GATE Metallurgical Engineering Metallurgical Thermodynamics Syllabus

Define the chemical potential of component $i$ in terms of Gibbs free energy.

$$\mu_i = \left(\frac{\partial G}{\partial n_i}\right)_{T,P,n_{j\neq i}}$$ It is the partial molar Gibbs free energy of component $i$.

Write the Gibbs free energy differential for an open multicomponent system including chemical potential.

$$dG = -S\,dT + V\,dP + \sum_i \mu_i\,dn_i$$

What is the equilibrium condition for a component distributed between two coexisting phases $\alpha$ and $\beta$?

The chemical potential of each component must be equal in all phases: $$\mu_i^{\alpha} = \mu_i^{\beta}$$ (along with uniform $T$ and $P$ throughout).

State the Gibbs–Duhem equation at constant $T$ and $P$.

$$\sum_i n_i\,d\mu_i = 0 \quad\text{or}\quad \sum_i X_i\,d\mu_i = 0$$ It links the changes in chemical potentials of the components, so they are not all independent.

Express the chemical potential of a component in a solution in terms of its activity.

$$\mu_i = \mu_i^{\circ} + RT \ln a_i$$ where $\mu_i^{\circ}$ is the standard-state chemical potential and $a_i$ is the activity of component $i$.

Define activity and activity coefficient for a solution component.

Activity $a_i = \gamma_i X_i$, where $X_i$ is the mole fraction and $\gamma_i$ is the activity coefficient measuring deviation from ideality. For an ideal solution $\gamma_i = 1$, so $a_i = X_i$.

State Raoult's Law and Henry's Law for activity in solutions.

Raoult's Law (solvent, $X_i \to 1$): $a_i = X_i$, so $\gamma_i \to 1$. Henry's Law (solute, $X_i \to 0$): $a_i = \gamma_i^{\circ} X_i$ with constant $\gamma_i^{\circ}$, i.e. activity is proportional to mole fraction.

What defines an ideal solution thermodynamically?

An ideal solution obeys Raoult's Law over all compositions with $\gamma_i = 1$ and $a_i = X_i$. Its enthalpy of mixing is zero ($\Delta H_{mix} = 0$) and volume of mixing is zero ($\Delta V_{mix} = 0$).

Give the entropy and Gibbs free energy of mixing for an ideal binary solution.

$$\Delta S_{mix} = -R(X_A \ln X_A + X_B \ln X_B)$$ $$\Delta G_{mix} = RT(X_A \ln X_A + X_B \ln X_B)$$ with $\Delta H_{mix}=0$.

Define a regular solution and give its enthalpy and Gibbs free energy of mixing.

A regular solution has ideal entropy of mixing but non-zero enthalpy of mixing. $$\Delta H_{mix} = \Omega X_A X_B$$ $$\Delta G_{mix} = \Omega X_A X_B + RT(X_A\ln X_A + X_B\ln X_B)$$ where $\Omega$ is the interaction parameter.

In the regular solution model, what does the sign of the interaction parameter $\Omega$ indicate?

$\Omega > 0$: like-atom bonds favoured, tendency to cluster/phase-separate (positive deviation, possible miscibility gap). $\Omega < 0$: unlike-atom bonds favoured, tendency toward ordering/compound formation (negative deviation). $\Omega = 0$ gives an ideal solution.

Give the activity coefficient expression for a regular solution.

$$RT\ln\gamma_A = \Omega X_B^{2}, \qquad RT\ln\gamma_B = \Omega X_A^{2}$$ The terminal (Henrian) value is $RT\ln\gamma_i^{\circ} = \Omega$ as $X_i \to 0$.

State the Gibbs Phase Rule and define each term.

$$F = C - P + 2$$ where $F$ = degrees of freedom, $C$ = number of components, $P$ = number of phases, and $2$ accounts for temperature and pressure.

How does the Gibbs Phase Rule change for a condensed system at fixed pressure (typical metallurgical phase diagram)?

Pressure is treated as constant, removing one variable: $$F = C - P + 1$$ This is the 'condensed' or reduced phase rule used for binary alloy diagrams.

For a binary system ($C=2$) at constant pressure, how many phases can coexist at an invariant point, and what is $F$ there?

Using $F = C - P + 1 = 3 - P$. At an invariant point $F = 0$, so $P = 3$ — three phases coexist (e.g. eutectic: liquid + two solids), at a fixed temperature and composition.

State the Lever Rule for the fraction of two phases in a binary tie-line.

For an alloy of composition $C_0$ between phases $\alpha$ (at $C_\alpha$) and liquid (at $C_L$): $$f_\alpha = \frac{C_L - C_0}{C_L - C_\alpha}, \qquad f_L = \frac{C_0 - C_\alpha}{C_L - C_\alpha}$$ The fraction of a phase is proportional to the length of the opposite arm of the tie-line.

