In the theory of active intermediates, decomposition of the intermediate does not occur instantaneously after internal activation of the molecule; rather, there is a time lag, although infinitesimally small, during which the species remains activated. Zewail’s work was the first definitive proof of a gas-phase active intermediate that exists for an infinitesimally short time. Because a reactive intermediate reacts virtually as fast as it is formed, the net rate of formation of an active intermediate (e.g., A*) is zero, i.e.,
This condition is also referred to as the Pseudo-Steady-State Hypothesis (PSSH). If the active intermediate appears in n reactions, then
To illustrate how rate laws of this type are formed, we shall first consider the gas-phase decomposition of azomethane, AZO, to give ethane and nitrogen:
Experimental observations4 show that the rate of formation of ethane is first order with respect to AZO at pressures greater than 1 atm (relatively high concentrations)
rC2H6 ∝ CAZO
and second order at pressures below 50 mmHg (low concentrations):
We could combine these two observations to postulate a rate law of the form
To find a mechanism that is consistent with the experimental observations, we use the following steps.
Table 9-1. Steps to Deduce a Rate Law
In reaction 1, two AZO molecules collide and the kinetic energy of one AZO molecule is transferred to internal rotational and vibrational energies of the other AZO molecule, and it becomes activated and highly reactive (i.e., AZO*). In reaction 2, the activated molecule (AZO*) is deactivated through collision with another AZO by transferring its internal energy to increase the kinetic energy of the molecules with which AZO* collides. In reaction 3, this highly activated AZO* molecule, which is wildly vibrating, spontaneously decomposes into ethane and nitrogen.
Because each of the reaction steps is elementary, the corresponding rate laws for the active intermediate AZO* in reactions (1), (2), and (3) are
[Let k1 = k1AZO*, k2 = k2AZO*, and k3 = k3AZO*]
These rate laws [Equations (9-3) through (9-5)] are pretty much useless in the design of any reaction system because the concentration of the active intermediate AZO* is not readily measurable. Consequently, we will use the Pseudo-Steady-State-Hypothesis (PSSH) to obtain a rate law in terms of measurable concentrations.
We first write the rate of formation of product
To find the concentration of the active intermediate AZO*, we set the net rate of formation of AZO* equal to zero,5 rAZO* ≡ 0.
Solving for CAZO*
At low AZO concentrations,
for which case we obtain the following second-order rate law:
At high concentrations
in which case the rate expression follows first-order kinetics,
In describing reaction orders for this equation, one would say that the reaction is apparent first order at high azomethane concentrations and apparent second order at low azomethane concentrations.
The PSSH can also explain why one observes so many first-order reactions such as
(CH3)2O → CH4 + H2 + CO
Symbolically, this reaction will be represented as A going to product P, that is,
A → P
with
–rA = kCA
The reaction is first order but the reaction is not elementary. The reaction proceeds by first forming an active intermediate, A*, from the collision of the reactant molecule and an inert molecule of M. Either this wildly oscillating active intermediate, A*, is deactivated by collision with inert M, or it decomposes to form product.
Figure 9-2. Collision and activation of a vibrating A molecule.
The mechanism consists of the three elementary reactions:
Writing the rate of formation of product
rP = k3CA*
and using the PSSH to find the concentrations of A* in a manner similar to the azomethane decomposition described earlier, the rate law can be shown to be
Because the concentration of the inert M is constant, we let
to obtain the first-order rate law
–rA = kCA
Consequently, we see the reaction
A → P
follows an elementary rate law but is not an elementary reaction.
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