Download 6. Kinetic salt effects and the derivation of the

Derivation of the Brønsted-Bjerrum Equation for the Effect
of Added Salt on Rate Constants in Ionic Reactions
Introduction: When reactions involving ionic species occur in solution,
addition of salt (e.g. NaCl) can speed up the reaction, slow it down, or have no
effect, as compared to when no salt is added. In those cases where there is an
effect, it becomes greater as the concentration of added salt is increased. These
salt effects can be understood qualitatively and quantitatively using an equation
called the Brønsted-Bjerrum equation, named after those physical chemists
instrumental in its discovery. In this Supplement, we present the derivation of the
equation that explains these effects. It follows closely the development given in
lecture.
The Derivation: Our approach will be similar to that used in deriving the
Eyring equation. However, because the reactants may be charged (Z representing
charge), we will have to introduce appropriate modifications at each step. The
overall bimolecular reaction we will be studying is the following, where the nature
of the products P is undefined:
Z
A
A
+ B
k
ZB
P
Once again, we write the reaction in two steps, noting that charges appear on both
reactants and transition state.
Z
AA
+
Z
B B
k1
‡ ZA + ZB
k'
(AB)
k -1
P
(1)
In this equation ZA and ZB correspond to the charges on the reactants A and
B, while ZA + ZB, by the law of conservation of charge, must be the charge on the
transition state. Now, we can write, as done previously when we derived the
Eyring equation, the following relationship.
Z
‡ ZA + ZB
]
Z
d[P]/dt = k[A A ][B B] = k'[AB
(2)
If we assume a quasi-equilibrium, we can express the concentration of the
activated complex as in (3).
K ‡eq
=
[AB‡
Z + Z
A
B
Z
A
[A
Z
][B B ]
]
g
‡
AB
g g
A
(3)
B
This equation corresponds to the expression for the thermodynamic equilibrium
constant, which you should recall from Chemistry 111 includes the activity
coefficient quotient as part of its expression. In this case, gA, gB and gAB‡ are the
activity coefficients for A, B and (AB)‡. Now, we solve for the concentration of
the activated complex as follows:
[AB
‡ ZA + ZB
g g K ‡ [AZA ][BZB ]
] = A B eq
g ‡
(4)
AB
Thus, we can write equation (5) as an extended version of equation (2).
Z
‡
g
A ][BZB ]
g
A
K
[A
Z
ZA
B
eq
B
[
d[P]/dt = k [A ][B ] = k'
g
‡
AB
]
(5)
Therefore, we can write equation (6) for the bimolecular rate constant k.
k =
[
k' K ‡eq
g g
A B
g ‡
AB
]
(6)
Recall from Chemistry 111 that, according to Debye-Huckel theory, the activity
coefficient for an ion in dilute aqueous solution is given by the expression
log gi = -0.509Zi2I1/2 where Zi is the charge on the ion and I is the ionic strength of
the solution in which the ion finds itself immersed. Also recall that I is given by
equation (7); [Ci] is the concentration of the i-th ion.
I = 0.5 [C i]Z 2i
(7)
S
Thus, we can write for each ion an expression for log gi, as below:
2 1/2
2 1/2
log gA = -0.509ZA I ; log gB = -0.509ZB I ;
2 1/2
log gAB‡ = -0.509 (ZA + ZB ) I
The last expression can be rewritten as log gAB‡ = -0.509(ZA2 + 2ZAZB + ZB2)I1/2.
Now, let us take the log of both sides of equation (6). After doing so, we obtain
the following equation:
‡
log k = log k'K
+ log g
A
+ log g
B
- log gAB ‡
Substituting in the expressions given above for the various values of log gi and
simplifying, we obtain as a final result the equation shown in (8).
1/2
‡
log k = log k'K eq
+ 1.018ZAZ BI
= log k 0 + 1.018ZAZ BI1/2 (8)
This is the Brønsted-Bjerrum equation. Note that this equation has the form y =
mx + b, where y = log k, x = I1/2, m = 1.018ZAZB and b = log k0. This implies that
by plotting the rate constant k versus the square root of the ionic strength of the
solution, we can obtain the product of the charges on the two reaction ions from
the slope. From the intercept, we can obtain the value of k that would be observed
if we were able to completely eliminate any ions from solution.
Comments
(1) In calculating the ionic strength, contributions from all ions present in solution
must be included, including those from reactants and products. However, when
studies of the primary kinetic salt effect on reaction rate constants are made in
practice, the concentration of the added salt is usually made high enough so that
contributions from reactants and products to the ionic strength can be neglected.
(2) Equation (8) is quantitatively applicable only in aqueous solutions at 298 K.
In non-aqueous solutions (or in solutions that are mixtures of aqueous and organic
solvents), the expression for log gi is different, which means the final expression of
equation (8) will be somewhat different. The general expression for the log of the
activity coefficient is log gi = -AZi2I1/2, where A is a constant containing the
dielectric constant and density of the solution, the temperature, and a number of
constants of nature (e.g. charge of the electron, the permittivity of vacuum, etc.).
Since the dielectric constant and density will vary with the nature of the solvent(s)
contained in the solution, A will also vary. (Note that A is not the Arrhenius preexponential factor!) The end result: if you should want to apply equation (8) to
reactions in a solution that is not completely aqueous or at a temperature other than
298 K, you must replace 1.018 with number that is appropriate to the composition
and temperature of the solution being studied. (If your curiosity is driving you
farther, see the physical chemistry texts given on the Reserve Book list to find an
expression for A. It will most likely be indexed under activity coefficients for
ionic species or under Debye-Huckel theory.)
(3) Be sure that you understand that ZA and ZB are the charges on the reacting
ions, not the charges on the ions contained in the added salt!
Questions: Will the reaction rate between two positively charged ions increase,
decrease or stay the same as we increase the ionic strength? What about the
reaction rate between an ion with a positive charge with an uncharged species?
What will happen to the rate of reaction between a positively charged ion and a
negatively charged ion as the ionic strength is decreased?