Phase transitions are key in thermodynamics, marking shifts between different states of matter. First-order transitions involve abrupt changes, like ice melting, while continuous transitions show gradual changes, like magnetism fading at high temperatures.

Understanding phase transitions helps explain everyday phenomena and advanced physics concepts. They reveal how materials behave under different conditions, from boiling water to superconductivity, highlighting the interplay between energy, temperature, and material properties.

Phase Transitions

First-order vs continuous phase transitions

• First-order phase transitions
  • Involve discontinuous change in first derivatives of free energy (entropy, volume)
  • Exhibit coexistence of two distinct phases at transition point
  • Examples: solid-liquid (melting), liquid-gas (boiling), solid-solid transitions (allotropic transformations)
• Continuous (second-order) phase transitions
  • Involve continuous change in first derivatives of free energy
  • Exhibit divergence in second derivatives of free energy (heat capacity, compressibility)
  • Examples: ferromagnetic-paramagnetic transition (Curie point), superconducting transition (critical temperature), superfluid transition (lambda point)

Discontinuities in thermodynamic quantities

• Entropy
  • Discontinuous change in entropy at transition point
  • Latent heat $L$ related to entropy change $\Delta S$ by $L = T \Delta S$, where $T$ is transition temperature
• Volume
  • Discontinuous change in volume at transition point
  • Density difference between two coexisting phases
• Gibbs free energy
  • Remains continuous during transition
  • First derivatives (entropy and volume) exhibit discontinuities

Latent heat in phase transitions

• Latent heat is energy required for substance to undergo phase transition without changing temperature
• During first-order phase transition, system absorbs or releases latent heat while temperature remains constant
• Latent heat related to entropy change by $L = T \Delta S$
• Examples:
  • Latent heat of fusion: energy required for solid to melt into liquid (ice to water at 0℃)
  • Latent heat of vaporization: energy required for liquid to vaporize into gas (water to steam at 100℃)

Continuity in second-order transitions

• First derivatives of free energy (entropy, volume) remain continuous during transition
• Second derivatives of free energy (heat capacity, compressibility) diverge at critical point
• Order parameter (magnetization in ferromagnet) continuously goes to zero as system approaches critical point
• Correlation length (distance over which fluctuations in order parameter are correlated) diverges near critical point
• Critical exponents characterize power-law behavior of various thermodynamic quantities near critical point
  • Example: specific heat capacity $C \propto |T - T_c|^{-\alpha}$, where $T_c$ is critical temperature and $\alpha$ is critical exponent