Chapter 5: Simple Mixtures
The Thermodynamic Description of Mixtures
• Binary Mixtures: mixtures of two components, A and B; χ A + χ B = 1.
• The partial molar volume of a substance is the contribution to the volume that
a substance makes when it is part of a mixture.
o Partial Molar Volume: Vj = (δ V/δ nj)p,T,n’, where n’ signifies that the
amounts of all other substances present are constant.
o The partial molar volume is the slope of the plot of the total volume as
the amount of J is changed, the pressure, temperature and amount of
other components being constant.
o dV = (δ V/δ nA)p,T,nB • dnA + (δ V/δ nB)p,T,nA
o V = VAnA + VBnB
o Molar volumes are always positive but partial molar quantities need
• The chemical potential is the partial molar Gibbs energy and enables us to
express the dependence of the Gibbs energy on the composition of a mixture.
o Chemical Potential: μ j = (δ G/δ nj)p,T,n’
o The chemical potential is the slope of a plot of Gibbs energy against
the amount of the component J, with the pressure and temperature
o Total Gibbs Energy of a Binary Mixture: G = nAμ A + μ Bμ B, where μ A
and μ B are the chemical potentials at the composition of the mixture.
o Chemical potentials depend on composition, pressure and
temperature, so the Gibbs energy of a mixture may change when
these variables change.
o Fundamental Equation of Chemical Thermodynamics: dG = Vdp – SdT
+ μ AdnA + μ BdnB + …
o At constant pressure and temperature, dG = μ AdnA + μ BdnB + …
o Because at constant temperature and pressure, dG = dw(max),
dw(max) = μ AdnA + μ BdnB + …
• The chemical potential also shows how, under a variety of different conditions,
the thermodynamic functions vary with composition.
o dU = -pdV + TdS + μ AdnA + μ BdnB + … from the equation dU =
-pdV –Vdp + SdT + TdS + dG
o μ j = (δ U/δ nj)S,V,n’
o μ j = (δ H/δ nj)S,p,n’
o μ j = (δ A/δ nj)T,V,n’
• The Gibbs-Duhem equation shows how the changes in chemical potential of
the components of a mixture are related.
o Gibbs-Duhem Equation: Σ njdμ j = 0
o The chemical potential of one component of a mixture cannot change
independently of the chemical potentials of the other components.
o dμ B = -(nA/nB) • dμ A
The Thermodynamics of Mixing
• The Gibbs energy of mixing is calculated by forming the difference of the
Gibbs energies before and after mixing: the quantity is negative for perfect
gases at the same pressure.
o Variation of Chemical Potential of a Perfect Gas with Pressure: μ = μ°
+ RTln(p/p°), where μ° is standard chemical potential, chemical potential of the pure gas at 1 bar.
o μ = μ° + RTlnp, where p = (p/p°).
o Gibbs Energy of Mixing of Perfect Gases: Δ mixG = nRT(χ Alnχ A +
χ Blnχ B); Δ mixG < 0 so perfect gases mix spontaneously in all
• The entropy of mixing of perfect gases initially at the same pressure is
positive and the enthalpy of mixing is zero.
o Entropy Mixing of Perfect Gases: Δ mixS = (δ Δ mixG/δ T)p,nA,nB =
-nR(χ Alnχ A + χ Blnχ B); because lnχ < 0, Δ mixS > 0 for all
o Enthalpy of Mixing of Perfect Gases: Δ mixH = 0; there are no
interactions between the molecules forming the gaseous mixture.
The Chemical Potentials of Liquids
• Raoult’s Law provides a relation between the vapor pressure of a substance
and its mole fraction in a mixture; it is the basis of the definition of an ideal
o μ *A is the chemical potential of pure A, where A is a liquid.
o p*A is the vapor pressure of the pure liquid.
o Raoult’s Law: pA = χ Ap*A
o Ideal Solution: mixtures that obey the law throughout the composition
range from pure A to pure B.
o Chemical Potential of Component of an Ideal Solution: μ A = μ *A +
o In the pure solvent, the molecules have a certain disorder and a
corresponding entropy; the vapor pressure then represents the
tendency of the system and its surroundings to reach a higher entropy.
o When a solute is present, the solution has a greater disorder than the
pure solvent because we cannot be sure that a molecule chosen at
random will be a solvent molecule.
o Vapor pressure of the solvent in the solution is lower than that of the
o The law is a good approximation for properties of the solvent if a
solution is dilute.
• Henry’s Law provides a relation between the vapor pressure of a solute and its
mole fraction in a mixture; it is the basis of the definition of an ideal-dilute
o For real solutions at low concentrations, although the vapor pressure of
the solute is proportional to its mole fraction, the constant of
proportionality is not the vapor pressure of the pure substance.
o Henry’s Law: pB = χ B