What is a tie-line, and what does it connect on a binary phase diagram?

A tie-line is a horizontal (constant-temperature) line drawn across a two-phase region connecting the compositions of the two phases in equilibrium at that temperature. Its endpoints give the phase compositions used in the Lever Rule.

Define a eutectic reaction and write its general form.

A eutectic reaction is an invariant reaction in which a liquid transforms on cooling into two solid phases: $$L \rightarrow \alpha + \beta$$ It occurs at a single fixed temperature and composition.

Distinguish eutectoid, peritectic, and peritectoid invariant reactions.

Eutectoid: $\gamma \rightarrow \alpha + \beta$ (one solid to two solids). Peritectic: $L + \alpha \rightarrow \beta$ (liquid + solid to a new solid). Peritectoid: $\alpha + \beta \rightarrow \gamma$ (two solids to a new solid).

What is the relationship between the molar Gibbs free energy of a phase and the chemical potentials read off a free-energy vs. composition diagram?

For a binary solution phase, the tangent to the $G$–$X$ curve at composition $X$ intercepts the pure-component axes ($X_B=0$ and $X_B=1$) at $\mu_A$ and $\mu_B$ respectively. This is the common-tangent / intercept construction.

Explain the common tangent construction for two-phase equilibrium on a free-energy vs. composition diagram.

When two phases ($\alpha$, $\beta$) coexist, a single line is tangent to both $G$–$X$ curves. The tangent points give the equilibrium compositions, ensuring $\mu_A^{\alpha}=\mu_A^{\beta}$ and $\mu_B^{\alpha}=\mu_B^{\beta}$ — equal chemical potentials in both phases.

On a regular-solution free-energy curve with $\Omega>0$, what feature indicates a miscibility gap, and how is the spinodal defined?

A double-well $G$–$X$ curve (two minima) indicates a miscibility gap; the binodal is found by common tangent. The spinodal is where the curvature changes sign: $$\frac{\partial^{2} G}{\partial X^{2}} = 0.$$ Inside the spinodal the solution is unstable to spinodal decomposition.

Relate the standard Gibbs free energy change of a reaction to its equilibrium constant.

$$\Delta G^{\circ} = -RT \ln K$$ where $K$ is the equilibrium constant expressed in terms of activities of products and reactants.

Write the equilibrium constant in terms of activities for the reaction $\ce{aA + bB <=> cC + dD}$.

$$K = \frac{a_C^{c}\, a_D^{d}}{a_A^{a}\, a_B^{b}}$$ where each $a_i$ is the activity of the species at equilibrium.

State the van't Hoff equation relating the equilibrium constant to temperature.

$$\frac{d(\ln K)}{dT} = \frac{\Delta H^{\circ}}{RT^{2}} \quad\text{or}\quad \frac{d(\ln K)}{d(1/T)} = -\frac{\Delta H^{\circ}}{R}$$ A plot of $\ln K$ vs $1/T$ has slope $-\Delta H^{\circ}/R$.

How does the actual Gibbs free energy change $\Delta G$ relate to $\Delta G^{\circ}$ and the reaction quotient $Q$?

$$\Delta G = \Delta G^{\circ} + RT\ln Q$$ where $Q$ has the same form as $K$ but uses instantaneous (non-equilibrium) activities. At equilibrium $Q=K$ and $\Delta G=0$.

What does an Ellingham diagram plot, and what is the significance of a line's position?

An Ellingham diagram plots the standard Gibbs free energy of formation of oxides (per mole of $\ce{O2}$), $\Delta G^{\circ}$, versus temperature $T$. A metal whose oxide line lies lower (more negative) can reduce the oxide of any metal whose line lies above it.

Why do most oxide lines on an Ellingham diagram slope upward (positive slope) with temperature?

Because $\Delta G^{\circ} = \Delta H^{\circ} - T\Delta S^{\circ}$ and the slope equals $-\Delta S^{\circ}$. Oxidation consumes gaseous $\ce{O2}$, decreasing entropy ($\Delta S^{\circ}<0$), so $-\Delta S^{\circ}>0$ gives a positive slope.

What is special about the line for the reaction $\ce{2C + O2 -> 2CO}$ on the Ellingham diagram?

It has a negative slope (entropy increases because 1 mole of gas becomes 2 moles of gas). It therefore crosses other oxide lines, meaning carbon becomes an increasingly powerful reducing agent at high temperatures — the basis of carbothermic reduction.

Define surface (interfacial) energy and relate it to the work of creating new surface.

Surface energy $\gamma$ is the reversible work required to create a unit area of new surface: $$\gamma = \left(\frac{\partial G}{\partial A}\right)_{T,P,n_i}$$ It has units of $\text{J/m}^2$ (equivalently $\text{N/m}$) and drives processes that reduce total interfacial area.