4.2 Chemical Kinetics
53
k = Ae
−E/RT
(4.18)
A is the frequency factor and E is known as the energy of activation, meaning
that there is an activation barrier to the reaction, A measures the collision rate and
e −E/RT is the probability that a collision will be sufficiently energetic to get over
the barrier. The ideas considered in Eq. 4.18 have been extended in the theory of
absolute reaction rates.
4.2.1 Absolute Reaction Rate Theory
The first concept that must be considered is that of the reaction coordinate, we
will denote by ξ . For the reaction Eq. 4.15, ξ(t) is the distance between the ion
(H + ) and the amino acid charge, A − . The spatial scale of this coordinate is much
smaller (i.e. angstroms) than the spatial scale of the motor’s motion. In Fig. 4.3 is
plotted the energy diagram corresponding to Eq. 4.16, at the top of the potential
barrier along this coordinate is the activated complex. We represent this complex
by a shallow potential. The width of the well is δ and its height may be represented
as about
1
2 k B T , the mean thermal energy per degree of freedom. The complex may
be visualized as a harmonic oscillator with the reaction coordinate being a degree
of vibrational freedom. The complex can be assumed to have an effective reduced
mass m with respect to the vibrational and a force constant of κ (Fig. 4.2).
Following with the amino acid hydrolis we write the reaction as
Reactants Activated Complex → Products
A
−
+ H
+
M
‡
→ P roducts
(4.19)
The rate of the forward reaction will be given by
Rate = (Concentration of activated complexes)
× (Rate of crossing the barrier)
(4.20)
= C
‡
× (Rate of crossing)
We use the double dagger notation (‡) to indicate parameters associated with the
activated complex. We assume a partial equilibrium; the reactants are in equilibrium
with the activated complex, namely
C ‡
[A − ][H + ]
= K
‡
= exp
−
G
‡
1
RT
(4.21)
53
k = Ae
−E/RT
(4.18)
A is the frequency factor and E is known as the energy of activation, meaning
that there is an activation barrier to the reaction, A measures the collision rate and
e −E/RT is the probability that a collision will be sufficiently energetic to get over
the barrier. The ideas considered in Eq. 4.18 have been extended in the theory of
absolute reaction rates.
4.2.1 Absolute Reaction Rate Theory
The first concept that must be considered is that of the reaction coordinate, we
will denote by ξ . For the reaction Eq. 4.15, ξ(t) is the distance between the ion
(H + ) and the amino acid charge, A − . The spatial scale of this coordinate is much
smaller (i.e. angstroms) than the spatial scale of the motor’s motion. In Fig. 4.3 is
plotted the energy diagram corresponding to Eq. 4.16, at the top of the potential
barrier along this coordinate is the activated complex. We represent this complex
by a shallow potential. The width of the well is δ and its height may be represented
as about
1
2 k B T , the mean thermal energy per degree of freedom. The complex may
be visualized as a harmonic oscillator with the reaction coordinate being a degree
of vibrational freedom. The complex can be assumed to have an effective reduced
mass m with respect to the vibrational and a force constant of κ (Fig. 4.2).
Following with the amino acid hydrolis we write the reaction as
Reactants Activated Complex → Products
A
−
+ H
+
M
‡
→ P roducts
(4.19)
The rate of the forward reaction will be given by
Rate = (Concentration of activated complexes)
× (Rate of crossing the barrier)
(4.20)
= C
‡
× (Rate of crossing)
We use the double dagger notation (‡) to indicate parameters associated with the
activated complex. We assume a partial equilibrium; the reactants are in equilibrium
with the activated complex, namely
C ‡
[A − ][H + ]
= K
‡
= exp
−
G
‡
1
RT
(4.21)