Chemical kinetics
Enzymes are proteins that are specialized to catalyze biological reactions. Catalysts are entities that
participate in, but are not components of the reactions. That is, CATALYSTS ARE NOT USED UP
IN THE REACTION.
Enzymes are characterized by their extremely high degree of substrate specificity
(SUBSTRATE=REACTANT IN A REACTION) and their amazingly high catalytic power. Some
enzymes accelerate the rate of their reactions by ~106 over that observed in solution.
Enzymes catalyze (facilitate) the same type of reactions as occur in solution-only they make them
occur faster! This high degree of rate acceleration is obtained by bringing the substrates close
together (PROXIMITY) and in the proper ORIENTATION. This is effected by the fact that
enzymes bind there substrates, there is an intermediate complex formed between E and S, before
catalysis. This discussion suggests that ENZYME CATALYZED REACTIONS OCCUR IN
MULTIPLE STEPS.
YOU HAVE TO REALIZE-ENZYMES DO NOT ALTER THE EQUILIBRIUM OF THE
REACTION!!! THEY CATALYZE ONLY REACTIONS THAT OCCUR SPONTANEOUSLY!!!
Some enzymes utilize tricks, coupling favorable and unfavorable reactions to give the desired
results, but the free energy of an enzyme catalyzed reaction is always negative.
Kinetics- determination and study of the velocity of chemical reactions. Some reactions are very
slow others very fast. In order for a chemical reaction to occur, the reactants must collide with each
other and with enough energy to result in a reaction. Any condition that creates a greater number of
collisions with the minimum activation energy required will result in a faster reaction.
The speed of a chemical reaction is dependent on all of the following:
1. The temperature at which the reaction occurs
An increase in temperature will result in an increase in kinetic energy. Since the kinetic energy
increases, the velocity of the particles will also increase. Since the speed of the particles increases,
they should collide more often (therefore speed of reaction increases). The particles will also have
more energy thereby speeding up the reaction even more.
2. The concentration of reactants
The more reactant particles there are per unit volume (concentration of reactants) the more chances
there are for collisions. This should result in an increase in the speed of the reaction.
3. The presence of a catalyst
A catalyst will speed up a reaction by providing a better geometric arrangement for reactivity
(reaction sites line up better) or by providing an alternate pathway for the reaction (one with a lower
reaction energy). Although a catalyst may take an active part in the reaction, it can be recovered at
the end of the reaction chemically unchanged
4. The nature of the reactants
The nature of the reactants will also affect the speed of reaction. Since reacting particles must come
close to each other in order to react, oppositely charged ions would be expected to react very
quickly. Large molecules may be slow to react if the reaction site on each reactant is not aligned
correctly.
The purpose of the study of enzyme kinetics is to deduce the reaction mechanism. The reaction
mechanism determination involves identifying the intermediate steps by which the reactants are
converted into products. Having established the reaction mechanism, we examine the effect of the
environment on the rate of intermediate formation and breakdown. This probing gives insight into
the chemical and thermodynamic mechanisms of catalysis
We will begin with an exploration of the thermodynamic principles that determine the rate constants
and finally then a discussion of how catalysts (enzymes) affect the rates of reactions. This will be
followed by a discussion of how the experimental determination gives of the # of steps and reactants
involved in the reactions.
Kinetic Energy and Chemical Reactions
Consider the conversion of 1 base pair of double-stranded DNA to the unpaired or singlestranded
state:
dsDNA⇄ssDNA
A⇄B
In order to achieve this transition requires the input of energy. Where does the energy come
from?
Solvent!
Solvent molecules transfer their energy to the DNA. The energy inherent in a solvent molecule at
any temperature is its kinetic energy. The kinetic energy within a water molecule is equal to:
ke= ½ mv2 (Eqn. 1)
where v2 equals the average squared velocity and m equals the mass. To describe the distribution
of kinetic energies in a large collection of atoms or molecules, we must resort to statistics.
Maxwell and Boltzmann discovered that this distribution may be described by plotting the
fraction of molecules in a container with a given kinetic energy vs kinetic energy. Alternately,
we can plot the probability that water molecules will have a given kinetic energy vs. kinetic
energy
Figure 1
The reason for this odd distribution is that for a collection of particles, for example solution of
H2O molecules, each has a set of energy states available to it. The energy states are quantized
states accessible to any particle. If a collection has the same energy, the overall distribution of
particles (molecules) is the one that is most degenerate, i.e., the largest number of ways of
Figure 2
arranging the particles over the states, is the most likely distribution.
