9.1.1 Pseudo-Steady-State Hypothesis (PSSH)

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.,

PSSH

9-1

image

This condition is also referred to as the Pseudo-Steady-State Hypothesis (PSSH). If the active intermediate appears in n reactions, then

9-2

image

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:

image

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)

rC2H6CAZO

and second order at pressures below 50 mmHg (low concentrations):

image

We could combine these two observations to postulate a rate law of the form

image

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

images

  1. Propose an active intermediate. We will choose as an active intermediate an azomethane molecule that has been excited through molecular collisions, to form AZO*, i.e., [(CH3)2N2]*.
  2. Propose a mechanism.

    image

    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.

  3. Write rate laws.

    Because each of the reaction steps is elementary, the corresponding rate laws for the active intermediate AZO* in reactions (1), (2), and (3) are

    9-3

    image

    9-4

    image

    9-5

    image

    Note: The specific reaction rates, k, are all defined wrt the active intermediate AZO*.

    [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.

  4. Write rate of formation of product.

    We first write the rate of formation of product

    9-6

    image

  5. Write net rate of formation of the active intermediate and use the PSSH.

    To find the concentration of the active intermediate AZO*, we set the net rate of formation of AZO* equal to zero,5 rAZO* ≡ 0.

    9-7

    image

    Solving for CAZO*

    9-8

    image

  6. Eliminate the concentration of the active intermediate species in the rate laws by solving the simultaneous equations developed in Steps 4 and 5. Substituting Equation (9-8) into Equation (9-6)

    9-9

    image

  7. Compare with experimental data.

    At low AZO concentrations,

    image

    for which case we obtain the following second-order rate law:

    image

    At high concentrations

    image

    in which case the rate expression follows first-order kinetics,

    image

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.

Apparent Reaction Orders

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.

image

The mechanism consists of the three elementary reactions:

Reaction pathways

image

image

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

9-10

image

Because the concentration of the inert M is constant, we let

9-11

image

to obtain the first-order rate law

rA = kCA

Consequently, we see the reaction

A → P

First-order rate law for a nonelementary reaction

follows an elementary rate law but is not an elementary reaction.

..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.
Reset
52.91.255.225