Phase Transitions and Phase Equilibria Study Guide

Phase Transitions and Changes of State

• Matter can transition between three primary states: solid, liquid, and gas. These transformations are known as phase transitions, phase changes, or changes of state.
• Common examples of phase transitions include:
  • Evaporation: Liquid water left uncovered in a glass eventually transitions into the gaseous state.
  • Melting: An ice cube transitions into liquid water when placed in a warm room.
  • Sublimation: Solid $CO_2$ (dry ice) changes directly from a solid to a gas at room temperature.
• The stability of a phase is predicted by its chemical potential. The most stable form of a substance at a specific temperature and pressure is the one with the minimum chemical potential.
• For a system to achieve equilibrium, the chemical potential of the compound must be identical in every phase present.

Energy Changes Accompanying Phase Changes

• Every phase change requires a change in the energy of the system. In solids, particles (atoms, ions, or molecules) are arranged in fixed positions to minimize energy. As temperature increases, these particles vibrate more energetically.
• Melting (Fusion): When a solid melts, particles gain enough energy to move relative to one another, typically increasing the average distance between them.
  • Heat of Fusion (Enthalpy of Fusion, $\Delta H_{fus}$): The energy required to melt a substance. For ice, this value is $6.01\,kJ/mol$.
• Boiling and Vaporization: As liquid temperature increases, gas-phase particles above the surface increase in concentration, exerting vapor pressure. When vapor pressure equals external (atmospheric) pressure, the liquid boils.
  • Heat of Vaporization (Enthalpy of Vaporization, $\Delta H_{vap}$): The energy required for the liquid-to-gas transition. For water, this is $40.7\,kJ/mol$.
  • Values of $\Delta H_{vap}$ are generally larger than $\Delta H_{fus}$ because boiling requires severing almost all interparticle attractive interactions, while melting leaves many interactions intact.
• Sublimation: The transition of a solid directly into a gaseous state.
  • Heat of Sublimation ($\Delta H_{sub}$): The enthalpy change for this transition. It is the sum of the heat of fusion and heat of vaporization ($\Delta H_{sub} = \Delta H_{fus} + \Delta H_{vap}$). For water, $\Delta H_{sub} \approx 47\,kJ/mol$.
• Latent Heat (Molar Heat): The heat necessary for $1\,mole$ of a substance to change to another phase. Phase transitions cannot occur without the gain or loss of latent heat. During a phase transition, this heat is exchanged without a change in temperature.

Critical Temperature and Pressure

• Gases typically liquefy when pressure is applied, but this ability is limited by temperature. For example:
  • At $100^\circ C$, water vapor liquefies at $706\,torr$.
  • At $110^\circ C$, liquid forms at $1075\,torr$.
  • At $374^\circ C$, the liquid phase forms only at $1.655 \times 10^5\,torr$ ($217.7\,atm$). Above this temperature, no amount of pressure can form a distinct liquid phase.
• Critical Temperature ($T_c$): The highest temperature at which a distinct liquid phase can exist. Above this, kinetic energy overcomes intermolecular attractive forces regardless of compression.
• Critical Pressure ($P_c$): The pressure required to bring about liquefaction at the critical temperature.
• Substance Trends:
  • Nonpolar, low-molecular-weight substances (e.g., $O_2$ with $T_c = 154.4\,K$) have lower critical values.
  • Polar substances or those with high molecular weight have higher critical values.
  • Water and ammonia have exceptionally high critical temperatures and pressures due to strong hydrogen bonding. Ammonia ($NH_3$) has a $T_c = 405.6\,K$, meaning it can be liquefied at room temperature ($\approx 295\,K$) with sufficient pressure.
• Supercritical Fluids: A state occurring when both temperature and pressure exceed their critical values. Liquid and gas phases become indistinguishable. Supercritical fluids, such as supercritical $CO_2$, are used as solvents in chemical, food, and pharmaceutical industries for extraction due to being inexpensive, non-toxic, and leaving no residue.