Note that the average kinetic energy of the favored distribution is 1.67e, but the most probable
kinetic energy is 1e. This distribution of energies is known at the Boltzmann distribution.
As we all know, kinetic energy is the energy of motion, and as described in Eq. 1, ke is
directly proportional to the square of the speed of how fast the molecules are moving. One way
to get things moving faster is to heat them, i.e., change the temperature. In fact the temperature
of a solution is directly proportional to the average kinetic energy of molecules. Thus:
ke = 3/2 nRT (Eqn. 2)
where n is the number of moles, T is the temperature in Kelvin, and R is the gas constant usually
expressed in units of 1.98 cal/deg mole.
If T is proportional to the average kinetic energy, then we'd expect for this change in the average
kinetic energy to be reflected in changes in the shape of the Boltzmann distribution. Since in our
example we're using probability on the vertical axis, the area under the curve must be exactly
equal to 1.0 at all times (there must a unit probability of the water having some kinetic energy).
As temperature increases, the curve will spread to the right and the value of the most probable
kinetic energy will decrease. This is illustrated in Figure 3 for several temperatures.
Figure 3
It is often useful to fix our attention on some value of the kinetic energy and ask what happens to
the fraction of molecules with kinetic energies equal to or greater than this value as we change
the temperature. Let's reexamine the curves from above with this in mind. These curves are
illustrated below. I have selected a reference kinetic energy in Red.
Figure 4
Notice that as T increases, the fraction of molecules with energies greater than the red line
increase. This fact is of fundamental importance to properties of matter such as vapor pressures,
and we will see that these ideas underlie the effect of temperature on the rates of chemical
processes.
To begin evaluating the effect of temperature on a particular process, it is useful to think
about the effect of temperature on the kinetic energy of a particular molecule. This is done by
scaling Eqn 2 to a per molecule:
Molecular ke = 3/2 kBT (Eqn. 3)
where kB is the Boltzmann constant. This constant is version of R, the gas constant, scaled on a
per molecule basis. Thus:
R= N kB = 6.02 X 1023 * kB
where N is Avogadro's number. Thus, we can rewrite Eqn. 2 for 1 mole of molecules
Molar ke = 3/2 NRT (Eqn. 4)
for any molecule at room temperature (298 K), the average (kinetic) energy a molecule has is
0.89 kcal/mol. This is the amount of energy any one mole of solution has available for transfer.
According to the first law of thermodynamics, simply stated as the conservation of energy, each
molecule may have different energies, but the total energy, regardless of distribution, must be
0.89 kcal/mol. Rexamining Figures 1-3 and considering our goal of finding out where the energy
for breaking a base pair H-bond comes from, we can see that it derives from the transfer of
energy from water molecules that have higher ke that the energy of the base pair bond.
This example, while simplistic, illustrates the basic thermodynamic concept for energy
transfer in chemical reactions. Thus, whenever a question arises about the effect of temperature
on some chemical process, you should refer back to the Maxwell-Boltzmann concepts.
Evaluation of the Thermodynamics of Activation Parameters
Transition State Theory.
As stated above, before molecules can react, they must attain a particular energetic state, allowing
the enthalpic processes to occur, i.e., bond breaking and formation and rearrangement and/or collide
with each other, as in a second order reaction, and in doing so, overcome unfavorable entropic
factors, orientation and proximity considerations. Thus, there is an energy barrier between reactants
and products. That this is true is illustrated by the fact that complex material forms exist, we are not
all decomposed, there is some barrier to decomposition. We can illustrate this on an energy
diagram, with the energy barrier separating reactants and products called the activation energy.
For the reaction AB
Figure 5
Note that:
E describes energy difference equilibrium between reactant (A) and product (B). Since the
second law of thermodynamics predicts that the energy of the system at equilibrium, the reaction
always flows down hill (-E), i.e., the energy of the products is lower than the energy of the
reactant
Ea describes the height of the barrier between the reactant and the transition state.
Since the rate constant of a given reaction actually describes the number of events in a given
amount of time (products formed or reactants consumed), the rate constant is inversely proportional
to the height of the activation barrier. To more easily overcome this barrier, the reactants must be
"energized".
