The mechanical lever analogy
Fulcrum balances at overall composition zState coordinates
System T = 80 °CThermodynamic state
Evaluating...Theoretical foundation and derivation of the lever rule
Phase equilibrium in the binary Benzene (A) – Toluene (B) system
1. Raoult's law and boundaries
Benzene (A, normal b.p. 80.1 °C) and Toluene (B, normal b.p. 110.6 °C) form an ideal solution. At isothermal T = 80 °C:
- P*A = 1.000 atm (pure Benzene)
- P*B = 0.380 atm (pure Toluene)
Bubble-point line (liquidus): The equilibrium vapour pressure of liquid composition xA: P = xA P*A + (1 − xA) P*B This boundary is straight because partial vapour pressures are additive following Raoult's law.
Dew-point line (vapourus): Condensation pressure of vapour yA: P = 1 / [ yA / P*A + (1 − yA) / P*B ]
2. The horizontal tie-line
Within the two-phase envelope, the mixture splits into coexisting liquid and vapour at identical system pressure and temperature:
- Point L on the liquidus: gives the liquid phase composition xA.
- Point V on the vapourus: gives the vapour phase composition yA.
- State point (z, P): sits directly on this horizontal tie line.
Because Benzene is more volatile than Toluene, the vapour phase is always enriched in Benzene compared to the equilibrium liquid (yA > xA).
3. Mathematical lever rule
Conserving total moles n = nL + nV with molar fractions fL = nL/n and fV = nV/n: z = fL xA + fV yA Substituting fL = 1 − fV yields: fV = (z − xA) / (yA − xA) and for the liquid phase: fL = (yA − z) / (yA − xA)