Phase Diagrams and Relative Stability of Solids, Liquids, and Gases

Phase: A form of matter uniform throughout in chemical composition and physical state.
Examples: - Liquid water in a beaker: Single phase. - Ice and water mixture: Two distinct phases.
Allotropy in Metals: Metals often exist in multiple solid phases. Pure Iron can exist as BCC ( $\alpha$ ) or FCC ( $\gamma$ ), which are treated as distinct states.

General Rule: Solids are stable at low temperature ( $T$ ), liquids at intermediate $T$ , and gases at high $T$ .
Equilibrium: Occurs at constant $T$ and Pressure ( $P$ ) when Gibbs Energy ( $G$ ) is minimised ( $dG = 0$ ).
Chemical Potential ( $\mu$ ): Defined as the molar Gibbs energy: $\mu = \frac{G}{n}$ $d\mu = -S_m dT + V_m dP$
Stability Rule: The stable phase is the one with the lowest chemical potential at given conditions.

Temperature Dependence (at constant $P$ ): $\left( \frac{\partial \mu}{\partial T} \right)_P = -S_m$ - Since molar entropy ( $S_m$ ) is always positive, $\mu$ decreases as $T$ increases. - Slope steepness: Solid (shallow) < Liquid (steeper) < Gas (steepest). - Melting/boiling points occur where chemical potential lines intersect.
Pressure Dependence (at constant $T$ ): $\left( \frac{\partial \mu}{\partial P} \right)_T = V_m$ - Since molar volume ( $V_m$ ) is positive, $\mu$ increases with $P$ . - Gases have the steepest upward slope; solids have the shallowest/most "stubborn" slope.
Application: High-temperature alloys use structures with low molar entropy to keep the $\mu$ slope flat and delay melting.

Definition: A map combining $T$ and $P$ to show physical and chemical equilibria.
Regions: Represent the state where atoms have the lowest chemical potential.
Phase Boundaries: Lines where the chemical potential of two phases is equal ( $\mu_1 = \mu_2$ ).
Triple Point: The specific conditions where solid, liquid, and vapour are equally stable.
Critical Point: The state beyond which the distinction between liquid and vapour disappears, forming a "Supercritical fluid."
Sulfur Phases: Features Monoclinic solid, Rhombic solid ( $95.31^\circ C$ , $5.1 \times 10^{-6}\,atm$ ), liquid, and gas phases.

Formula: $F = C - P + 2$ - $C$ : Number of chemically independent components. - $P$ : Number of phases in equilibrium. - $F$ : Degrees of freedom (independent variables like $P, T$ ).
Pure Metal Example ( $C = 1$ ): - Single phase region ( $P=1$ ): $F = 2$ (Bivariant). - Phase boundary ( $P=2$ ): $F = 1$ (Univariant). - Triple point ( $P=3$ ): $F = 0$ (Invariant).
Reduced Rule: In metallurgy (constant pressure), the formula becomes $F = C - P + 1$ .

Clapeyron Equation: Describes the slope of boundary lines where $d\mu_1 = d\mu_2$ : $\frac{dP}{dT} = \frac{\Delta S_m}{\Delta V_m}$
Clausius-Clapeyron Equation: Applied when one phase is a gas (assuming ideal gas behavior and ignoring liquid volume): $\ln\left( \frac{P_2}{P_1} \right) = -\frac{\Delta H_{vap}}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)$
Application: In aerospace metallurgy, dropping pressure allows "boiling off" harmful trace elements (like Lead or Bismuth) at lower temperatures while the alloy remains liquid.

Gibbs Energy with Surface Term: $dG = -SdT + VdP + \gamma d\sigma$ - $\gamma$ : Surface tension. - $d\sigma$ : Change in surface area.
Laplace Pressure: For a spherical droplet of radius $r$ , internal pressure is higher than external: $\Delta P = \frac{2\gamma}{r}$
Impact: - Smaller particles ( $r \rightarrow 0$ ) have massive internal pressure and higher $\mu$ . - Nanoparticles melt at lower temperatures than bulk metals. - Surface energy "debt" explains why liquids don't freeze immediately at the melting point; creating a crystal surface requires extra energy.