If an intermediate is much higher in energy than either the reactants or intermediates, where
does this energy come from? In line with our discussion of the Boltzmann distribution, the easiest
way to energize reactants is to change the temperature, or increase the kinetic energy of the system.
Kinetic Energy and Potential Energy
Let's think of a simple situation in which a car is speeding toward a wall. The car has some kinetic
energy that is equal to one half of its mass times its velocity squared. When the car hits the wall, its
velocity goes to zero, where does its kinetic energy go to? Obviously it goes into the energy
A
B
E
necessary to bend, break and otherwise rearrange the bonds in the frame of the car and the
passengers. Modern automobiles have specially designed components that are designed to absorb as
much of this energy as possible so that it doesn't go into breaking up the passengers. In summary, the
kinetic energy is converted into potential energy at the moment of impact, and this potential energy
is available for bond making and breaking. The situation with two molecules is the same.
Let's simplify things and assume that they collide head-on with exactly the same kinetic
energy. At the moment of collision, all of the kinetic energy will be converted to potential energy,
and this potential energy can be used to make and break bonds leading to a new product. We can
illustrate this process in Figure 6.
Figure 6
In a real collision, some of the energy may remain in kinetic energy, and some of the energy may go
into molecular vibrations that don't result in bond breaking and making, thus not every collision
necessarily leads to product formation.
If Ea is the minimum energy necessary to get over the energy hill, then we can go back to the
Boltzman curve and imagine that only those molecules with energy in excess of Ea will make it over
the hill. Figure 7
As shown in Figure 7, an increase in temperature will increase the number of molecules with
energies in excess of Ea and hence increase the rate of the reaction.
ArrheniusEquation:
In 1889, Arrhenius suggested the simple relationship between the specific rate constant for a
reaction and the temperature of the system.
k = Qe-E
a
/RT
the constant Q is a factor that describes the probability that molecules that have sufficient energy to
jump the Ea barrier will react, and R is the gas constant and T is the absolute temperature. The more
useful form of the Arrhenius equation is given as:
ln k = -Ea/RT + ln Q
Thus if the rate constant of a reaction is measured at several different temperatures and is plotted vs
1/T, a straight line should result, the slope of which is Ea/R.
Arrhenius theorized correctly that the temperature dependence of a reaction must indicate
that before products can be formed, the reactants must be in an activated complex or as we also call
it, transition state.
Eyring Theory and Eyring Equation.
Strictly speaking, the Arrhenius equation works only for gas phase, purely inelastic
collisions. For mixed phase reactions or those that involve elastic collisions, Eyring’s theory and
equations are used. The basis of Eyring’s theory stems from a thermodynamic understanding of
equilibrium (Gibb’s) processes. Using Eyring’s theory, therefore, we can use the concepts of
transition state theory to describe the velocity of a reaction in thermodynamic terms.
In the transition state, chemical bonds are in the process of being made or broken. A simple
way then of understanding the rate constant for a reaction is to realize that the transition state and the
ground state (reactants) are in thermodynamic equilibrium:
A
1
1
k
k A*
so that the concentration of the transition state is calculated from their difference in energies, the
overall rate is then the concentration of the transition state multiplied by the velocity of its
productive decomposition.
A
1
1
k
k A* k‡→P
k=K‡ . k‡
We will treat the two terms of this equation separately. For K+, the equilibrium constant, as we
know from standard equilibrium thermodynamics is
G‡ = -RT lnK‡,
solving for K and taking the antilog:
K‡ = e-(G‡/RT).
The frequency at which a transition state decomposes is the same as the vibrational frequency of the
bond which is breaking. According to physical principles, this frequency , is equal to E/h, where h
is Planck's constant, and E is the average vibrational energy of a bond.Thus:
k‡ = =E/h
At temperature T, the vibration of an excited bond in a transition state, according to classical
physics and our discussions of kinetic energy above, has the value of kBT, where kB is Boltzmann's
constant (which relates mean kinetic energy to absolute temperature). Since
E= kBT,
then
k‡ = kBT/h
Thus, putting this all together gives:
K‡ = kBT/h
Therefore, the rate constant, k=K‡*k‡ can be expressed in classical physics as:
k = kBT/h e-(G‡/RT)
or taking the ln and rearranging:
G‡ = -RT * ln (k *h/kBT)
Since the free energy of a the formation of a transition state can be described by equilibrium
thermodynamics equations:
G‡ = H‡ - TS‡
Thus, the reaction rate constant can be given as:
k = kBT/h*e(-H‡/RT)*e(S‡/R)
or taking the ln:
ln (k/T) = -H‡/RT + S‡/R + ln kB/h
y m x + b
(the ln kB/h term is constant & small over
a small range of T)
Thus a plot of ln k vs 1/T allows evaluation of the H‡ from the slope and S‡ from the intercept
Significance and application of transition state theory
The significance of the transition state theory lies in the understanding of the factors that
affect equilibrium constant between the activated complex and the ground state. The rate constant
for a reaction is directly proportional to this term; i.e., the slower the reaction, smaller the
equilibrium constant, indicating a high energy barrier (large positive G‡). By assessing various
factors, e.g., structure of the substrate or solvent conditions, that affect the rate constant for a
particular reaction, we can get insights into the reaction's chemical mechanism.
