MCEN90014 · Materials Engineering
Thermodynamic Basis of Phase Equilibrium
The Thermodynamic Basis of Phase Equilibrium is the Week-4 core of MCEN90014 Materials Engineering at the University of Melbourne, the point where the subject stops describing microstructures and starts predicting which phase is stable and why. Building from the first and second laws to internal energy, entropy from heat capacity and the Gibbs free energy G = H − TS, this chapter shows how ΔG = ΔH − TΔS sets a transformation's driving force, how the critical temperature T_c = ΔH/ΔS is found, and how regular-solution mixing energies shape a phase diagram.
What this chapter covers
- 01State the first law dU = dQ − P dV and identify internal energy, heat and PV work with SI units
- 02Get internal energy and entropy from heat capacity: U = U0 + integral of C dT and S = S0 + integral of (C/T) dT
- 03State the second law dS >= dQ/T and explain why spontaneous change raises the entropy of an isolated system
- 04Define enthalpy H = U + PV and the combined statement dU <= T dS − P dV
- 05Assemble the Gibbs free energy G = H − TS and use dG <= 0 at constant T and P as the equilibrium criterion
- 06Compute a transformation's driving force with ΔG = ΔH − TΔS and read off spontaneity from its sign
- 07Find the critical (transition) temperature T_c = ΔH/ΔS where two phases are in equilibrium
- 08Evaluate regular-solution mixing: ΔH_mix = k X_A X_B and ΔS_mix = −R(X_A ln X_A + X_B ln X_B)
- 09Explain how the sign of the interaction parameter k drives mixing, ordering or a miscibility gap
- 10Read a Gibbs-energy-versus-composition plot and use the common tangent to locate a two-phase field
Gibbs free energy of a transformation and its critical temperature
- +1Match units first: work in joules per mole, so dH = -8.0 kJ/mol = -8000 J/mol and dS = -6.4 J/mol.K.
- +1Apply dG = dH - T dS at T = 1000 K: dG = -8000 - (1000)(-6.4).
- +1Evaluate: dG = -8000 + 6400 = -1600 J/mol = -1.6 kJ/mol.
- +1Read the sign: dG = -1.6 kJ/mol < 0, so at 1000 K the ordered phase has the lower Gibbs energy and the transformation is spontaneous.
- +1Set dG = 0 for equilibrium: T_c = dH/dS = (-8000)/(-6.4) = 1250 K.
- +1Interpret the direction: both dH and dS are negative, so dG rises with T; the transformation is favoured below 1250 K and stops above it, consistent with dG < 0 at 1000 K.
Key terms
- First law of thermodynamics
- Energy conservation for a closed system, dU = dQ - P dV: the change in internal energy U equals the heat added Q minus the PV work done by the system. It conserves energy but does not, by itself, give the direction of a spontaneous process.
- Entropy
- A measure of disorder, S (J/K). The second law states dS >= dQ/T, so an isolated system evolves toward higher entropy. Entropy is built from heat-capacity data via S = S0 + integral of (C/T) dT.
- Enthalpy
- The heat content at constant pressure, H = U + PV (J or J/mol). Its change equals the heat exchanged at constant pressure, which is why reaction and transformation energies are quoted as enthalpies.
- Gibbs free energy
- The potential a material minimises at constant temperature and pressure, G = H - TS. A process is spontaneous when dG <= 0, and equilibrium is the state of lowest G.
- Driving force (dG = dH - T dS)
- The Gibbs-energy change of a transformation, split into an enthalpy term dH and an entropy term T dS. dG < 0 means the change is spontaneous; dG = 0 means the two states are in equilibrium.
- Critical (transition) temperature
- The temperature at which dG = 0 and two phases are in equilibrium, T_c = dH/dS. For a real transition dH and dS share a sign, so T_c is positive.
- Regular-solution mixing
- A model for the free energy of mixing, dG_mix = dH_mix - T dS_mix, with dH_mix = k X_A X_B and dS_mix = -R(X_A ln X_A + X_B ln X_B), where k is the interaction parameter (J/mol), R = 8.314 J/mol.K, and X are mole fractions.
- Common tangent
- The single line tangent to two phases' Gibbs-energy-versus-composition curves. Its two contact points fix the equilibrium compositions of a two-phase field, where each component has the same chemical potential in both phases.
Thermodynamic Basis of Phase Equilibrium FAQ
Why does a material minimise Gibbs free energy rather than just internal energy?
Because real processing happens at constant temperature and pressure, and nature balances two competing tendencies: the drive to lower enthalpy and the drive to raise entropy. The Gibbs free energy G = H - TS packages both into one quantity, so the stable state is simply the one with the lowest G. At low temperature the enthalpy term dominates and ordered phases win; at high temperature the -TS term dominates and disordered, higher-entropy phases win.
What does the sign of the interaction parameter k tell me in regular-solution mixing?
It sets the enthalpy of mixing dH_mix = k X_A X_B. When k is negative the components attract and mix freely with a tendency to order; when k is zero the solution is ideal and mixing is driven entirely by entropy; when k is positive the enthalpy penalty fights the entropy, and on cooling far enough the solution can unmix into A-rich and B-rich phases, opening a miscibility gap. Because the entropy of mixing is always positive, higher temperature always favours mixing.
Can AI help me with the thermodynamic basis of phase equilibrium in MCEN90014?
Yes, within the University of Melbourne academic-integrity rules. An AI tutor like Sia can explain the first and second laws step by step, walk through how G = H - TS and dG = dH - T dS are assembled, and check your reasoning on a critical-temperature or regular-solution calculation. It cannot sit your invigilated exam or guarantee a mark, and MCEN90014's assessed work must be your own, so always confirm what tools are permitted on Canvas.
Studying with AI? Sia — free AI mechanical engineering tutor works through MCEN90014 step by step.
Exam move
Treat this chapter as the engine room behind every phase diagram, and learn the logical chain rather than isolated formulas: the first law conserves energy, the second law adds a direction through entropy, and Gibbs free energy G = H - TS combines them into the quantity a material minimises at constant temperature and pressure. Drill the mechanical steps until they are automatic: put dH and dS in consistent units, assemble dG = dH - T dS, read spontaneity from its sign, and find the critical temperature from T_c = dH/dS. For regular-solution problems, remember the entropy of mixing is always positive and keep the leading minus sign outside the bracket. Because the final exam is worth 50 percent, is a hurdle you must pass, and provides a formula sheet, the highest-value habit is showing method and carrying units and signs through every line; with ten equally weighted questions, budget about one-tenth of the paper per question and confirm the total exam duration and format on Canvas.
