Lecture 21: Complex reactions (continued)
Review
o Methods for determining reaction order.
o Overall composition analysis
o Using Excess of components
o Method of initial rates
o Complex processes
o Bifurcation
o Chain Reactions
o Example: Blood clotting
Today
o Reversible reactions
o Concept of microscopic reversibility
o Analysis of complex reactions and approximations
o Initial rates
o Prior Equilibrium approximation
o Steady state approximation
o Effect of temperature
o Arrhenius theory
o Eyring theory: Transition State Complex
o Electron transfer reaction (Marcus theory)
o Effect of salts
Reversible Reactions
So far we have considered kinetics of irreversible reactions.
But most biological reactions are controlled through
reversibility. That is, when concentration of products increases
significantly the rate of its formation is reduced. To describe a
simple first order reversible kinetics :
AB
Let the rate of forward reaction be k1 and reverse reaction be
k-1.. Now the let initial concentration of A and B be A0 and B0
and the equilibrium concentrations be Aeq and Beq. That is,
after sufficient time is allowed to pass (5-10 half lives), the
concentration of A and B will become independent of time.
From mass balance considerations, we can write:
x  A  Aeq  Beq  B
dA
 k1 A  k 1 B  k1 [ Aeq  x]  k 1 [ x  Beq ]
dt
dA
dx


 (k1  k 1 ) x
dt
dt
 x
 ln    (k1  k 1 )t
 x0 

 A  Aeq
ln 
A A
eq
 0

 B  Beq
  ln 

B B
eq

 0

  (k1  k 1 )t


This is still a first order equation, except that concentrations
are measured with respect to equilibrium concentration.
Concept of microscopic reversibility
One of the key notions we must keep in mind concerning the
reversible reactions is that it must be consistent with the
reversibility principle we studied in thermodynamics. The
principle can be adapted in following way. Every reaction, or
set of reactions must be microscopically reversible at every
stage. Consider a following thermodynamic path for a system
involving three components A, B and C.
This scheme is not consistent with the principle of
microscopic reversibility since B cannot be converted back to
A in the first step. Similar argument can be applied to C. So,
to be consistent we must write:
Analysis of complex reactions and approximations
Thus, the main problem for the kinetic experiments is the
number of independent rate constants that must be
determined. Even for simple reactions, as seen above, this
poses significant challenges. As expected, we can study the
reaction for very short initial duration to determine the kinetic
order. The method also yields information about how the rate
of reactant disappearance depends on the concentrations of
the reactants. This is, the so-called method of initial rates,
useful in the initial stages and is inadequate for the later
stages of complex kinetics.
Therefore, to study multi-step reactions two important
assumptions are evoked, namely, (1) prior equilibrium
(2) steady state approximations. For a two- step reaction:
A B  X
step 1
X P
step 2
In prior equilibrium approximation method, we set step 1 to
be a fast reaction and assume that step 2 is the ratedetermining step. In steady state approximation, we suggest
that the concentration of the intermediate X is small (implying
now that the step 1 is slow and step 2 is fast) or constant.
Using these of assumptions, it is possible to simplify the
kinetic equations greatly.
These are such broad sweeping assumptions that leave us with
several possible schemes that are consistent with a given set
of experimental data. Therefore, design and selection of
experimental methods to elucidate kinetics of complex
reactions takes time as well as tenacity.
Prior Equilibrium Approximation
A B  X
( fast )
X P
( slow)
since the first reaction is fast, a “quasi-equilibrium’’ is
established and a finite concentration of X is developed. This
implies the rates of forward and reverse reactions in the first
step become equal and using this equilibrium we can estimate
the concentration of X readily.
k1 A.B  k 1 X  K eq 
k1
X

k 1 A.B
dP
  k 2 X  k 2 K eq . A.B
dt
v  v 2  v1  v 2
v
Steady State Approximation
A B  X
( slow)
X P
( Fast )
In the case of steady state approximation, we have low
concentration of X and setting it’s rate of formation to be
zero:
dX
 k1 A.B  k 2 X  0
dt
dP

 k 2 X  k1 A.B
dt
Note the difference between the prior equilibrium case and
the steady state case.
Temperature dependence of rate constant
Arrhenius model
Transition state theory
M  N  ( fast ) MN *  ( slow ) P
k T
dP
 k 2 K eq M .N k 2  B
dt
h
Gcomplex   RT ln{ K eq )
Marcus theory of electron transfer reactions
One of the most common chemical reactions is the transfer of
electron (i.e. oxidation-reduction). Marcus envisioned the
structure of free energy of the reaction state complex, in terms
of stabilization of the complex brought about by the solvent
and electrostatic interactions. Consider a simplest reaction,
where the free energy of the product and the reactant is
identical.
Fe 3  *Fe 2  Fe 2  *Fe 3
where the star specifies a radioactive Fe species. In this case,
the suggested form of the free energy of the electron Fe/Fe
complex is:
  G 0 
1
1  1 1 
2 1



G  1 
  (e) 

  2  

4
 

 2a1 2a2 R  n
where G0 is standard free energy of the reaction, and  is the
reorganization energy of the complex including solvent and
reacting molecular contributions. n and  are solvent
refractive index and dielectric constant respectively. a’s are
the radii of the ions and R is the center to center distance
between the reactants. e is the charge transferred.
Thus, unlike the empirical Arrhenius equation, Marcus theory
gives more insight about how electron transfer and solvent
reorganization effects influence the rate of electron transfer
processes. This approach is especially valuable in the studies
of photosynthesis.
Ionic Reactions and Salt effects.
Eyring’s thermodynamic model for the rate constant relates
the rate constant to the equilibrium constant for reactant and
transition state complex. We can develop, how a given
reaction will be influenced by ionic species that are
commonly present in biological fluids. Here, we will borrow
from our development of activities and activity coefficients
from the last quarter.
M  N  ( fast ) MN *  ( slow ) P
dP
 k 2 K eq M .N  k 2 MN *
dt
cMN *  MN *
MN *
K eq 

M .N c N  N c M  M
v
v
k2 
k BT
h
k BT  N  M
cM c N
h  MN *
log( k )  log( k 0 )  2.0.51Z M Z N I
Thus as ionic strength is increased, the k value increases
exponentially!