MODULE 24_03
REACTIONS IN SOLUTION: DIFFUSION AND CONDUCTION
In liquids there is a very limited amount of free space; each molecule interacts strongly with
neighboring molecules; the term "mean free path" has almost no meaning, unlike in the gas
phase where the average distance between collisions is large compared to reactant radii. In
solution phase processes, molecules are driven to move down concentration gradients and when
ionic species are involved, potential gradients are also important. We shall consider these
separately, at first.
DIFFUSION
In a dilute, homogeneous, isotropic solution of molecules A and B in an inert solvent, we can
think of a single molecule of A as being at the center of a microscopic volume of solution in
which the species B predominates. At A the concentration of B is zero; therefore a gradient
exists which favors B molecules to approach A through normal diffusive processes (Brownian
motion). Diffusion tends to equal out concentration gradients.
Fick's first law of diffusion tells us that solute molecules at a point of high concentration will
tend to move towards regions of lower concentration:

dni
dt

A dci
dx
(24.1)
where ni is the excess of species i moving through a plane of area A cm2 in a direction x that is
perpendicular to the plane and
dci
is the concentration gradient along the linear dimension, x.
dx
The flux, i, in units of (quantity) cm-2s-1 is given by
i 
 dni 1
A
dt
(24.2)
For a spherically symmetrical situation
i   Di
dci
dr
(24.3)
where dci is the spherically symmetrical concentration gradient about molecule A and Di is the
dr
1
diffusion coefficient (in cm2 s-1) of the species i. This is Fick's law for a radial distribution. The
negative sign assures that Di is a positive quantity. This law applies to all species, neutrals and
ions, but for the latter another factor is important.
IONIC CONDUCTANCE
Another definition of flux is given by the product of the concentration of the species (in
molecules cm-3) and it's velocity (in cm s-1)
i  vi ci
(24.4)
For a spherically symmetric potential gradient dE about an ion at an origin
dr
vi  ui
dE
dr
(24.5)
where ui is the field dependent mobility of the ion (cm2 V-1 s-1). Then
z
dE
i   i ui ci
zi
dr
(24.6)
where zi is the ionic charge and the factor  zi / zi accounts for the direction of the ionic motion
in the field. In liquids the mobility of ions is not dependent on the potential gradient they
experience since the energy gained by an ion from the field between collisions with the
surrounding molecules is rapidly dissipated in the next few collisions.
TOTAL MOTION
An ionic species has motion derived from both concentration gradients and potential gradients.
The total flux is then

dc
z
dE 
itot    Di i  i ui ci

dr 
 dr zi
(24.7)
In a chemical reaction both motions are clearly important since without the diffusion-driven
component, interaction between ions of like charge would never occur.
2
Using the relationships
ui zi e

Di k BT
(24.8)
V (r )  zi eE
(24.9)
and
where e is the electronic charge, kB is the Boltzmann constant, T is absolute temperature, and
V(r) is the potential energy of an ion of charge zie Coulombs in a potential field of strength E
volts, equation (24.6) becomes
 dc
c dV (r ) 

dr 
itot   Di  i  i
 dr kBT
(24.10)
RAPID BIMOLECULAR REACTIONS
The maximum rate of a bimolecular reaction in solution is limited by the rate at which the
reactants can get together, i.e., by diffusion. Once the molecules approach to their reaction
distance, the reaction itself becomes instantaneous.
A+B

(A...B)
where the species in parenthesis is the encounter (reaction) complex.
Suppose we consider a photochemical reaction, initiated by a brief flash of light. Initially only A
is present in solution, and then suddenly the flash produces a low concentration of B species in a
homogeneous distribution. When the concentrations of both A and B are low, we can imagine
that immediately after the flash the concentration of B in a small volume that contains a single A
is zero. Thus a flux of B molecules towards A will be set up. If A is also an ion, we can regard
each A species as being at the center of radial concentration and potential gradients.
Let the total flux of B species moving towards A be B; then from equation (27.9)
 d [ B]r [ B]r dV (r ) 


kBT dr 
 dr
B   ( DA  DB ) 
(24.11)
where [B]r is the local concentration of B at a distance r from A. A steady state condition will be
achieved when the rate of transport of B molecules into the environment of an average A
molecule becomes equal to the rate at which they are removed by the chemical reaction. The
3
total inward flux through a spherical shell of radius r centered on A, of area 4r2 is 4 r 2 B. .
2
For all A molecules this is 4 r B [ A] . Thus the reaction rate is given by
k[A][B]   4 r 2B [ A]
(24.12)
Using equation (24.10) then we obtain
 d [ B]r [ B]r dV (r ) 
k[ B]  4 r 2 ( DA  DB ) 


kBT dr 
 dr
(24.13)
Making the generalization to all values of r by integrating equation (24.12) between the limits of
r = R and r =  where R is the contact distance leads to
[ B]#  [ B]#R 
k[ B] 1
4 ( DA  DB )
(24.14)
where [ B]#r  [ B]r exp  V (r )  is used as a substituent to assist the integration and
 kBT 

