What is condensed phase rule equation? Explain its application with the help of phase diagram to two component lead-silver system.

Condensed (reduced) phase rule: Gibbs' phase rule is F = C − P + 2. In solid–liquid systems such as alloys the vapour phase is negligible, and the system is studied at constant (atmospheric) pressure. Pressure is therefore not a variable, which reduces the degrees of freedom by one:

\( F' = C - P + 1 \)

For a two-component system (C = 2): F' = 3 − P. The variables are temperature and composition, so the system is shown on a temperature–composition diagram.

Lead–silver system (simple eutectic system):

 Temp (°C)                                              Temp (°C)
     |                                                  |  961  A (m.p. of Ag)
     |                       Liquid (melt), F' = 2     /|
     |                                               /  |
     |                    curve AO: solid Ag + liquid  |
 327 B (m.p. of Pb)                          /         |
     | \                                  /            |
     |  \  BO: solid Pb               /                |
     |   \  + liquid           /    Solid Ag + liquid   |
 303 |----O------------------------------------------- | eutectic line (F' = 0 at O)
     |    |  Solid Ag + eutectic                        |
     |Pb +|                                             |
     |eut.|                                             |
     |____|_____________________________________________|
   100% Pb  2.6% Ag  ---- composition (% Ag) ---->    100% Ag
   (B on the Pb axis, A on the Ag axis; curves BO and AO meet at the eutectic O: 303 °C, 97.4% Pb + 2.6% Ag)

• Curve AO (freezing-point curve of Ag): Ag melts at 961 °C (point A). Adding Pb lowers its freezing point along AO. Solid Ag is in equilibrium with the melt: P = 2, so F' = 2 − 2 + 1 = 1 (univariant).
• Curve BO (freezing-point curve of Pb): Pb melts at 327 °C (point B). Adding Ag lowers its freezing point along BO. Solid Pb is in equilibrium with the melt, so F' = 1.
• Eutectic point O: the two curves meet at 303 °C and 2.6% Ag (97.4% Pb). Solid Ag, solid Pb and the melt coexist: P = 3, so F' = 2 − 3 + 1 = 0 (invariant). This is the lowest temperature at which any liquid mixture of Pb and Ag can exist.
• Area above AOB: one liquid phase, P = 1, so F' = 2 (bivariant: temperature and composition can both be varied).
• Areas below the curves: solid Ag + liquid; solid Pb + liquid; and below the eutectic line, solid Pb + eutectic or solid Ag + eutectic.

Application – Pattinson's process (desilverisation of lead): argentiferous lead containing a little Ag (less than 2.6%, e.g. 0.1%) is melted and cooled slowly. Its composition lies on the Pb side of the eutectic, so pure Pb crystallises out along curve BO and is removed with ladles. The remaining melt grows richer in Ag until it reaches the eutectic composition (2.6% Ag). This enriched alloy is then treated (e.g. by cupellation) to recover the silver.