A useful guide in the application of transition state theory to understanding solution effects
or structure-reactivity data on rate processes is the Hammond Postulate. This states that if there is
an unstable intermediate along the reaction pathway, then the transition state will resemble this
intermediate. This is a useful way to guess the structure of the transition state and thereby predict
the type of stabilization it will require. For understanding enzymic reactions, this is of critical
importance for two reasons:
1) enzymes accelerate reactions by lowering the energy of the transition state. That is, they
decrease G‡. An insight into transition state structure tells the reaction mechanism.
2) the best enzyme inhibitors are often those that mimic the transition state, because they are
so specific and bind very tightly. This then points out one of the uses of enzyme studiesdevelopment
of new enzyme inhibitors which may be able to function as drugs or
biochemical tools.
To understand the transition state and how catalysts may increase the rate by stabilizing this state, let
us consider the hydrolysis of an ester:
The uncatalyzed attack by water on an ester (as present in proteins) leads to a transition state which
has a carboxonium ion as an intermediate:
The formation of this transition state is very unfavorable because of the unstable charges that are
developed. Clearly, the transition state would be stabilized if the partial negative charge on the
carboxyl oxygen were stabilized by a positively-charged proton donor and if one proton of the
attacking water were drawn off by a proton acceptor.
A partial solution to transition state stabilization in solution can be obtained by performing the
hydrolysis reaction in the presence of acetate.
This type of catalysis is known as general base catalysis, since one of the protons from water is
transferred to a base during the reaction.
Entropy and the role of intramolecular catalysis So this is how reactions in solution are
catalyzed-stabilization of transition states. But enzyme catalyzed reactions are characterized not
only by this process, but also by their incredible efficiency. The question at issue then becomes, why
are enzymes such efficient catalysts? The answer lies in the principle of effective concentration. In
a solution, the catalyst, which in our example above is the acetate molecule, must find the substrate
(the ester) and the water, and all must form a complex having the proper orientation of the reactants.
Enzymes on the other hand contain the catalysts (for the ester hydrolysis reaction, the general acid
and base catalysts) present in one molecule-and by virtue of its highly defined tertiary structure,
when the substrate binds to the enzyme, they are in the proper orientation for reaction. These
proximity and orientation effects can combine to increase the effective concentration of a catalyst to
13M, in esterases.
This binding of substrate and catalysts into a tightly bound complex alters the reaction which
forms the transition state from one which is wholly intermolecular to one which is intramolecular.
This change from inter- to intramolecular leads to a rate acceleration due to changes in the
entropy of the transition state. In a solution reaction, the water, substrate and catalyst must come
together in proper orientation, leading to a huge decrease in the entropy of the system-rendering the
G‡ huge, thereby slowing the reaction tremendously. In an enzyme reaction however, the bound
substrate and catalytic groups are already properly oriented, thus there are little, if any, changes in
entropy during the catalytic reaction, its rate then is mostly dominated by enthalpy considerations of
bond breaking or making.
Of course most of you are saying, 'but there is an entropy decrease upon binding substrate,
how is that paid for?' First, you must realize that the entropy of substrate binding occurs in previous,
thermodynamically distinct, separate step. The 'entropy cost' is paid for by complimentary
interactions between groups on the enzyme and substrate-i.e., a number of interactions with
favorable H pay for the substrate binding entropy.
Therefore, enzymes are poised to be effective catalysts for two reasons:
1) they pay for substrate entropy loss with favorable enthalpic interactions in a binding step.
2) they eliminate any unfavorable entropy effects in the formation of a transition state by
having the catalytic groups positioned in a favorable orientation and in close proximity.