 1   exp (V (r ) / k BT )r 2 dr
(24.15)
R
 has the dimensions of length and the properties of a reaction radius. Removing the substitution
we obtain
[ B] eV / kBT  [ B]R eVR / kBT 
k[ B] 1
4 ( DA  DB )
(24.16)
As r  , V ( R)  0 and [ B]  [ B] , the stoichiometric concentration, when


k
[ B]R  [ B] eVR / kBT 1 

 4 ( DA  DB )  
(24.17)
The two species A and B come together to the encounter distance R where they can now react.
The rate constant, kR, for reaction at R can take any value from 0 (no reaction) to  (instant
reaction) depending on the species involved

d
[B]  k[A][B]  k R [A][B]R
dt
(24.18)
or


k
k[ A][ B]  kR [ A][ B]eVR / kBT 1 

 4 ( DA  DB ) 
4
(24.19)
i.e.,
k
4 ( DA  DB ) 
1  [4 ( DA  DB )  / k R e VR / kBT ]
(24.20)
kD
1  [k D / k R exp(V ( R) / k BT )]
(24.21)
or
k
where
kD  4 ( DA  DB ) 
(24.22)
From equation (24.20) we see that k (the overall rate constant) exhibits a maximum value of kD
even if kR  (i.e, the reaction is instantaneous at r = R). Thus kD given by equation (24.21) is
the rate constant of a bimolecular reaction in solution that is governed solely by the diffusion
together of the reactants, that is the rate is infinite once the reactants are within the collision
radius.
In mol-1s-1 units
k D  4 ( DA  DB )
N0
1000
(24.23)
where N0 is Avogadro's number.
DIFFUSION-LIMITED REACTIONS
Equation (24.20) can be used in a predictive way when we know R and the form of the potential
V(r).
(i)
Uncharged reactants
 
1

 exp(V (r ) / k T )r
B
2
dr
(24.24)
R
which, for V (r )  0 reduces to

 1   r 2 dr  R 1
(24.25)
R
i.e.
k Diff 
4 ( DA  DB ) R N 0
1000
5
(24.26)
Thus kDiff can be computed for non-ionic species when Di and R are known (see later).
(ii) Ionic reactants:
The potential energy, V(r), is given by a Coulomb term
V (r ) 
z A zB e2
r
(24.27)
where  is the dielectric constant of the medium. Then

   exp(
1
R
i.e.
z A zB e2 2
)r dr
 rkBT
(24.28)

 z A z B r0
1  exp ( z A z B r0 / R )
(24.29)
r0 
e2
 k BT
(24.30)
where
The radius r0 is the critical radius around an ion where the Coulomb and the thermal energies are
equal. In H2O (at 25oC, r0 = 0.71 nm; in hexane ( = 2) at 25oC, r0 = 28.0 nm. Then
kD 
4 ( DA  DB ) z A zB r0 N 0
(24.31)
1000(1  exp[ z A zB r0 / R]
or
kD
ions
 kD
neutrals
z A z B r0
1  exp[ z A z B r0 / R]
(24.32)
The second term on the right hand side is called the Debye factor, and it takes into account the
Coulomb effect that modifies the neutral-neutral diffusion controlled rate constant.
For zA = 1, zB = -1

r0
1  exp (r0 / R)
since z A zB  1 . For small ions (R << r0 ) in low  media 
kD
ions

4 ( DA  DB ) r0 N 0
1000
6
(24.33)
r0 and
(24.34)
CALCULATION OF Di
An Einstein equation tells us that
Di  kBT / 
(24.35)
where  is the viscous resistance of the medium. Stokes showed that for spherical particles of
radius r in a continuous medium of viscosity 
 1  2 /  v r 

 1  3 /  v r 
  6r 
(24.36)
where v is the coefficient of sliding friction. For large solute molecules in small solvent
molecules,  v   and
  6 r
(24.37)
kBT
6r
(24.38)
i.e.
Di 
For neutral species we showed
k Diff  4 ( DA  DB ) R
and putting R = rA + rB and DAB  DA  DB 
k Diff 
(24.39)
2k BT
then
6 ( R / 2)
8k BT 8RT

3 3000
(24.40)
and the diffusion limited rate constant for neutral reactants depends only on the solvent viscosity
and the temperature. The following table shows the kDiff values calculated for some solvents
using equation (24.39).
7
 20 C / cP
0
Solvent
10-10 kDiff/ M-1s-1
diethyl ether
0.24
4.1
n-hexane
0.31
3.1
CH 3CN
0.36
2.7
CH 3OH
0.55
1.8
C6 H 6
0.65
1.5
H 2O
1.00
0.97
glycerol
1412
0.0007
SOME COMPARISONS
The next two tables show a comparison between calculated and experimental rate constants for
some chemical reactions.
Reaction
Solvent
I  I  I2
CCI 4
4.2
OH  OH  H 2O2
H 2O
2.6
__
300
0.6
0.5
OH  C6 H 6  C6 H 6OH
H 2O
2.6
1.1
300
0.8
0.33
C2 H 5  C2 H 5  C4 H10
C2 H 6
0.7
__
200
0.1
0.02
a
in cm2 s-1
b
in pm
c
105DAa 105DBa
__
in M-1s-1
8
Rb
400
10-10k (calc)c 10-10k (obs)c
1.3
0.82
10-10 k (calc)
Reaction

eaq
 Co( NH 3 )6

eaq
 Cr ( H 2O)6
H3O  SO4
2

eaq
 Ag 
OH   H 3O 
3
2
10-10 k (obs)
9.3
8.2
6.5
6.2
6.5
10
4.1
3.2
11
14
9