Phase Diagrams of Pure Substances

• A phase diagram summarizes the temperature and pressure conditions under which equilibria exist between states of matter. It consists of three primary curves:
  1. Vapor-Pressure Curve (Liquid-Gas): Represents equilibrium between liquid and gas. It ends at the critical point (C). The temperature where vapor pressure equals $1\,atm$ is the normal boiling point.
  2. Sublimation Curve (Solid-Gas): Represents the change in vapor pressure of a solid as it sublimes.
  3. Melting Curve (Solid-Liquid): Separates solid and liquid phases. For most substances, this curve slopes right because the solid is denser than the liquid, and higher pressure favors the compact solid phase. The melting point at $1\,atm$ is the normal melting point.
• Triple Point (T): The unique temperature and pressure where all three phases (solid, liquid, gas) coexist in equilibrium. For air-free water, this is $0.0098^\circ C$.
• Phase Stability Regions:
  • Gas: Stable at high temperatures and low pressures.
  • Solid: Stable at low temperatures and high pressures.
  • Liquid: Stable in the intermediate region.

The Phase Rule

• Formulated by J. Willard Gibbs, the phase rule determines the least number of intensive variables (independent variables like temperature, pressure, density, concentration) required to define the state of a system or change without altering the equilibrium.
• The formula is:     $\text{F} = \text{C} - \text{P} + 2$
  • $F$: Number of degrees of freedom.
  • $C$: Number of components. This is the smallest number of constituents needed to express the composition of each phase. For water/vapor/ice mixtures, $C = 1$. For the system $CaCO_3 = CaO + CO_2$, $C = 2$.
  • $P$: Number of phases. A phase is a homogenous, physically distinct portion of a system (e.g., a bottle with liquid water and vapor has $P = 2$).

Application of the Phase Rule to Single-Component Systems

• Gas Phase Only ($P = 1$, $C = 1$):     $F = 1 - 1 + 2 = 2$ (Bivariant). Temperature and pressure must both be known to define the system.
• Two Phases in Equilibrium (e.g., Liquid + Vapor, $P = 2$, $C = 1$):     $F = 1 - 2 + 2 = 1$ (Univariant). Identifying either temperature or pressure automatically defines the other.
• Three Phases in Equilibrium (Triple Point, $P = 3$, $C = 1$):     $F = 1 - 3 + 2 = 0$ (Invariant). The conditions are fixed and unique; changing temperature or pressure causes the loss of a phase.
• Specific Case: For water, the freezing point (liquid saturated with air at $1\,atm$) is $0^\circ C$, whereas the triple point is $0.0098^\circ C$.

Condensed Systems and Two-Component Systems

• Condensed Systems: In many liquid or solid systems, the vapor phase is ignored, and the system is held at a constant pressure (usually $1\,atm$). This reduces the degrees of freedom by one. For two-component condensed systems, the variables are temperature and concentration.
• Binary Liquid Systems (Partial Miscibility):
  • Phenol and Water: Below certain temperatures, they form two conjugate phases: a water-rich phase (A) and a phenol-rich phase (B).
  • Tie Line: A line drawn across a two-phase region (e.g., line $bc$ in the phenol-water diagram). All systems on a tie line have phases of constant composition.
  • Upper Consolute Temperature (Critical Solution Temperature): The maximum temperature at which two phases can coexist. For phenol-water, this is $66.8^\circ C$.
  • Lower Consolute Temperature: The temperature below which components are miscible in all proportions (e.g., triethylamine-water).
  • Nicotine and Water: This system exhibits both upper and lower consolute temperatures.
• Lever Rule for Phase Weight:
  • The relative weights of two phases are determined by the position on the tie line. If point $d$ is the system composition, and $b$ and $c$ are the phase compositions:     $\frac{\text{Weight of Phase A}}{\text{Weight of Phase B}} = \frac{\text{length } dc}{\text{length } bd}$