Enzymes are proteins that are specialized to catalyze biological reactions. Catalysts are entities that
participate in, but are not components of the reactions. That is, CATALYSTS ARE NOT USED UP
IN THE REACTION.
Enzymes are characterized by their extremely high degree of substrate specificity
(SUBSTRATE=REACTANT IN A REACTION) and their amazingly high catalytic power. Some
enzymes accelerate the rate of their reactions by ~106 over that observed in solution.
Enzymes catalyze (facilitate) the same type of reactions as occur in solution-only they make them
occur faster! This high degree of rate acceleration is obtained by bringing the substrates close
together (PROXIMITY) and in the proper ORIENTATION. This is effected by the fact that
enzymes bind there substrates, there is an intermediate complex formed between E and S, before
catalysis. This discussion suggests that ENZYME CATALYZED REACTIONS OCCUR IN
MULTIPLE STEPS.
YOU HAVE TO REALIZE-ENZYMES DO NOT ALTER THE EQUILIBRIUM OF THE
REACTION!!! THEY CATALYZE ONLY REACTIONS THAT OCCUR SPONTANEOUSLY!!!
Some enzymes utilize tricks, coupling favorable and unfavorable reactions to give the desired
results, but the free energy of an enzyme catalyzed reaction is always negative.
Kinetics- determination and study of the velocity of chemical reactions. Some reactions are very
slow others very fast. In order for a chemical reaction to occur, the reactants must collide with each
other and with enough energy to result in a reaction. Any condition that creates a greater number of
collisions with the minimum activation energy required will result in a faster reaction.
The speed of a chemical reaction is dependent on all of the following:
1. The temperature at which the reaction occurs
An increase in temperature will result in an increase in kinetic energy. Since the kinetic energy
increases, the velocity of the particles will also increase. Since the speed of the particles increases,
they should collide more often (therefore speed of reaction increases). The particles will also have
more energy thereby speeding up the reaction even more.
2. The concentration of reactants
The more reactant particles there are per unit volume (concentration of reactants) the more chances
there are for collisions. This should result in an increase in the speed of the reaction.
3. The presence of a catalyst
A catalyst will speed up a reaction by providing a better geometric arrangement for reactivity
(reaction sites line up better) or by providing an alternate pathway for the reaction (one with a lower
reaction energy). Although a catalyst may take an active part in the reaction, it can be recovered at
the end of the reaction chemically unchanged
4. The nature of the reactants
The nature of the reactants will also affect the speed of reaction. Since reacting particles must come
close to each other in order to react, oppositely charged ions would be expected to react very
quickly. Large molecules may be slow to react if the reaction site on each reactant is not aligned
correctly.
The purpose of the study of enzyme kinetics is to deduce the reaction mechanism. The reaction
mechanism determination involves identifying the intermediate steps by which the reactants are
converted into products. Having established the reaction mechanism, we examine the effect of the
environment on the rate of intermediate formation and breakdown. This probing gives insight into
the chemical and thermodynamic mechanisms of catalysis
We will begin with an exploration of the thermodynamic principles that determine the rate constants
and finally then a discussion of how catalysts (enzymes) affect the rates of reactions. This will be
followed by a discussion of how the experimental determination gives of the # of steps and reactants
involved in the reactions.
Kinetic Energy and Chemical Reactions
Consider the conversion of 1 base pair of double-stranded DNA to the unpaired or singlestranded
state:
dsDNA⇄ssDNA
A⇄B
In order to achieve this transition requires the input of energy. Where does the energy come
from?
Solvent!
Solvent molecules transfer their energy to the DNA. The energy inherent in a solvent molecule at
any temperature is its kinetic energy. The kinetic energy within a water molecule is equal to:
ke= ½ mv2 (Eqn. 1)
where v2 equals the average squared velocity and m equals the mass. To describe the distribution
of kinetic energies in a large collection of atoms or molecules, we must resort to statistics.
Maxwell and Boltzmann discovered that this distribution may be described by plotting the
fraction of molecules in a container with a given kinetic energy vs kinetic energy. Alternately,
we can plot the probability that water molecules will have a given kinetic energy vs. kinetic
energy
Figure 1
The reason for this odd distribution is that for a collection of particles, for example solution of
H2O molecules, each has a set of energy states available to it. The energy states are quantized
states accessible to any particle. If a collection has the same energy, the overall distribution of
particles (molecules) is the one that is most degenerate, i.e., the largest number of ways of
Figure 2
arranging the particles over the states, is the most likely distribution.
