§9.6 Rate Theories of elementary reaction

Unimolecular reaction Two important empirical rule: Rate equation (law of mass action) Arrhenius equation Type of reaction Unimolecular reaction Bimolecular reaction Termolecular A 1013 s 1011 mol-1dm3s-1 109 mol-2dm6s-1 A seems related to collision frequency. Boltzmann distribution term

Basic consideration and history It is obvious that a molecule of A cannot react with a molecule of B unless the two reactant molecules can somehow interact. This interaction can only take place if they come within a certain distance of each other, i.e., collides with each other. A reaction can take place only if the molecules of the reactants collide. Therefore, the rate constant of the reaction may be predicted by calculation of the collision frequency of the reactants. During 1920s, M. Trautz, W. Lewis, C. Hinshelwood et al. finally established a theory based on the collision, which is called the collision theory.

6.1 Fundamental assumptions of SCT for gaseous bimolecular reaction 1) The reaction rate of reaction is proportional to the collision frequency (Z); 2) The collision can be either non-reactive (elastic) collision or reactive collision. Only the molecules posses energy excess to a critical value (Ec) can lead to reactive collision. The reaction rate should be in proportion to the fraction of reactive collision (q). reaction rate can be expressed as: where ZAB is the collision frequency of A with B per unit cubic meter per second, q is the portion of effective collision.

6.2 Calculation of ZAB dA dB Definition: mean collision diameter: dAB SCT assumes that molecules can be taken as rigid ball without inner structure. dA dB dA and dB are the diameter of A and B molecule, respectively. The way to collide: 撞个满怀、擦肩而过，失之交臂 Definition: mean collision diameter: dAB

Definition:collision cross-section motionless

When the concentration of A is NA/V (molecm-3): When both A and B moves, the relative velocity VAB should be used.

according to the kinetic theory of gases (reduced mass)

? For example Decomposition of HI: 2HI = H2 + I2 At 1.0  105 Pa and 700 K, d = 3.50  10-10 m, Z HI-HI = ? Generally, ZAB of gaseous reactions at ambient temperature and pressure is of the magnitude of 1035 m-3s-1.

If reaction takes place whenever the molecules collides: because k = 7.88 104 mol-1dm3s-1 When c0 = 1.00 mol dm-3, the half-life of HI is 1.27  10-5 s. This result differs greatly from the experimental fact. In 1909, Max Trantz introduced fraction of reactive collision (q) to solve this great discrepancy.

6.3 Calculation of q Only the molecules posses energy excess to a critical value (Ec) can lead to reactive collision. It is apparent that E of translational energy of motion is related to the relative motion of two molecules. And Ec is thus the minimum translational energy of motion along the connecting line between the mass-point of the two molecules which are to collide.

If the energy exchange between colliding molecules is much rapid than reaction, the energy distribution of molecules may still obey the Maxwell-Boltzmann distribution equation. The fraction of the collision with the energy equal to or greater than Ec is: Boltzmann factor If Ec = 120 kJmol-1, T = 300 K, then q = 1.27 10-21 This suggest than among 7.8  1020 collision only one collision is effective.

6.4 Calculation of k B is a constant independent of T.

The experimental activation energy (Ea) depends on temperature. Using Ea for substitution of Ec, The pre-exponential factor corresponds to the collision frequency. This is the reason why A is also named as frequency factor.

6.5 Comment on SCT (1) Successfulness 1) The expression for the rate coefficient given by SCT conforms qualitatively to the Arrhenius equation observed experimentally. This suggests that SCT reveal the principal features of the reaction, i.e., in order to react, molecules have to collide (the pre-exponential term) and the collision should be sufficiently energetic (the exponential term) SCT gives a vivid physical image of the reaction process:

2) As pointed out by SCT, the pre-exponential factor, dependent only on the masses of the species involved in the collision, can be calculated easily. SCT reveals the physical meaning of the pre-exponential factor, i.e., the collision frequency. 3) SCT demonstrated theoretically that experimental activation energy depends on temperature.

(2) Shortcomings 1) For calculating k, Ec is needed. However, SCT can not give Ec. Calculation of k depends on the experimental determination of Ea. Therefore, SCT can not predict the kinetic features of the reaction theoretically. 2) The quantitative agreement between SCT and experiments is poor. Reaction Ea Acal Aexp Acal./Aexp. 2NOCl2NO+Cl2 107.8 2.95109 3.23109 0.91 H+Br2 HBr+Br 3.76 4.61010 6.76109 6.76 NO+O3NO2+O2 9.61 7.94109 6.31107 1.25102 CH3+CHCl3 CH4+CCl3 24.2 1.51010 1.26106 1.19104 2-cyclopentadiene dimer 60.6 8.13109 2.45103 3.32106

In some cases, the agreement between experimental and calculated A values can be quite good. However, in many cases, the observed rate is definitely too small. It was found that the more complex of the reactant molecules, the greater the discrepancy between Acal and Aexp. In fact, the reactant is of complex molecular structure. To take reactant molecules as rigid balls without inner structure will spontaneously result in systematic error. 2 ONBr  Br2 + 2 NO ?

Substitution CH3+CHCl3 CH4+CCl3 OH¯ + CH3Br  CH3OH + Br¯ The colliding molecules must be suitably oriented. OH¯ + CH3Br  CH3OH + Br¯

The great discrepancies between experimental and calculated A were recognized around 1925. The equation was then modified by introduction of an empirical factor P called the steric factor / probability factor. Steric factor (P), ranging between 1~10-9, represents the fraction of energetically suitable collisions for which the orientation is also favorable, can be only determined experimentally. SCT can not give any clue to calculate P.