Two-Component Solid-Liquid Systems: Eutectic Mixtures

• These systems involve components miscible in the liquid state but completely immiscible as solids (forming pure crystals). Examples: salol-thymol, salol-camphor, acetaminophen-propyphenazone.
• Eutectic Point: The lowest temperature at which a liquid phase can exist. For salol-thymol, this is $13^\circ C$ with a composition of $34\%$ thymol.
• At the eutectic point, three phases (Liquid, Solid A, Solid B) coexist, making it invariant ($F = 0$ for a condensed system).
• Contact Melting: The phenomenon where an intimate mixture of crystals exhibits a melting temperature lower than either individual component.

Three-Component (Ternary) Systems

• For ternary systems ($C = 3$), the phase rule for a non-condensed system is $F = 3 - P + 2$. If isothermal and condensed, $F = 3 - P$.
• Triangular Coordinate Graphs:
  • Apexes: Represent $100\%$ of one component (A, B, or C).
  • Lines: Represent two-component mixtures.
  • Area: Represents all possible three-component combinations.
  • Adding a third component (C) to a mixture of A and B moves the system composition along a line toward apex C, maintaining a constant ratio of A to B.
• Ternary Systems with One Partially Miscible Pair (e.g., Benzene, Water, and Alcohol):
  • Binodal Curve: Marks the boundary of the two-phase region within the triangle.
  • Adding alcohol (a mutual solvent) to a water-benzene mixture promotes miscibility similar to the effect of increasing temperature.
• Effect of Temperature on Ternary Systems:
  • Increasing temperature generally decreases the area of the binodal curve (promoting miscibility) until it vanishes.
  • Systems with three pairs of partially miscible liquids may have a central triangular region (D) where three conjugate liquid phases coexist ($F = 0$).

Questions & Discussion

Calculation of Degrees of Freedom ($F = C - P + 2$):

• Scenario A: Liquid water + vapor
  • $C = 1$, $P = 2$
  • $F = 1 - 2 + 2 = 1$ (Univariant). One variable must be fixed.
• Scenario B: Liquid ethyl alcohol + vapor
  • $C = 1$, $P = 2$
  • $F = 1 - 2 + 2 = 1$ (Univariant).
• Scenario C: Liquid water + liquid ethyl alcohol + vapor mixture
  • Note: Components are completely miscible.
  • $C = 2$, $P = 2$
  • $F = 2 - 2 + 2 = 2$ (Bivariant). Two variables must be fixed to define the system.
• Scenario D: Liquid water + liquid benzyl alcohol + vapor mixture
  • Note: Water and benzyl alcohol form two separate liquid phases.
  • $C = 2$, $P = 3$
  • $F = 2 - 3 + 2 = 1$ (Univariant).

Phase Transition Classification:

• Liquid to Gas: Vaporization; Decreasing IMFs; Endothermic.
• Liquid to Solid: Freezing; Increasing IMFs; Exothermic.
• Gas to Solid: Deposition; Increasing IMFs; Exothermic.
• Solid to Liquid: Melting; Decreasing IMFs; Endothermic.
• Solid to Gas: Sublimation; Decreasing IMFs; Endothermic.
• Gas to Liquid: Condensation; Increasing IMFs; Exothermic.

Frequently Asked Questions:

• Does all matter change its state? Yes, all matter can move between states, though some may require extreme temperatures or pressures.
• What is the importance of Gibbs' Phase Rule? It provides a thermodynamic foundation for characterizing the chemical state of a system and predicting phase equilibrium relations based on physical conditions.
• When is a system in equilibrium? A system is in equilibrium if no changes are observed over time, the rate of the forward reaction equals the backward reaction, the chemical potential of all components in all phases is equal, and the Gibbs free energy of the reaction is zero (minimum energy state).