Note that the average kinetic energy of the favored distribution is 1.67e, but the most probable
kinetic energy is 1e. This distribution of energies is known at the Boltzmann distribution.
As we all know, kinetic energy is the energy of motion, and as described in Eq. 1, ke is
directly proportional to the square of the speed of how fast the molecules are moving. One way
to get things moving faster is to heat them, i.e., change the temperature. In fact the temperature
of a solution is directly proportional to the average kinetic energy of molecules. Thus:
ke = 3/2 nRT (Eqn. 2)
where n is the number of moles, T is the temperature in Kelvin, and R is the gas constant usually
expressed in units of 1.98 cal/deg mole.
If T is proportional to the average kinetic energy, then we'd expect for this change in the average
kinetic energy to be reflected in changes in the shape of the Boltzmann distribution. Since in our
example we're using probability on the vertical axis, the area under the curve must be exactly
equal to 1.0 at all times (there must a unit probability of the water having some kinetic energy).
As temperature increases, the curve will spread to the right and the value of the most probable
kinetic energy will decrease. This is illustrated in Figure 3 for several temperatures.
Figure 3
It is often useful to fix our attention on some value of the kinetic energy and ask what happens to
the fraction of molecules with kinetic energies equal to or greater than this value as we change
the temperature. Let's reexamine the curves from above with this in mind. These curves are
illustrated below. I have selected a reference kinetic energy in Red.
Figure 4
Notice that as T increases, the fraction of molecules with energies greater than the red line
increase. This fact is of fundamental importance to properties of matter such as vapor pressures,
and we will see that these ideas underlie the effect of temperature on the rates of chemical
processes.
To begin evaluating the effect of temperature on a particular process, it is useful to think
about the effect of temperature on the kinetic energy of a particular molecule. This is done by
scaling Eqn 2 to a per molecule:
Molecular ke = 3/2 kBT (Eqn. 3)
where kB is the Boltzmann constant. This constant is version of R, the gas constant, scaled on a
per molecule basis. Thus:
R= N kB = 6.02 X 1023 * kB
where N is Avogadro's number. Thus, we can rewrite Eqn. 2 for 1 mole of molecules
Molar ke = 3/2 NRT (Eqn. 4)
for any molecule at room temperature (298 K), the average (kinetic) energy a molecule has is
0.89 kcal/mol. This is the amount of energy any one mole of solution has available for transfer.
According to the first law of thermodynamics, simply stated as the conservation of energy, each
molecule may have different energies, but the total energy, regardless of distribution, must be
0.89 kcal/mol. Rexamining Figures 1-3 and considering our goal of finding out where the energy
for breaking a base pair H-bond comes from, we can see that it derives from the transfer of
energy from water molecules that have higher ke that the energy of the base pair bond.
This example, while simplistic, illustrates the basic thermodynamic concept for energy
transfer in chemical reactions. Thus, whenever a question arises about the effect of temperature
on some chemical process, you should refer back to the Maxwell-Boltzmann concepts.
Evaluation of the Thermodynamics of Activation Parameters
Transition State Theory.
As stated above, before molecules can react, they must attain a particular energetic state, allowing
the enthalpic processes to occur, i.e., bond breaking and formation and rearrangement and/or collide
with each other, as in a second order reaction, and in doing so, overcome unfavorable entropic
factors, orientation and proximity considerations. Thus, there is an energy barrier between reactants
and products. That this is true is illustrated by the fact that complex material forms exist, we are not
all decomposed, there is some barrier to decomposition. We can illustrate this on an energy
diagram, with the energy barrier separating reactants and products called the activation energy.
For the reaction AB
Figure 5
Note that:
E describes energy difference equilibrium between reactant (A) and product (B). Since the
second law of thermodynamics predicts that the energy of the system at equilibrium, the reaction
always flows down hill (-E), i.e., the energy of the products is lower than the energy of the
reactant
Ea describes the height of the barrier between the reactant and the transition state.
Since the rate constant of a given reaction actually describes the number of events in a given
amount of time (products formed or reactants consumed), the rate constant is inversely proportional
to the height of the activation barrier. To more easily overcome this barrier, the reactants must be
"energized".
If an intermediate is much higher in energy than either the reactants or intermediates, where
does this energy come from? In line with our discussion of the Boltzmann distribution, the easiest
way to energize reactants is to change the temperature, or increase the kinetic energy of the system.
