A Theoretical approach to reaction
In this section we will be deriving from first principles the rate law for the chemical equation A + B -> P, given empirically by v = k[A][B]. This will be done through:(2-3) The activated complex theory and reactions between ions.
In this section we shall attempt to understand the origin of the Arrhenius parameters by studying a class of gas-phase reactions in which reaction occurs when two molecules meet (collision theory).
Under this theory, reaction occurs only if two molecules collide with a certain minimum kinetic energy along their line of approach (Fig. 1). In fact, in the collision theory, a reaction resembles the collision of two defective billiard balls: the balls bounce apart if they collide with only a small energy, but might smash each other into fragments (products) if they collide with more than a certain minimum kinetic energy. This model of a reaction is a reasonable first approximation to the types of process that take place in planetary atmospheres and govern their compositions and temperature profiles.
NOTE: The importance of the collision theory is that it provides
a 'proof' from first principles that reactions (or, strictly speaking,
gas phase reactions!) can indeed display Arrhenius-like behaviour.
Fig. 1: In the collision theory of gas-phase chemical reactions, reaction occurs when two molecules collide, but only if the collision is sufficiently vigorous, (a) An insufficiently vigorous collision: the reactant molecules collide but bounce apart unchanged, (b) A sufficiently vigorous collision results in a reaction.
A reaction profile in collision theory is a graph showing the variation
in potential energy as one reactant molecule approaches another and the
products then separate (Fig 2). (NOTE: Recall that the potential energy
of an object is the energy arising from its position (not speed), in this
case the separation of the two reactant molecules as they approach, react,
and then separate as products.)
This profile shows:
(1-2) Derivation of the rate law through the Collision theory:
With the reaction profile in mind, the collision theory predics that the rate law for a reaction:
is proportional to:
(a) The encounter rate;where:
(a) The encounter rate:
The rate of collisions between species A and B is proportional to both their concentrations: if the concentration of B is doubled, then the rate at which A molecules collide with B molecules is doubled, and, if the concentration of A is doubled, then the rate at which B molecules collide with A molecules is also doubled. The kinetic theory of gasses goes even a step further and allows us to write:
where NA is Avogadro's constant, kB is Boltzmann's constant, T is the temperature, s is the collision cross-section (see fig. 3), and m is the reduced mass given by:
Fig. 3: The collision cross-section for two molecules can be regarded to be the area within which the centre of the projectile molecule (A) must enter around the target molecule (B) in order for a collision to occur. If the diameters of the two molecules are dA and dB, the radius of the target area is d = 1/2 (dA + dB) and the cross-section is pd2.
Next, we need to multiply the collision rate by a factor f that represents the fraction of collisions that occur with at least a kinetic energy Ea along the line of approach (Fig 4), for only these collisions will lead to the formation of products. Molecules that approach with less than a kinetic energy Ea will behave like a ball that rolls toward the activation barrier, fails to surmount it, and rolls back.
However, it is often found that the experimental value of A is smaller than that calculated from the kinetic theory so far, i.e.:
One possible explanation is that, not only must the molecules collide with sufficient kinetic energy, but they must also come together in a specific relative orientation (Fig 5).
The activated complex theory (ACT), formerly known as the transition state theory, is a more sophisticated theory of reaction rates that offers the following advantages over the collision theory:
(1) The concepts used in the derivation of the rate constants in terms of the ACT are based on statistical thermodynamics.
(2) The reason why P appears automatically in the ACT framework is that the orientation requirements are carried in the entropy of activation. Thus, if there are strict orientation requirements (for example, in the approach of a substrate molecule to an enzyme), then the entropy of activation will be strongly negative (representing a decrease in disorder when the activated complex forms), and the pre-exponential factor will be small. In practice, it is occasionally possible to estimate the sign and magnitude of the entropy of activation and hence to estimate the rate constant.
(3) The general importance of activated complex theory is that it
shows that even a complex series of events (not only a collisional encounter
in the gas phase) displays Arrhenius-like behaviour, and that the concept
of activation energy is applicable.
(2-1) The reaction profile in the ACT:Activated complex theory is an attempt to identify the principal features governing the magnitude of a rate constant in terms of a model of the events that take place during the reaction. Let us have a look at a typical the reaction profile (i.e.a plot of the 'potential energy' vs. 'reaction coordinate'), for a bimolecular reaction between A and B forming products P (fig. 6). This plot illustartes the following features:
Fig. 6: A reaction profile. The horizontal axis is the reaction coordinate, and the vertical axis is potential energy. The activated complex is the region near the potential maximum, and the transition state corresponds to the maximum itself.
In other words we have:
which in terms for the ACT, we have:
i.e. the reaction between A and B proceeding through the formation of an activated complex, C‡, that falls apart by unimolecular decay into products, P. Note that we not all (AB)‡ will proceed to products, and in fact, some will decay back to A and B, i.e.
The scope now becomes to derive an expression for k2 from first principles, which as we have said before, is done through concepts from statistical thermodynamics. This formulation is known as the Eyring equation.
In a simple form of activated complex theory, we suppose that the activated complex is in equilibrium with the reactants (pre-equilibrium), and that we can express its abundance in the reaction mixture in terms of an equilibrium constant, K‡.
Then, if we suppose that the rate at which products are formed is proportional to the concentration of the activated complex, we can write:
The full activated complex theory gives an estimate of the constant of proportionality to give:
Thus, if we compare this expression with the form of the rate law, i.e.:
we would find:
But since we may think of K‡ as an equilibrium constant, then it may be expressed in terms of the standard reaction Gibbs energy, or in this context, the activation Gibbs energy, and written D‡G. It follows that:
This expression has the form of the Arrhenius expression, i.e.:
if we identify the enthalpy of activation, D‡H, with the activation energy and the term in square brackets, which depends on the entropy of activation, D‡S, with the pre-exponential factor.
(2-3) The activated complex theory and reactions between ions.
The simplified (thromodynamic) approach to the activated complex theory simplifies the discussion of reactions in solution. We can combine the rate law:
with the thermodynamic equilibrium constant:
Let us now define ko2 as the rate constant with the activity coefficients equal to 1, i.e.:
At low concentrations the activity coefficients can be expressed in terms of the ionic strength, I, of the solution by using the Debye-Huckel limiting law (see P.W. Atkins, Physical Chemistry, 6th Ed. Section 10.2c, particularly eqn 10.19) in the form:
where A = 0.509 in aqueous solution at 298 K and zJ is the charge numbers on particle J.
Conversely, if reactant ions have opposite same signs, the charges in the activated complex cancel out and the complex has a less favourable interaction with its atmosphere than the separated ions with a solution of a high ionic strength.
>> TECHNICAL NOTE: A plot of log(k/k0)
vs. l1/2 should be linear wheer the gradient gtives infomation
about the charge type of the activated complex for the rate determining
step. (see Fig 5 below).
Fig. 6: Experimental tests of the kinetic salt effect for reactions in water at 298 K. The ion types are shown as spheres, and the slopes of the lines are those given by the Debye-Hückel limiting law and the Kinetic salt effect equation above (Eqn. K.Salt.Effect).