Chemistry · Analytical and physical chemistry · Year 2

Alloys: phase diagrams

Why does solder melt at a lower temperature than either tin or lead?

The situation

A radio technician is soldering a board: the solder wire (≈ 62 % tin and 38 % lead) melts all at once at 183 °C, although pure tin melts at 232 °C and lead at 327 °C. The Almalyk Mining and Metallurgical Complex produces copper; alloys of copper with nickel (cupronickel) are used in coins and pipes — and these solidify not at one temperature but over a whole interval. In the laboratory, mixtures of naphthalene and diphenylamine are cooled and the thermometer reading is recorded; when a mixture of phenol and water is heated, the cloudy solution turns clear in an instant. One map explains all of this — the phase diagram. How do we construct it and read it?

Topic

Multicomponent systems: thermal analysis and phase diagrams. A radio technician's solder (Pb–Sn) and coins minted from a copper–nickel alloy; in the laboratory, cooling curves of naphthalene–diphenylamine mixtures and phenol–water…

Key ideas

Cooling curves
Eutectic and the lever rule
Gibbs phase rule

Formulas

f = K - F + 2, f_{p = const} = K - F + 1

The Gibbs phase rule: $f$ is the number of degrees of freedom, $K$ the number of components, $F$ the number of phases. In a condensed system the pressure is constant — a “conditional” rule.

\ln x_{A} = - \frac{Δ H_{fus, A}}{R} (\frac{1}{T} - \frac{1}{T_{A}})

The Schröder–van Laar equation: the solubility of substance A in an ideal solution (the liquidus line), $x_{A}$ — mole fraction.

w_{liq} = \frac{x_{0} - x_{α}}{x_{liq} - x_{α}}, \frac{m_{liq}}{m_{α}} = \frac{x_{0} - x_{α}}{x_{liq} - x_{0}}

The lever rule: the representative point ( $x_{0}$ ) divides the tie line into arms inversely proportional to the masses of the two phases.

\frac{d p}{d T} = \frac{Δ H}{T Δ V}

The Clausius–Clapeyron equation: the slope of the line of a first-order phase transition (for vaporisation $\ln p = - Δ H / R T + C$ ).

\frac{d p}{d T} = \frac{Δ c_{p}}{T V Δ α} = \frac{Δ α}{Δ κ}

The Ehrenfest equations: in a second-order transition there is no latent heat; the heat capacity $c_{p}$ , the expansivity $α$ and the compressibility $κ$ jump.

Theory

Phase, component, degrees of freedom

A phase is a homogeneous part of a system separated from the other parts by a boundary surface (a liquid solution, each kind of crystal). A component is the smallest number of independent substances sufficient to define the composition. The degrees of freedom are the number of parameters (temperature, pressure, composition) that can be changed freely without changing the number of phases. The Gibbs phase rule: $f = K - F + 2$ ; in condensed systems such as alloys the effect of pressure is small — $f = K - F + 1$ . The principle of continuity: as long as the set of phases does not change, the properties change continuously; the principle of correspondence: to each set of phases there corresponds one geometric element of the diagram (a region, a line, a point). In one-component systems the boundaries between phases obey the Clausius–Clapeyron equation: from the dependence of vapour pressure on temperature the heat of vaporisation is found, and from the two lines the heat of sublimation ( $Δ H_{sub} = Δ H_{fus} + Δ H_{vap}$ , at the triple point). The p–T diagrams of water and CO₂ are in the “Phase transitions” experiment.

Thermal analysis (laboratory)

The mixture is melted and cooled slowly, and the temperature is recorded at equal time intervals — the cooling curve. If only the liquid cools, the curve falls smoothly ( $f = 2$ ). When crystals begin to separate, the heat of crystallisation slows the cooling — a break in the curve (the liquidus temperature, $f = 1$ ). When a pure substance solidifies ( $K = 1$ , two phases: liquid + crystal) or a eutectic solidifies ( $K = 2$ , three phases: liquid + two kinds of crystal), $f = 0$ — the temperature does not change, and the curve has a horizontal plateau. The breaks in the curves of different compositions give the liquidus line, and the plateaus give the eutectic line. The Tammann method: if all the samples have the same mass, the duration of the plateau is proportional to the amount of eutectic — its dependence on composition is a triangle, whose apex shows the eutectic composition. In the naphthalene–diphenylamine work the test tube is placed in ice water inside an air “jacket” (a wide test tube) — the cooling is slow and even, and the mixture is stirred with the thermometer. Real curves often show supercooling: the temperature falls below the freezing point and jumps back when crystallisation begins. Two-component solid alloys are studied in the same way — with a thermocouple and a furnace. Safety: work with molten metals and crucibles using special tongs and gloves; lead is poisonous — wash your hands after work and do not eat in the laboratory; naphthalene vapour and diphenylamine — in a fume cupboard.

