Solutions of electrolytes

          Solutions of electrolytes are nonideal at relatively low concentrations.  The activities of ions in solution is relatively large compared to neutral compounds.  Ions interact through a Coulombic potential that varies as 1/r (r is the distance between ions).  Neutral solutes interact through London dispersion forces that vary as 1/r6.  The greater the charge on the ions the larger the deviations from ideality.  For example, per mole of CaCl2 dissolved the deviation from ideal behavior is larger than for NaCl due to the 2+ charge of calcium ion.

          We consider a general salt Cn+An-, which dissociates into n+ cations and n- anions per formula unit

 

Cn+An- (s) ® n+Cz+ (aq) +  n+Az- (aq)

Where n+z+ + n-z- = 0 by electroneutrality.  We write the chemical potential of the salr in terms of the chemical potentials of its constituent ions according to

m2 = n+m+ + n-m -

where the subscript 2 refers to the ionic solute.

As for neutral solutes we have

m2 = m2o + RT ln a2

but for ionic solutes we have that

m+ = m+o + RT ln a+

and

m- = m-o + RT ln a- .

n+ ln a+ + n- ln a- = ln a2

which implies that a2 = a+n+ a-n-

We can use this development of introduce the mean  ionic activity.

a ± n= a+n+ a-n- where n = n+ + n-.

We cannot define the activity coefficients of individual ions, but we can determine the mean activity coefficients by the same means used to determine the activity coefficients of other substances.  The mean activity coefficients are defined based on single-ion activity coefficients

a+ = m+g+ and a- = m-g-

where m+ and m- are the molalities of the individual ions given by

m+ = n+m and m- = n -m.

In analogy with the definition of the mean ionic activity a± we define a mean ionic molality, m± by

m± n = m+n+ m-n-

and a mean ionic activity coefficient, g±  by

g± n = g +n+ g -n-.

Given these definitions we can write

a± n = m± ng± n

Mean ionic activity coefficients can be determined experimentally by the same methods used for the activity coefficients of nonelectrolytes.  For example, we can use the vapor pressure of the solvent to define an osmotic coefficient for an aqueous electrolyte solution.

ln a1 = nmf/55.5 mol/kg

Using the Gibbs-Duhem equation one derive


Which integrates to


Example: the vapor pressure of water is measured at various concentrations of NaCl.  Using the known mole fractions the activity of water is calculated.  Osmotic coefficients are calculated using the equation

ln a1 = nmf/55.5 mol/kg

where n = 1for NaCl.  The mean activity coefficient of NaCl is calculated from these data using a fit of f to the data to obtain a functional form

f = 1 + am1/2 + bm + cm3/2 + ….

 

The data are exemplified in the Table below

m(mol/kg) PH2O(torr)   aw      f       lng±

0.000         23.76         1.000         1.000         0.000

0.200         23.60         0.993         0.924         -0.308

0.400         23.44         0.9868       0.920         -0.369

0.600         23.29         0.980         0.923         -0.398

0.800         23.13         0.974         0.929         -0.414

1.000         22.97         0.967         0.935         -0.423

1.400         22.64         0.953         0.950         -0.426

1.800         22.30         0.939         0.972         -0.416

2.200         21.96         0.924         0.994         -0.397

2.600         21.59         0.909         1.019         -0.370

3.000         21.22         0.893         1.045         -0.340

3.400         20.83         0.876         1.072         -0.305

3.800         20.43         0.860         1.102         -0.267

4.400         19.81         0.834         1.145         -0.205

5.000         19.17         0.807         1.191         -0.138

 

The formulae derived for the colligative properties of nonelectrolytes take on a slightly different form for solutions of electrolytes.  For a strong electrolyte that dissociates into n+ cations and n- anions per formula unit, the mole fraction of solute particles is given by

 


The factor n is the sum of n+ and n-.  The factor n also appears in the formulae for the colligative properties

DTfus = nKfm

DTvap = nKbm

P = ncRT

 

Debye-Hückel Theory

          At low concentrations of ionic solute the mean activity coefficient goes as

Here co is 1 mol/L.  This is the concentration in the standard state.  This standard concentration cancels the units of ionic strength.  The ionic strength is calculated using:

where z+ and z- are the charge of the positive and negative ions, respectively and c+ and c- are their concentrations. 

Debye-Hückel theory is valid only in the limit of low concentrations.  However, as the concentration increases the theory breaks down.  For intermediate concentrations the Extended Debye-Hückel theory gives the activity coefficient as