Kinetic Energy and Potential Energy
Let's think of a simple situation in which a car is speeding toward a wall. The car has some kinetic
energy that is equal to one half of its mass times its velocity squared. When the car hits the wall, its
velocity goes to zero, where does its kinetic energy go to? Obviously it goes into the energy
A
B
E
necessary to bend, break and otherwise rearrange the bonds in the frame of the car and the
passengers. Modern automobiles have specially designed components that are designed to absorb as
much of this energy as possible so that it doesn't go into breaking up the passengers. In summary, the
kinetic energy is converted into potential energy at the moment of impact, and this potential energy
is available for bond making and breaking. The situation with two molecules is the same.
Let's simplify things and assume that they collide head-on with exactly the same kinetic
energy. At the moment of collision, all of the kinetic energy will be converted to potential energy,
and this potential energy can be used to make and break bonds leading to a new product. We can
illustrate this process in Figure 6.
Figure 6
In a real collision, some of the energy may remain in kinetic energy, and some of the energy may go
into molecular vibrations that don't result in bond breaking and making, thus not every collision
necessarily leads to product formation.
If Ea is the minimum energy necessary to get over the energy hill, then we can go back to the
Boltzman curve and imagine that only those molecules with energy in excess of Ea will make it over
the hill. Figure 7
As shown in Figure 7, an increase in temperature will increase the number of molecules with
energies in excess of Ea and hence increase the rate of the reaction.
ArrheniusEquation:
In 1889, Arrhenius suggested the simple relationship between the specific rate constant for a
reaction and the temperature of the system.
k = Qe-E
a
/RT
the constant Q is a factor that describes the probability that molecules that have sufficient energy to
jump the Ea barrier will react, and R is the gas constant and T is the absolute temperature. The more
useful form of the Arrhenius equation is given as:
ln k = -Ea/RT + ln Q
Thus if the rate constant of a reaction is measured at several different temperatures and is plotted vs
1/T, a straight line should result, the slope of which is Ea/R.
Arrhenius theorized correctly that the temperature dependence of a reaction must indicate
that before products can be formed, the reactants must be in an activated complex or as we also call
it, transition state.
Eyring Theory and Eyring Equation.
Strictly speaking, the Arrhenius equation works only for gas phase, purely inelastic
collisions. For mixed phase reactions or those that involve elastic collisions, Eyring’s theory and
equations are used. The basis of Eyring’s theory stems from a thermodynamic understanding of
equilibrium (Gibb’s) processes. Using Eyring’s theory, therefore, we can use the concepts of
transition state theory to describe the velocity of a reaction in thermodynamic terms.
In the transition state, chemical bonds are in the process of being made or broken. A simple
way then of understanding the rate constant for a reaction is to realize that the transition state and the
ground state (reactants) are in thermodynamic equilibrium:
A
1
1
k
k A*
so that the concentration of the transition state is calculated from their difference in energies, the
overall rate is then the concentration of the transition state multiplied by the velocity of its
productive decomposition.
A
1
1
k
k A* k‡→P
k=K‡ . k‡
We will treat the two terms of this equation separately. For K+, the equilibrium constant, as we
know from standard equilibrium thermodynamics is
G‡ = -RT lnK‡,
solving for K and taking the antilog:
K‡ = e-(G‡/RT).
The frequency at which a transition state decomposes is the same as the vibrational frequency of the
bond which is breaking. According to physical principles, this frequency , is equal to E/h, where h
is Planck's constant, and E is the average vibrational energy of a bond.Thus:
k‡ = =E/h
At temperature T, the vibration of an excited bond in a transition state, according to classical
physics and our discussions of kinetic energy above, has the value of kBT, where kB is Boltzmann's
constant (which relates mean kinetic energy to absolute temperature). Since
E= kBT,
then
k‡ = kBT/h
Thus, putting this all together gives:
K‡ = kBT/h
Therefore, the rate constant, k=K‡*k‡ can be expressed in classical physics as:
k = kBT/h e-(G‡/RT)
or taking the ln and rearranging:
G‡ = -RT * ln (k *h/kBT)
Since the free energy of a the formation of a transition state can be described by equilibrium
thermodynamics equations:
G‡ = H‡ - TS‡
Thus, the reaction rate constant can be given as:
k = kBT/h*e(-H‡/RT)*e(S‡/R)
or taking the ln:
ln (k/T) = -H‡/RT + S‡/R + ln kB/h
y m x + b
(the ln kB/h term is constant & small over
a small range of T)
Thus a plot of ln k vs 1/T allows evaluation of the H‡ from the slope and S‡ from the intercept
Significance and application of transition state theory
The significance of the transition state theory lies in the understanding of the factors that
affect equilibrium constant between the activated complex and the ground state. The rate constant
for a reaction is directly proportional to this term; i.e., the slower the reaction, smaller the
equilibrium constant, indicating a high energy barrier (large positive G‡). By assessing various
factors, e.g., structure of the substrate or solvent conditions, that affect the rate constant for a
particular reaction, we can get insights into the reaction's chemical mechanism.
