become equal, establishing a dynamic equilibrium.

• The vapor pressure measured under these conditions is called the equilibrium vapor pressure (often just "vapor pressure").

• Vapor pressure is constant at constant temperature and increases with increasing temperature.

Molar Heat of Vaporization (ΔHvap): The energy required to vaporize one mole of a liquid. This value is directly related to the strength of intermolecular forces in the liquid.

• Stronger intermolecular forces result in higher ΔHvap and lower vapor pressure at a given temperature.

Boiling Point: The temperature at which the vapor pressure of a liquid equals the external pressure.

• Normal Boiling Point: The boiling point when the external pressure is 1 atm.

• Both the boiling point and ΔHvap are related to the strength of intermolecular forces.

Example: Water has a high boiling point and ΔHvap due to strong hydrogen bonding.

The Clausius-Clapeyron Equation

The Clausius-Clapeyron equation describes the relationship between vapor pressure and temperature for a pure substance:

• The equation is: $\ln P = -\frac{\Delta H_{\text{vap}}}{R} \left( \frac{1}{T} \right) + C$ where $P$ is vapor pressure, $\Delta H_{\text{vap}}$ is the molar heat of vaporization, $R$ is the gas constant, $T$ is temperature in Kelvin, and $C$ is a constant.

• Alternatively, for two temperatures: $\ln \left( \frac{P_2}{P_1} \right) = -\frac{\Delta H_{\text{vap}}}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)$

• This equation allows calculation of vapor pressure at a new temperature if $\Delta H_{\text{vap}}$ and the vapor pressure at another temperature are known.

Example: Calculate the vapor pressure of water at 210°C using the Clausius-Clapeyron equation (requires known values for $\Delta H_{\text{vap}}$ and vapor pressure at a reference temperature).

Solid-Liquid Equilibrium

• Freezing: The phase transformation from liquid to solid.

• Melting: The phase transformation from solid to liquid.

• Melting Point: The temperature at which the solid and liquid phases are in equilibrium for a substance.

• Molar Heat of Fusion (ΔHfus): The energy required to melt (or freeze, with a negative sign) one mole of a substance under standard conditions.

Solid-Vapor Equilibrium

• Sublimation: The phase transformation from solid directly to vapor.

• Deposition: The phase transformation from vapor directly to solid.

• Heat of Sublimation (ΔHsub): The energy required to sublime one mole of a solid.

Relationship: $\Delta H_{\text{sub}} = \Delta H_{\text{fus}} + \Delta H_{\text{vap}}$

Phase Diagrams

Phase Diagram Overview

A phase diagram is a graph that shows the conditions of temperature and pressure under which a substance exists as a solid, liquid, or gas. Typically, it is a plot of pressure (y-axis) vs. temperature (x-axis).

• Regions on the diagram represent the stable phase (solid, liquid, or gas).

• Lines (phase boundaries) indicate equilibrium between two phases.

• The point where all three lines meet is the triple point.

Triple Point

The triple point is the unique set of conditions at which all three phases (solid, liquid, and gas) coexist in equilibrium.

• Each substance has a specific triple point (e.g., for water: 0.01°C and 0.006 atm).

Critical Temperature and Pressure

• There are two ways to condense a gas: decrease temperature or increase pressure.

• Critical Temperature (Tc): The highest temperature at which a substance can exist as a liquid, regardless of pressure.

• Critical Pressure (Pc): The minimum pressure required to liquefy a gas at its critical temperature.

• Note: $T_c \neq T_b$ (critical temperature is not the same as boiling point, except in special cases).

Heating and Cooling Curves

A heating or cooling curve is a plot of temperature as a function of time as a substance is heated or cooled. These curves show the temperature changes and phase transitions (plateaus) as energy is added or removed.

• During phase changes (melting, boiling), temperature remains constant while the substance absorbs or releases heat.

• The slopes of the curve correspond to temperature changes within a single phase.

Supercooling

Supercooling occurs when a liquid is cooled below its freezing point without solidification. Molecules are unable to arrange into the ordered structure of a solid.

• Example: Water can be supercooled below 0°C if undisturbed and free of impurities.

Summary Table: Key Phase Changes and Terms

Process	Phase Change	Term for Energy Change
Melting (Fusion)	Solid → Liquid	Heat of Fusion ($\Delta H_{\text{fus}}$)
Freezing	Liquid → Solid	Heat of Fusion (negative)
Vaporization	Liquid → Gas	Heat of Vaporization ($\Delta H_{\text{vap}}$)
Condensation	Gas → Liquid	Heat of Vaporization (negative)
Sublimation	Solid → Gas	Heat of Sublimation ($\Delta H_{\text{sub}}$)
Deposition	Gas → Solid	Heat of Sublimation (negative)

Additional info: The notes above are expanded with standard definitions, equations, and examples to provide a self-contained study guide for phase equilibria, phase diagrams, and heating/cooling curves, as covered in General Chemistry (Ch. 11 - Liquids & Phase Changes).