© A.W.Marczewski 2002

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Activity coefficients of ions in aqueous solutions:

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Activity coefficients of ions in solutions may be calculated by Debye-Hückel formula (here average coeffcient for Mez+ : Xz- electrolyte, for individual ions use single z value):
Debye-Hückel formula - average activity coefficient
γ is activity coefficient given as the ratio of activity and concentration (or analogously for molality, m):
activity coefficient
and average coefficient for Mev+Xv- electrolyte may be calculated from individual ionic coefficients by:
average activity coefficient
A, B, C are constants (see below), a is ionic size (of the hydrated ion!) and I is ionic strength (identical with concentration - or molality - for 1:1 electrolytes):
Ionic strength for molar concentrations [mol/dm3] or Ionic strength for molality [mol/kg]

C is determined experimentally, however A and B may be calculated from Debye-Hückel theory (below formulas for ionic strength defined as molar concentration, for B the formulas depend also on the units of ion size, e.g. for Na+ a = 450pm or 4.5A):
A in Debye-Hückel formula for molar concentration
B in Debye-Hückel formula for molar concentration and pm
B in Debye-Hückel formula for molar concentration and Angstrom

Why do we often use molality m (ratio of solute quantity [mol] to the amount of solvent [kg]) instead of molar concentration (solute quantity to solution volume)? Molality is temperature independent value, whereas molar concentration changes with temperature along solution density changes:
If we express concentration through molality, a dependence on solution temperature appears, too:
This formula may be simplified for dilute solutions:
(For commonly used water solution temperatures and units ( 1 g/cm3 = 1 kg/dm3 ), molar concentration and molality is expressed by the same numbers.)
Then, concentrations are temperature dependent exactly like solution density:

Then it is easy to show, that the above A, B constants defined for molar concentrations may be recalculated for use with ionic strength, I, expressed as molality, m [mol/kg]:
Am=f(Ac) and Bm=f(Bc)
where index "c" denotes values calculated for I [mol/dm3] and d is solution density d [g/cm3] or d [kg/dm3].

Both A and B depend on temperature and dielectric constant of solution (for dilute solutions, the dielectric constant is the same as dielectric constant of water - see below).
A(T), A(T9/2) (better polynomial fit in 0..370°C range) and B(T) graphs with polynomial fits:
A in Debye-Hückel formula for molar concentration - temperature dependence: A(T) A in Debye-Hückel formula for molar concentration - temperature T^(9/2) power dependence - better polynomial fit in 0..370°C range B in Debye-Hückel formula for molar concentration - temperature dependence: B(T)

The dielectric constant of water depends on temperature (see left fig.). Various powers of the product (εT) are also presented as functions of temperature (see also tables of water properties):
dielectric constant of water, ε(T) dielectric constant of water: (εT)^p = f(T), where p = 1 or 3/2

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