23CHMA08_HARD-SPHERE COLLISION THEORY OF GAS PHASE REACTIONS.pptx

The document discusses Bravais lattices, which are the 14 possible arrangements of points in a crystal structure

1. HARD SPHERE COLLISION THEORY DHARA SHREE B V 23CHMA08 II – MSC CHEMISTRY BHARATHIAR UNIVERSITY

2. The hard sphere collision theory assumes the following to arrive at an expression for the rate constant of an elementary bimolecular gas-phase reaction. The molecules are hard spheres . For a reaction to occur between molecules B and C, the two molecules must collide. Not all collisions produce reaction. The Maxwell–Boltzmann equilibrium distribution of molecular velocities is maintained during the reaction.

3. Line-of-centres components V C,lC and V B,IC of the velocities of two colliding molecules.    

4. The equilibrium distribution of molecular speeds is maintained by molecular collisions. In most reactions, the threshold energy is much greater than the mean molecular translational energy kT. (Typical gas-phase bimolecular activation energies are 3 to 30 kcal/mole compared with 0.9 kcal/mole for RT at room temperature ). Consequently, only a tiny fraction of collisions produces reaction.  

5. Since the collision rate is usually far greater than the rate of depletion of high-energy reactant molecules, the redistribution of energy by collisions is able to maintain Maxwell distribution of speeds during the reaction . For the opposite extreme, Where E thr ≈ 0,virtually all B-C collisions lead to reaction, so the mixture is being depleted of low-energy as well as high-energy molecules.

6. T he rate of collisions for which the line-of- centre relative translational energy E lc exceeds E thr . Z BC , the total number of B-C collisions per unit time per unit volume when the gas is in equilibrium . Z BC was found from the Maxwell distribution of speeds, The number of molecules reacting per unit time per unit volume equals Z BC multiplied by the fraction of collisions for which E lc ≥ E thr . Since the rigorous calculation of this fraction is complicated a plausible argument, rather than a derivation . There are two components of velocity involved (namely, the component of velocity of each molecule along the line of centres), so we might suspect that equals the fraction of molecules in a hypothetical two-dimensional gas whose translational energy E = E x + E y = 1/2 mv x 2 + 1/2 mv y 2 = 1/2 mv 2 exceeds E thr .  

7. T hree-dimensional gas shows that the fraction of molecules with speed between v and v + dv in a two-dimensional gas is (1 ) (m/2 × 2π vdv = (m/ kT ) vdv (1) The use of E = 1/2mv 2 and dE = mv dv gives the fraction of molecules with energy between E and E + dE as (1/ kT ) dE . Integration of this expression from E thr to ∞ gives the fraction with translational energy exceeding E thr as e - E thr / KT . The rigorous derivation shows that does equal  

8. Therefore, the number of B molecules reacting per unit volume per second in the elementary bimolecular reaction B + C → products is Z BC e - E thr / RT . where E thr N A is the threshold energy on a per-mole basis, and N A is the Avogadro constant . The reaction rate r is defined in terms of moles, so r = Z BC e - E thr / RT /N A . Since r = k[B][C], the predicted rate constant is ,   k = Z BC e - E thr / RT N A [B ][C] (2)

9. The use of Z BC with N B /V = N A /V = N A [B] and N C /V = N A [C ] gives   K =N A ( r B + r C ) 2 [ ] 1/2 e - E thr / RT for B ≠ C (3)   For the bimolecular reaction 2B →products, the rate of reaction is given by r d[B]/ dt = k[B] 2 .  

10. The rate of disappearance of B is -d[B]/ dt = 2Z BB e - E thr / RT /N A . The factor 2 appears because two B molecules disappear at each reactive collision. Therefore k = - (d[B]/ dt )/ [B] 2 = Z BB e - E thr / RT /N A [B] 2 ) Substitute with N B /V = N A [B] gives     K = 1/ ( 8RT/ π M Equations (3) and (4) have the form ln k = constant + lnT – E thr /RT.  

11. The activation energy as Ea RT 2 dln k/ dT = RT 2 (1/2T + E thr /RT 2 ): E a = E thr + RT (5) Substitution of (5) and (3) in A = k gives the pre-exponential factor as     A =N A π ( r B + r C ) 2 [ ] 1/2 for B ≠ C (6)    

12. Since RTis small, the hard-sphere threshold energy is nearly the same as the activation energy. The simple collision theory provides no means of calculating E t but gives only the pre-exponential factor A, because of the crudities of the theory, the predicted T ½ dependence of A should not be taken seriously . For most reactions, the calculated A values are much higher than the observed values . Hence , the hard-sphere collision theory was modified by adding a factor p to the right sides of Equations. (2) to (4) and (6 ).  

13. The factor p is called the steric (or probability) factor. The argument is that the colliding molecules must be properly oriented for collision to produce a reaction; p (which lies between 0 and 1) represents the fraction of collisions in which the molecules have the right orientation.

14. Reference: physical chemistry sixth edition – IRA N. LEVINE

15. Thank you