Q3(b). Partial molal property and its significance

Definition

1. An extensive property X of a mixture depends on T, p and the amounts n₁,n₂,…: dX=((∂ X)/(∂ T))dT+((∂ X)/(∂ p))dp+Σᵢ Xᵢ dnᵢ
2. The coefficient Xᵢ=((∂ X)/(∂ nᵢ))_T,p,n_(j≠ i) is the partial molal property of i. It is an intensive quantity: it depends on composition, not on the amount of mixture.
3. At constant T, p and composition, integration gives the additivity rule: X=n₁ X₁+n₂ X₂+…
4. Differentiating and comparing gives the Gibbs–Duhem equation, Σ nᵢ d Xᵢ=0 at constant T, p: in a binary mixture the two partial molal quantities cannot change independently (n₁d X₁=-n₂d X₂).

Physical significance

1. Xᵢ is the share of X carried by one mole of i in the mixture. In a pure substance it is the molar property Xₘ*; in a mixture it differs because the molecular surroundings differ.
2. Partial molar volume: adding 1 mol H₂O to a large volume of water increases the volume by 18 cm³; adding it to a large volume of ethanol increases it by only 14 cm³. In ethanol the hydrogen-bonded network of water is lost and the molecules pack differently. So 50 cm³ water + 50 cm³ ethanol gives less than 100 cm³.
3. It can even be negative: the limiting partial molar volume of MgSO₄ in water is -1.4 cm³ mol⁻¹; the ions, as they are hydrated, collapse the open structure of water.
4. Partial molar Gibbs energy is the chemical potential: μᵢ= Gᵢ=(∂ G/∂ nᵢ)_(T,p,nⱼ). Matter flows from high to low μ; at equilibrium μᵢ is the same in every phase; for a reaction Σνᵢμᵢ=0. All of solution thermodynamics (Raoult's law, colligative properties, Δ G=Δ G°+RTln Q) is built on it.
5. In an ideal solution Vᵢ=Vₘ* and Hᵢ=Hₘ*, so there is no volume or heat change on mixing; departures of Xᵢ from the pure values measure the non-ideality.

Application. the partial molar enthalpy gives the integral and differential heats of solution; partial molar volumes are used to design mixtures of known final volume (alcohol–water dilution tables).