CHM 3410 - Physical Chemistry I
Chapter 5 - Supplementary Material
1. Chemical potential for ideal and non-ideal liquids and gases
Chemical potential plays a central role in thermodynamics. Chemical potential is defined as
J = (G/nJ)p,T,n’
(1.1)
where n' indicates that the moles of all substances except the J th substance are held constant. Note that for a pure
substance the chemical potential is equal to the molar Gibbs free energy. By analogy, the chemical potential for a
mixture is sometimes called the partial molar Gibbs free energy.
For chemical reactions and other processes we are usually only interested in changes in free energy.
Therefore, in working with substances, whether real or ideal, we choose convenient reference states, and then
express the chemical potential in terms of the value found for the reference state.
Expressions for the chemical potential for a number of common situations are given below.
Gases
General expression
J = J + RT ln fJ/p = J + RT ln fJ (division by p implied)
Reference state
J, the chemical potential at standard pressure (usually taken as 1 bar or
1 atm) and assuming ideal gas behavior. This therefore represents
a hypothetical state for the gas.
Fugacity
fJ, the "effective pressure" of the gas.
Fugacity coefficient
J, defined by the expression fJ = J pJ, where pJ is the partial pressure of
the gas. For a pure gas J can be found from the relationship
ln J = 0p (Z-1)/p dp , where Z = pV/nRT = compressibility factor
Ideal behavior
For ideal behavior fJ = pJ (that is, J = 1). For this case
J = J + RT ln (pJ/p) = J + RT ln pJ (division by p implied)
Solvent (For liquid solutions. A is usually used to indicate the solvent. Vapor phase is assumed to behave
ideally.)
General expression
A = A* + RT ln aA , where aA = pA/pA*. In this expression pA is the
partial pressure of A above the solution and pA* is the vapor
pressure of pure A.
Reference state
A*, the chemical potential of the pure liquid. This is a real reference state.
Activity
aA, the "effective mole fraction" of the solvent. aA = pA/pA*
Activity coefficient
A, defined by the expression aA = AXA, where XA is the mole fraction
of A in the solution.
Ideal behavior
For ideal behavior aA = XA (that is, A = 1). For this case
A = A* + RT ln XA
Ideal behavior corresponds to the solvent obeying Raoult's law
pA = XA pA*
Solute, mole fraction (For liquid solutions. B is usually used to indicate the solute.)
General expression
B = B+ + RT ln aB , where aB = pB/KB, KB is the Henry's law constant.
Reference state
B+ = B* + RT ln (KB/pB*) , the chemical potential of pure B under
conditions where only A-B interactions occur. This represents
a hypothetical reference state for B.
Activity
aB, the "effective mole fraction" of the solute. aB = pB/KB.
Activity coefficient
B, defined by the expression aB = B XB, where XB is the mole fraction
of B in the solution.
Ideal behavior
For ideal behavior aB = XB (that is, B = 1). For this case
B= B+ + RT ln XB
Ideal behavior corresponds to the solute obeying Henry's law
pB = KBXB
Solute, molality (For liquid solutions. Note: I have used bB for molality, the symbol used by Atkins,
though many books use mB for the solute molality. Also note that for dilute aqueous solutions molality and molarity
are approximately equal, in which case molarity is often used for solute concentration)
General expression
B = B + RT ln aB
Reference state
B= B+ + RT ln  , where  is the conversion factor between mole
fraction and molality in the dilute solution limit
XB =  bB/bB , where bB is solute molality. bB is the reference molality
of 1 mole/kg and ideal solute behavior. This therefore represents
a hypothetical reference state for B.
Activity
aB, the "effective molality" of the solute, aB = pB/KB’, where KB is the
constant from Henry's law, which in terms of molality is
pB = (bB/bB) KB , KB =  KB
Activity coefficient
B, defined by the expression aB = B bB/bB
Ideal behavior
For ideal behavior aB = bB/bB (that is, B = 1). For this case
B= B + RT ln(bB/bB). Ideal behavior corresponds to the solute
obeying Henry's law, in terms of molality.