Metallurgical Thermodynamics Syllabus

2.1 Laws of Thermodynamics

First Law of Thermodynamics

Energy Conservation

The first law states that energy cannot be created or destroyed, only converted from one form to another.

ΔU = q + w

Where:

ΔU = change in internal energy of the system
q = heat transferred to the system
w = work done on the system

Enthalpy

H = U + PV

ΔH = ΔU + PΔV (at constant pressure)

Heat Capacity

Type	Definition	Relation
Constant Volume (C_V)	(∂U/∂T)_V	C_V = (∂q/∂T)_V
Constant Pressure (C_P)	(∂H/∂T)_P	C_P = (∂q/∂T)_P

Applications to Metallurgical Systems

Heat effects in metallurgical processes:

ΔH = ΣH_products - ΣH_reactants

Kirchhoff's Law for temperature dependence:

(∂(ΔH)/∂T)_P = ΔC_P

ΔH(T₂) = ΔH(T₁) + ∫ΔC_PdT

Example: Heat of Formation

For the reaction: Fe + ½O₂ → FeO

ΔH₂₉₈ = -267 kJ/mol

Calculate ΔH at 500K given ΔC_P = 10 J/mol·K

Solution: ΔH₅₀₀ = -267,000 + 10×(500-298) = -265 kJ/mol

Second Law of Thermodynamics

Entropy

The second law introduces the concept of entropy as a measure of disorder and establishes the direction of spontaneous processes.

dS ≥ δq/T (Clausius Inequality)

For reversible processes:

dS = δq_rev/T

Carnot Cycle Efficiency

η = 1 - T_cold/T_hot

Entropy Changes

Process	Entropy Change
Isothermal expansion	ΔS = nRln(V₂/V₁)
Isochoric heating	ΔS = nC_Vln(T₂/T₁)
Isobaric heating	ΔS = nC_Pln(T₂/T₁)
Phase transition	ΔS = ΔH_transition/T_transition

Key Points

Entropy always increases for irreversible processes
The universe tends toward maximum entropy
Entropy is a state function

Thermodynamic Potentials

Gibbs and Helmholtz Free Energy

Potential	Definition	Natural Variables
Gibbs Free Energy (G)	G = H - TS	T, P, n
Helmholtz Free Energy (A)	A = U - TS	T, V, n

Chemical Potential

μ_i = (∂G/∂n_i)_{T,P,n_j≠i}

For a pure substance: μ = G_m (molar Gibbs free energy)

Maxwell's Relations

Relation	Derived From
(∂T/∂V)_S = -(∂P/∂S)_V	dU = TdS - PdV
(∂T/∂P)_S = (∂V/∂S)_P	dH = TdS + VdP
(∂S/∂V)_T = (∂P/∂T)_V	dA = -SdT - PdV
(∂S/∂P)_T = -(∂V/∂T)_P	dG = -SdT + VdP

Example Application

Using (∂S/∂P)_T = -(∂V/∂T)_P to find entropy change with pressure:

For an ideal gas: (∂V/∂T)_P = nR/P

Thus: (∂S/∂P)_T = -nR/P

ΔS = -nRln(P₂/P₁) for isothermal pressure change

2.2 Solutions and Phase Equilibria

Ideal and Regular Solutions

Raoult's Law

P_i = x_iP_i⁰

Where P_i⁰ is the vapor pressure of pure component i

Activity and Activity Coefficient

μ_i = μ_i⁰ + RTln(a_i)

a_i = γ_ix_i

Where γ_i is the activity coefficient (γ_i = 1 for ideal solutions)

Excess Properties

Solution Type	G^E	Behavior
Ideal	0	ΔH_mix = 0, ΔV_mix = 0
Regular	Ωx₁x₂	ΔH_mix ≠ 0, ΔS_mix = ideal
Athermal	-TΔS^E	ΔH_mix = 0, ΔS_mix ≠ ideal

Henry's Law

P_i = K_Hx_i (for dilute solutions)

Where K_H is Henry's constant

Phase Diagrams

Gibbs Phase Rule

F = C - P + 2

Where:

F = degrees of freedom
C = number of components
P = number of phases

Lever Rule

W_α/W_β = (C_β - C₀)/(C₀ - C_α)

Where W_α and W_β are weight fractions of phases α and β

Common Binary Phase Diagrams

Type	Features	Example System
Isomorphous	Complete solid solubility	Cu-Ni
Eutectic	Eutectic point, limited solubility	Pb-Sn
Peritectic	Peritectic reaction: L + α → β	Fe-C (δ-ferrite region)
Monotectic	Liquid immiscibility gap	Cu-Pb

Free Energy vs Composition

(∂G/∂x_B)_α = (∂G/∂x_B)_β (common tangent rule)

G_mix = RT(x_Alnx_A + x_Blnx_B) (ideal solution)

Thermodynamic Diagrams

Equilibrium Constant

ΔG° = -RTlnK

K = exp(-ΔG°/RT)

Ellingham Diagrams

ΔG° = A + BT

Key features:

Slope = -ΔS°
Y-intercept = ΔH°
Phase change points show slope changes

Reaction	ΔG° (J/mol) Equation
4/3Al + O₂ → 2/3Al₂O₃	-1,120,000 + 214T
2C + O₂ → 2CO	-223,000 - 175T
2Fe + O₂ → 2FeO	-529,000 + 125T