The eutectic, liquidus, solidus and the lever rule

The liquidus — above it there is only liquid; the solidus — below it there are only solid phases. In the Pb–Sn system at 183 °C a liquid with 61.9 % Sn turns at the same time into crystals of α (a solid solution based on Pb, ≤ 18.3 % Sn) and β (based on Sn, ≥ 97.8 % Sn): $L ⇌ α + β$ — the eutectic (Greek “easily melting”). The point in the diagram that shows the composition and temperature is the representative point; in a two-phase region the horizontal line through it is the tie line, and its ends give the compositions of the phases in equilibrium. The ratio of the masses of the phases is found by the lever rule. Eutectic solder solidifies all at once, with no “pasty” interval — convenient for electronics; alloys that solidify over an interval are used for pipe joints. Compounds: a congruently melting compound turns into a liquid of its own composition — a maximum in the diagram and two eutectics (Mg₂Si is such a compound); an incongruently melting compound decomposes without melting, at the peritectic temperature, into another crystal and a liquid of a different composition: $AB ⇌ A + L$ .

Solid solutions

In a substitutional solid solution atoms of one metal take places in the crystal lattice of the other. For unlimited solubility the atomic radii must differ by less than 15 %, the lattices must be the same and the electronegativities close (the Hume-Rothery rules): Cu (128 pm) and Ni (124 pm) are both face-centred cubic — they mix in any proportion. The diagram is a “lens”: between the liquidus and the solidus there is liquid + α; the cooling curve has two breaks and no plateau. On rapid cooling the composition of the centre and the edge of a crystal differ (segregation, or coring). In an interstitial solid solution small atoms (C, N, H) enter the spaces between the metal atoms: austenite is carbon in the iron lattice (≤ 2.14 %). Limited solubility: in Pb–Sn the solubility of α and β depends on temperature — the solvus lines. Alloys of copper with nickel (cupronickel) resist corrosion.

Polymorphism, first- and second-order transitions

Several crystal forms of one substance are called polymorphism. Sulfur: above 95.5 °C the rhombic form changes into the monoclinic one, and monoclinic sulfur melts at 119 °C; this reversible change is an enantiotropic transition (each form is stable in its own temperature range). If rhombic sulfur is heated quickly, it melts at 112.8 °C before it has time to change (metastable melting). In a monotropic transition one form is metastable at all temperatures and the change goes in one direction only (white phosphorus → red phosphorus). First-order transitions (melting, vaporisation, sublimation, polymorphic transitions) occur with latent heat and a jump in volume — the Clausius–Clapeyron equation. In second-order transitions the enthalpy and volume are continuous, but the heat capacity and the coefficients of expansion and compressibility jump — the Ehrenfest equations (superconductivity, the Curie point of a ferromagnet, the λ-transition of liquid helium). The physics of these transitions is in the “Phase transition and fluctuations” experiment.

Three-component systems and the model

The Gibbs–Roozeboom triangle: the corners of an equilateral triangle are the pure components, the sides are the two-component systems; the composition of an interior point is read with lines parallel to the sides (the sum is 100 %). Partially miscible liquids: phenol and water separate into two layers at room temperature (≈ 8 % phenol in the water, ≈ 28 % water in the phenol); on heating the mutual solubility increases and above ≈ 66 °C (at ≈ 34 % phenol) they mix in any proportion — the upper critical solution temperature. The Alekseev method: a test tube of known composition is heated and the temperature at which the cloudiness clears is recorded. A third substance that dissolves in only one phase (NaCl in water) raises the critical temperature, and one that dissolves in both lowers it. Salt systems: in the NaCl–KCl–H₂O triangle the eutonic point is the solution saturated with both salts (the eutonic). Sylvinite, mined at the Dehqonobod deposits, is a mixture of KCl and NaCl; the solubility of KCl depends strongly on temperature, and that of NaCl hardly at all — in the halurgical method sylvinite is dissolved in a hot solution, and on cooling KCl crystallises. Model: the naphthalene–diphenylamine liquidus comes from the ideal-solution equation (the eutectic follows from the model); the Pb–Sn lines are smooth curves through reference points; the Cu–Ni lens is in the ideal-solution model — narrower than the real one; cooling follows Newton's law, there is no supercooling, and crystallisation is at equilibrium; the heats of transition of sulfur, the phenol–water binodal and the eutonic points are illustrative; time is sped up (the × next to the clock).

Did you know?

In the European Union the RoHS directive restricted lead solders in electronics from 1 July 2006; Sn–Ag–Cu alloys are used instead (for example SAC305: 96.5 % Sn, 3 % Ag, 0.5 % Cu — it melts at about 217–220 °C).
The phase rule was derived by the American scientist Josiah Willard Gibbs in 1875–1878 in his work “On the Equilibrium of Heterogeneous Substances”; it was widely applied to alloy and salt systems by the Dutch chemist Bakhuis Roozeboom.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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