aH+ aA’

≈

Ka = , (16.71)

[HA]c0

aHA

where the brackets denote concentrations in molar, and c0 = 1 M.5 When

the acid is the solvent, as in the dissociation reaction of water itself:

H+ + OH’ ,

H2 O

the standard state is mole fraction x = 1 and the dissociation constant

Kw = 10’14 is simply the product of ionic concentrations in molar.

With the two de¬nitions

def

pH = ’ log10 aH+ ≈ ’ log10 [H+ ] (16.72)

def

pKa = ’ log10 Ka , (16.73)

we ¬nd that

”G0 = ’RT ln Ka = 2.3026 RT pKa . (16.74)

It is easily seen that the acid is halfway dissociated (activities of acid and

base are equal) when the pH equals pKa .

16.7.2 Electron transfer reactions

The general electron transfer reaction involves two molecules (or ions): a

donor D and an acceptor A:

D+ + A’

D+A .

In this process the electron donor is the reductant that gets oxidized and the

electron acceptor is the oxidant that gets reduced. Such reactions can be for-

mally built up from two separate half-reactions, both written as a reduction:

D+ + e’ D ,

A + e’ A’ .

The second minus the ¬rst reaction yields the overall electron transfer reac-

tion. Since the free electron in solution is not a measurable intermediate,6

Usually, c0 is not included in the de¬nition of K, endowing K with a dimension (mol dm’3 ),

5

and causing a formal inconsistency when the logarithm of K is needed.

6 Solvated electrons do exist; they can be produced by radiation or electron bombardment. They

have a high energy and a high reductive potential. In donor-acceptor reactions electrons are

transferred through contact, through material electron transfer paths or through vacuum over

very small distances, but not via solvated electrons.

440 Review of thermodynamics

one cannot attach a meaning to the absolute value of the chemical potential

of the electron, and consequently to the equilibrium constant or the ”G0

of half-reactions. However, in practice all reactions involve the di¬erence

between two half-reactions and any measurable thermodynamic quantities

involve di¬erences in the chemical potential of the electron. Therefore such

quantities as μe and ”G0 are still meaningful if a proper reference state is

de¬ned. The same problem arises if one wishes to split the potential di¬er-

ence of an electrochemical cell (between two metallic electrodes) into two

contributions of each electrode. Although one may consider the potential

di¬erence between two electrodes as the di¬erence between the potential of

each electrode with respect to the solution, there is no way to measure the

“potential of the solution.” Any measurement would involve an electrode

again.

The required standard is internationally agreed as the potential of the

standard hydrogen electrode, de¬ned as zero with respect to the solution (at

any temperature). The standard hydrogen electrode is a platinum electrode

in a solution with pH = 0 and in contact with gaseous hydrogen at a pres-

sure of 1 bar. The electrode reduction half-reaction is

2 H+ + 2 e ’ H2

As the electron is in equilibrium with the electrons in a metallic electrode

at a given electrical potential ¦, under conditions of zero current, the ther-

modynamic potential of the electron is given by

μe = ’F ¦, (16.75)

where F is the Faraday, which is the absolute value of the charge of a mole

electrons (96 485 C). 1 Volt corresponds to almost 100 kJ/mol. Here, the

electrical potential is de¬ned with respect to the “potential of the solution”

according to the standard hydrogen electrode convention.

We can now summarize the thermodynamic electrochemical relations for

the general half-reaction

ox + ν e’ red

as follows:

”G = μred ’ μox ’ νμe = 0, (16.76)

implying that

μ0 + RT ln ared ’ μ0 ’ RT ln aox + νF ¦ = 0. (16.77)

ox

red

16.8 Colligative properties 441

With the de¬nition of the standard reduction potential E 0 :

’νF E 0 = ”G0 = μ0 ’ μ0 , (16.78)

ox

red

we arrive at an expression for the equilibrium, i.e., current-free, potential7

of a (platinum) electrode with respect to the “potential of the solution”

(de¬ned through the standard hydrogen electrode)

RT ared

¦ = E0 ’ ln . (16.79)

νF aox

Values of E 0 have been tabulated for a variety of reduction“oxidation cou-

ples, including redox couples of biological importance. When a metal or

other solid is involved, the activity is meant with respect to mole fraction,

which is equal to 1. The convention to tabulate reduction potentials is now

universally accepted, meaning that a couple with a more negative standard

potential has “ depending on concentrations “ the tendency to reduce a cou-

ple with a more positive E 0 . A concentration ratio of 10 corresponds to

almost 60 mV for a single electron transfer reaction.

16.8 Colligative properties

Colligative properties of solutions are properties that relate to the combined

in¬‚uence of all solutes on the thermodynamic potential of the solvent. This

causes the solvent to have an osmotic pressure against the pure solvent and

to show shifts in vapor pressure, melting point and boiling point. For dilute

solutions the thermodynamic potential of the solvent is given by

μ = μ— + RT ln xsolv = μ— + RT ln(1 ’ xj ) (16.80)

j

≈ μ— ’ RT xj ≈ μ— ’ Msolv RT mj , (16.81)

j j

where μ— is the thermodynamic potential of the pure solvent at the same

pressure and temperature, and the prime in the sum means omitting the

solvent itself. Msolv is the molar mass of the solvent. A solution with a

thermodynamic potential of the solvent equal to an ideal dilute solution of

m molal has an osmolality of m.

The consequences of a reduced thermodynamic potential of the solvent

μ— ’ ”μ are the following:

7 The current-free equilibrium potential has long been called the electromotive force (EMF), but

this historical and confusing nomenclature is now obsolete. The inconsistent notation E (usual

for electric ¬eld, not potential) for the standard potential has persisted, however.

442 Review of thermodynamics

(i) The vapor pressure p is reduced with respect to the saturation va-

por pressure p— of the pure solvent (assuming ideal gas behavior)

according to

p

’”μ = RT ln — , (16.82)

p

or

p = p— (1 ’ xj ). (16.83)

j

This is a form of Raoult™s law stating that the vapor pressure of a

volatile component in an ideal mixture is proportional to the mole

fraction of that component.

(ii) The solution has an osmotic pressure Π (to be realized as a real pres-

sure increase after the pure solvent is equilibrated with the solution,