Phase Stability Diagrams

Showing stable phases as functions of thermodynamic variables (T, P, composition)

Example: Fe-O System

Stable phases at different oxygen partial pressures and temperatures:

Low pO₂: Metallic Fe
Intermediate pO₂: FeO (wüstite)
High pO₂: Fe₂O₃ (hematite)

2.3 Defects and Interfaces

Point Defects

Vacancies

n_v/N = exp(-ΔG_f/kT)

Where:

n_v = number of vacancies
N = number of lattice sites
ΔG_f = Gibbs free energy of formation

Schottky Defects

[V_M][V_X] = K_S = exp(-ΔG_S/kT)

For NaCl: [V_Na] = [V_Cl] = exp(-ΔG_S/2kT)

Frenkel Defects

[V_M][M_i] = K_F = exp(-ΔG_F/kT)

Where M_i is interstitial metal ion

Defect Type	Example Materials	Formation Energy (eV)
Vacancy	Metals (Cu, Fe)	0.5-2.0
Schottky	NaCl, MgO	2.0-4.0
Frenkel	AgCl, ZrO₂	1.5-3.0

Surfaces and Interfaces

Surface Energy

γ = (∂G/∂A)_{T,P,n_i}

Typical values:

Metals: 1-3 J/m²
Oxides: 0.5-1.5 J/m²
Water: 0.072 J/m² at 20°C

Gibbs Adsorption Equation

dγ = -ΣΓ_idμ_i

For binary systems:

Γ₂ = -(∂γ/∂μ₂)_T

Segregation Phenomena

X_i^surface/X_i^bulk = exp(-ΔG_seg/RT)

Where ΔG_seg is segregation free energy

Interface Type	Energy (J/m²)	Characteristics
Grain boundary	0.5-1.0	Depends on misorientation angle
Phase boundary	0.2-0.8	Depends on lattice mismatch
Solid-liquid	0.1-0.5	Important in wetting phenomena

2.4 Electrochemistry

Single Electrode Potential

Standard electrode potential measured against SHE (Standard Hydrogen Electrode)

Electrode Reaction	E° (V vs SHE)
Li⁺ + e^- → Li	-3.04
Al³⁺ + 3e^- → Al	-1.66
Fe²⁺ + 2e^- → Fe	-0.44
Cu²⁺ + 2e^- → Cu	+0.34
Ag⁺ + e^- → Ag	+0.80

Electrochemical Cells

E_cell = E_cathode - E_anode

ΔG = -nFE_cell

Where:

n = number of electrons transferred
F = Faraday's constant (96,485 C/mol)

Cell Type	Example	E° (V)
Galvanic	Zn\|Zn²⁺\|\|Cu²⁺\|Cu	+1.10
Concentration	Cu\|Cu²⁺(0.1M)\|\|Cu²⁺(1M)\|Cu	≈0.03

Nernst Equation

E = E° - (RT/nF)ln(Q)

At 298K:

E = E° - (0.0591/n)log(Q)

Example: Zn/Zn²⁺ Electrode

For Zn²⁺ + 2e^- → Zn, E° = -0.76V

At [Zn²⁺] = 0.01M:

E = -0.76 - (0.0591/2)log(1/0.01) = -0.76 - 0.0591 = -0.82V

Potential-pH Diagrams

Pourbaix Diagrams

Showing stable phases as functions of potential (E) and pH

Key Lines

Hydrogen evolution reaction (HER):

2H⁺ + 2e^- → H₂; E = -0.0591pH

Oxygen evolution reaction (OER):

O₂ + 4H⁺ + 4e^- → 2H₂O; E = 1.23 - 0.0591pH

Region	Fe Species	Condition
Immunity	Fe	Low potential
Corrosion	Fe²⁺, Fe³⁺	Intermediate potential, acidic pH
Passivation	Fe₂O₃, Fe₃O₄	High potential, neutral/alkaline pH

Study | Crack | Inspire

Building, TestUrSelf | IISc Bengaluru | BOD, Kartavya | Materials Engineering

Metallurgical Thermodynamics Syllabus

Table of Contents

2.1 Laws of Thermodynamics

First Law of Thermodynamics

Energy Conservation

Enthalpy

Heat Capacity

Applications to Metallurgical Systems

Example: Heat of Formation

Second Law of Thermodynamics

Entropy

Carnot Cycle Efficiency

Entropy Changes

Key Points

Thermodynamic Potentials

Gibbs and Helmholtz Free Energy

Chemical Potential

Maxwell's Relations

Example Application

2.2 Solutions and Phase Equilibria

Ideal and Regular Solutions

Raoult's Law

Activity and Activity Coefficient

Excess Properties

Henry's Law

Phase Diagrams

Gibbs Phase Rule

Lever Rule

Common Binary Phase Diagrams

Free Energy vs Composition

Thermodynamic Diagrams

Equilibrium Constant

Ellingham Diagrams

Phase Stability Diagrams

Example: Fe-O System

2.3 Defects and Interfaces

Point Defects

Vacancies

Schottky Defects

Frenkel Defects

Surfaces and Interfaces

Surface Energy

Gibbs Adsorption Equation

Segregation Phenomena

2.4 Electrochemistry

Single Electrode Potential

Electrochemical Cells

Nernst Equation

Example: Zn/Zn2+ Electrode

Potential-pH Diagrams

Pourbaix Diagrams

Key Lines

Example: Zn/Zn²⁺ Electrode