1. §10.2 Approximate treatment of rate equation
Levine: p

2. Necessary of approximation treatment
A mechanism is a hypothesis about the elementary steps through which chemical change occurs.
To certify the correctness of the proposed mechanism, the mechanism must undergo strict examination.
Whether or not the rate equation derived from the proposed mechanism is consistent to the experimental one is an important criterion. However, because of the complexity of the mechanism, the exact treatment of the mechanism is usually impossible and some approximation have to be introduced.

3. For reaction: H2 + Br2  2 HBr
2.1 Steady-state approximation
For reaction: H2 + Br2  2 HBr
In 1919, J. A. Christiansen, K. F. Herzfeld and M. Polanyi independently proposed the mechanism consisting of five elementary reactions as follows:
Is it correct?

4. Steady-state approximation
For consecutive reaction, once the reaction take place, active intermediate rapidly attains their cmax and their concentration remains unchanged during the whole reaction.
Steady-state approximation

5. adding these two equations yields:
Steady-state approximation gives several mathematical equations from which the concentration of active intermediate can be solved.

6. 
Therefore, the proposed mechanism is probably correct.

7. 2.2 Rate-determining step (r. d. s.) approximation
For reaction:
The experimental rate equation is:
The following mechanism was proposed:

8. The same as the experimental one.
Reaction (1) is a reaction between two molecules whose activation energy is much larger than that of reaction (2) between radical and molecules. Therefore, reaction (1) may be a eligible rate-determining step of the overall reaction and, therefore, the rate equation acquires the form:
The same as the experimental one.
rate-determining step approximation provides a easier way to write the rate equation from a single step (r. d. s.) from a complicated mechanism.

9. 2.3 Pre-equilibrium approximation
When rapid opposing reaction occurs in the series of mechanism, the equilibrium approximation applies.
Its mechanism is:
deduce the rate equation according to this mechanism

10. according to r.d.s. approximation:
Reaction 1) is a rapid reversible reaction, its equilibrium keeps during the reaction – equilibrium approximation:
Equilibrium approximation: provides a useful relation between the concentration of common substances and active intermediate.

11. 4) Apparent activation energy and Ea of elementary reactions
k, The apparent rate constant is a combination of rate constants for some elementary steps.
Multiple the above equation with RT2, we get:

12. This suggests that, for some reaction with certain reaction order, the apparent activation energy of overall reaction may be expressed as a activation energy combination of some elementary steps. The physical meaning of complex reaction is thus not very clear.
Whether or not the activation energy combination of elementary step is consistent to the apparent activation energy of the overall reaction is the other important criterion for examination on reaction mechanism.

13. 5) Relation of the three approximate treatments.
Reaction: A + B  P
Mechanism:
If the generation rate of C is much less than its depletion rate (i.e., active species):
stationary-state approximation

14. When k-1 >> k2 [B], i. e
When k-1 >> k2 [B], i.e., k-1[C] >> k2 [B][C], most C undergoes reverse reaction so that equilibrium can attain rapidly.
Reaction 1 is a rapid equilibrium.

15. Equilibrium approximation
r. d. s. approximation
stationary-state approximation