A useful guide in the application of transition state theory to understanding solution effects
or structure-reactivity data on rate processes is the Hammond Postulate. This states that if there is
an unstable intermediate along the reaction pathway, then the transition state will resemble this
intermediate. This is a useful way to guess the structure of the transition state and thereby predict
the type of stabilization it will require. For understanding enzymic reactions, this is of critical
importance for two reasons:
1) enzymes accelerate reactions by lowering the energy of the transition state. That is, they
decrease G‡. An insight into transition state structure tells the reaction mechanism.
2) the best enzyme inhibitors are often those that mimic the transition state, because they are
so specific and bind very tightly. This then points out one of the uses of enzyme studiesdevelopment
of new enzyme inhibitors which may be able to function as drugs or
biochemical tools.
To understand the transition state and how catalysts may increase the rate by stabilizing this state, let
us consider the hydrolysis of an ester:
The uncatalyzed attack by water on an ester (as present in proteins) leads to a transition state which
has a carboxonium ion as an intermediate:
The formation of this transition state is very unfavorable because of the unstable charges that are
developed. Clearly, the transition state would be stabilized if the partial negative charge on the
carboxyl oxygen were stabilized by a positively-charged proton donor and if one proton of the
attacking water were drawn off by a proton acceptor.
A partial solution to transition state stabilization in solution can be obtained by performing the
hydrolysis reaction in the presence of acetate.
This type of catalysis is known as general base catalysis, since one of the protons from water is
transferred to a base during the reaction.
Entropy and the role of intramolecular catalysis So this is how reactions in solution are
catalyzed-stabilization of transition states. But enzyme catalyzed reactions are characterized not
only by this process, but also by their incredible efficiency. The question at issue then becomes, why
are enzymes such efficient catalysts? The answer lies in the principle of effective concentration. In
a solution, the catalyst, which in our example above is the acetate molecule, must find the substrate
(the ester) and the water, and all must form a complex having the proper orientation of the reactants.
Enzymes on the other hand contain the catalysts (for the ester hydrolysis reaction, the general acid
and base catalysts) present in one molecule-and by virtue of its highly defined tertiary structure,
when the substrate binds to the enzyme, they are in the proper orientation for reaction. These
proximity and orientation effects can combine to increase the effective concentration of a catalyst to
13M, in esterases.
This binding of substrate and catalysts into a tightly bound complex alters the reaction which
forms the transition state from one which is wholly intermolecular to one which is intramolecular.
This change from inter- to intramolecular leads to a rate acceleration due to changes in the
entropy of the transition state. In a solution reaction, the water, substrate and catalyst must come
together in proper orientation, leading to a huge decrease in the entropy of the system-rendering the
G‡ huge, thereby slowing the reaction tremendously. In an enzyme reaction however, the bound
substrate and catalytic groups are already properly oriented, thus there are little, if any, changes in
entropy during the catalytic reaction, its rate then is mostly dominated by enthalpy considerations of
bond breaking or making.
Of course most of you are saying, 'but there is an entropy decrease upon binding substrate,
how is that paid for?' First, you must realize that the entropy of substrate binding occurs in previous,
thermodynamically distinct, separate step. The 'entropy cost' is paid for by complimentary
interactions between groups on the enzyme and substrate-i.e., a number of interactions with
favorable H pay for the substrate binding entropy.
Therefore, enzymes are poised to be effective catalysts for two reasons:
1) they pay for substrate entropy loss with favorable enthalpic interactions in a binding step.
2) they eliminate any unfavorable entropy effects in the formation of a transition state by
having the catalytic groups positioned in a favorable orientation and in close proximity.
No comments:
Post a Comment