436 Review of thermodynamics

The standard concentration c0 is usually 1 M (molar = mol dm’3 ), and

the standard molality m0 is 1 mole per kg solvent. For mole fraction x

the standard reference is the pure substance, x = 1. The γ™s are activity

coe¬cients and the products γc c/c0 , γm m/m0 , γx x are called activities; one

should clearly distinguish these three di¬erent kinds of activities. They are,

of course, related through the densities and molar masses.

Note that μ(c0 ) = μ0 and μ(m0 ) = μ0 , unless the activity coe¬cients

c m

happen to be zero at the standard concentration or molality. The de¬nition

of μ0 is

c

def

μ0 = lim μ(c) ’ RT ln 0 , (16.58)

c

c

c’0

and similarly for molalities and for mole fractions of solutes. For mole frac-

tions of solvents the standard state x = 1 represents the pure solvent, and

μ0 is now de¬ned as μ(x = 1), which is usually indicated by μ— . For x = 1

x

the activity coe¬cient equals 1.

Solutions that have γx = 1 for any composition are called ideal.

The reader is warned about the inaccurate use, or incomplete de¬nitions,

of standard states and activities in the literature. Standard entropies and

free energies of transfer from gas phase to solution require proper de¬nition

of the standard states in both phases. It is very common to see ln c in equa-

tions, meaning ln(c/c0 ). The logarithm of a concentration is mathematically

unde¬ned.

16.6.1 Virial expansion

For dilute gases the deviation from ideal behavior can be expressed in the

virial expansion, i.e., the expansion of p/RT in the molar density ρ = n/V :

p

= ρ + B2 (T )ρ2 + B3 (T )ρ3 + · · · . (16.59)

kB T

This is in fact an equation of state for the dilute gas phase. The second

virial coe¬cient B2 is expressed in m3 /mol. It is temperature dependent

with usually negative values at low temperatures and tending towards a

limiting positive value at high temperature. The second virial coe¬cient

can be calculated on the basis of pair interactions and is therefore an im-

portant experimental quantity against which an interaction function can be

calibrated.3 For dilute solutions a similar expansion can be made of the

3 See Hirschfelder et al. (1954) for many details on determination and computation of virial

coe¬cients.

16.7 Reaction equilibria 437

osmotic pressure (see (16.85)) versus the concentration:

Π

= c + B2 (T )c2 + B3 (T )c3 + · · · . (16.60)

kB T

The activity coe¬cient is related to the virial coe¬cients: using the ex-

pression

‚μ 1 ‚p

= , (16.61)

‚ρ T ρ ‚ρ T

we ¬nd that

3

μ(ρ, T ) = μideal + 2RT B2 (T )ρ + RT B3 (T )ρ2 + · · · . (16.62)

2

This implies that

3

ln γc = 2B2 ρ + B3 ρ2 + · · · . (16.63)

2

Similar expressions apply to the fugacity and the osmotic coe¬cient.

16.7 Reaction equilibria

Consider reaction equilibria like

A + 2B AB2 ,

which is a special case of the general reaction equilibrium

0 νi Ci (16.64)

i

Here, Ci are the components, and νi the stoichiometric coe¬cients, positive

on the right-hand side and negative on the left-hand side of the reaction.

For the example above, νA = ’1, νB = ’2 and νAB2 = +1. In equilibrium

(under constant temperature and pressure) the total change in Gibbs free

energy must be zero, because otherwise the reaction would still proceed in

the direction of decreasing G:

νi μi = 0. (16.65)

i

Now we can write

μi = μ0 + RT ln ai , (16.66)

i

where we can ¬ll in any consistent standard state and activity de¬nition we

desire. Hence

νi μ0 = ’RT νi ln ai . (16.67)

i

i i

438 Review of thermodynamics

The left-hand side is a thermodynamic property of the combined reactants,

usually indicated by ”G0 of the reaction, and the right-hand side can also

be expressed in terms of the equilibrium constant K:

def

νi μ0 = ’RT ln K,

”G0 = (16.68)

i

i

def

K = Πi aνi . (16.69)

i

The equilibrium constant depends obviously on the de¬nitions used for the

activities and standard states. In dilute solutions concentrations or molal-

ities are often used instead of activities; note that such equilibrium “con-

stants” are not constant if activity coe¬cients deviate from 1.

Dimerization 2 A A2 in a gas diminishes the number of “active” par-

ticles and therefore reduces the pressure. In a solution similarly the os-

motic pressure is reduced. This leads to a negative second virial coe¬cient

B2 (T ) = ’Kp RT for dilute gases (Kp = pA2 /p2 being the dimerization

A

constant on the basis of pressures) and B2 (T ) = ’Kc for dilute solutions

(Kc = cA2 /c2 being the dimerization constant on the basis of concentra-

A

tions).

A special case of an equilibrium constant is Henry™s constant, being the

ratio between pressure in the gas phase of a substance A, and mole fraction

xA in dilute solution.4 It is an inverse solubility measure. The reaction is

A(sol, x) A(gas, p)

with standard state x = 1 in solution and p = p0 in the gas phase. Henry™s

constant KH = p/x relates to the standard Gibbs free energy change as

”G0 = μ0 (gas; p0 ) ’ μ0 (sol; x = 1) = ’RT ln KH . (16.70)

Other special cases are equilibrium constants for acid“base reactions in-

volving proton transfer, and for reduction“oxidation reactions involving elec-

tron transfer, both of which will be detailed below.

16.7.1 Proton transfer reactions

The general proton transfer reaction is

H+ + A’ ,

HA

where the proton donor or acid HA may also be a charged ion (like NH+ 4

’ + stands for any form in which the proton may appear in

or HCO3 ) and H

4 The inverse of KH , expressed not as mole fraction but as molality per bar, is often tabulated

(e.g., by NIST). It is also called Henry™s law constant and denoted by kH .

16.7 Reaction equilibria 439

solution (in aqueous solutions most likely as H3 O+ ). A’ is the corresponding

proton acceptor or base (like NH3 or (CO3 )2’ ). The equilibrium constant

in terms of activities based on molar concentrations is the acid dissociation

constant Ka :