Figure 1. The shielded potential around a solvated ion

r = (e²f/kT)Sc_iz_i²

where the c's are the concentrations in ions.m^-3.

The charge density affects the potential via Poisson's equation, expressed here in the spherically symmetric case:

Ѳf   =   r^-2(d/dr)(r²(df/dr))   =   -r/e

 

The screening factor of the Debye-Hückel theory

which has the solution for an ion of charge ze:

f   =   (ze/4per)(e^-r/l )

i.e. the solution without ions multiplied by the exponential term e^-r/l . l is called the Debye length. Much closer to the ion than the Debye length, the potential is essentially what it would be in pure water. Much further away than l, the potential is screened. The Debye length is also symbolised r_D, and is given by:

l   =   Ö(ekT /e² Sc_iz_i²)

In water it is approximately 0.32 I ^-½ nm.

The stabilisation of each ion by its accompanying cloud of opposite charge is given by:

U   =   (z²e²/8pel)

so that the log activity coefficient for the ions of charge z is given by:

ln g_z   =  (U/kT)   =  (z²e²/8pelkT)

Note that the activity coefficient of individual ion species cannot be measured separately, because all the ions always change energy together. The Brønsted-Bjerrum formula is derived from the difference in stabilisation between the ground-state reactants and the activated complex, and the charge of the activated complex is always the sum of the individual charges. Since

(z_A + z_B)² - z_A² - z_B²   =   2z_A z_B

the change of reaction rate constant depends on the product of the two charges. This is usually specified in terms of half the change:

ln g_±   =  (z_A z_B e²/8pelkT)   =  A z_A z_B I^½

Summary. The ions in an electrolyte have a screening effect on the electric field from individual ions. The screening length is called the Debye length and varies as the inverse square root of the ionic strength. The resulting relative stabilisation of charge concentrations fully explains the Brønsted-Bjerrum formula for reaction kinetics.

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