SF Cheical Kinetics. Lecture 5. Microscopic theory of cheical reaction inetics. Microscopic theories of cheical reaction inetics. basic ai is to calculate the rate constant for a cheical reaction fro first principles using fundaental physics. ny icroscopic leel theory of cheical reaction inetics ust result in the deriation of an ression for the rate constant that is consistent with the epirical rrhenius equation. icroscopic odel should furtherore proide a reasonable interpretation of the pre-onential factor and the actiation energy E in the rrhenius equation. We will exaine two icroscopic odels for cheical reactions : The collision theory. The actiated coplex theory. The ain ephasis will be on gas phase biolecular reactions since reactions in the gas phase are the ost siple reaction types.
References for Microscopic Theory of Reaction Rates. Effect of teperature on reaction rate. urrows et al Cheistry 3, Section 8.7, pp.383-389. Collision Theory/ ctiated Coplex Theory. urrows et al Cheistry 3, Section 8.8, pp.39-395. tins, de Paula, Physical Cheistry 9 th edition, Chapter, Reaction Dynaics. Section.., pp.83-838. tins, de Paula, Physical Cheistry 9 th edition, Chapter, Section..4-.5, pp. 843-85. Collision theory of biolecular gas phase reactions. We focus attention on gas phase reactions and assue that cheical reactiity is due to collisions between olecules. The theoretical approach is based on the inetic theory of gases. Molecules are assued to be hard structureless spheres. Hence the odel neglects the discrete cheical structure of an indiidual olecule. This assuption is unrealistic. We also assue that no interaction between olecules until contact. Molecular spheres aintain size and shape on collision. Hence the centres cannot coe closer than a distance d gien by the su of the olecular radii. The reaction rate will depend on two factors : the nuber of collisions per unit tie (the collision frequency) the fraction of collisions haing an energy greater than a certain threshold energy E*.
Siple collision theory : quantitatie aspects. ( g) ( g) Products Hard sphere reactants Molecular structure and details of internal otion such as ibrations and rotations ignored. Two basic requireents dictate a collision eent. One ust hae an, encounter oer a sufficiently short distance to allow reaction to occur. Colliding olecules ust hae sufficient energy of the correct type to oercoe the energy barrier for reaction. threshold energy E* is required. Two basic quantities are ealuated using the Kinetic Theory of gases : the collision frequency and the fraction of collisions that actiate olecules for reaction. To ealuate the collision frequency we need a atheatical way to define whether or not a collision occurs. The collision cross section s defines when a collision occurs. s d r r d r r Effectie collision diaeter d r r rea = s The collision cross section for two olecules can be regarded to be the area within which the center of the projectile olecule ust enter around the target olecule in order for a collision to occur. 3
Maxwell-oltzann elocity distribution function J.C. Maxwell 83-879 F( ) 4 Gas olecules exhibit a spread or distribution of speeds. F() T 3/ T The elocity distribution cure has a ery characteristic shape. sall fraction of olecules oe with ery low speeds, a sall fraction oe with ery high speeds, and the ast ajority of olecules oe at interediate speeds. The bell shaped cure is called a Gaussian cure and the olecular speeds in an ideal gas saple are Gaussian distributed. The shape of the Gaussian distribution cure changes as the teperature is raised. The axiu of the cure shifts to higher speeds with increasing teperature, and the cure becoes broader as the teperature increases. greater proportion of the gas olecules hae high speeds at high teperature than at low teperature. The collision frequency is coputed ia the inetic theory of gases. We define a collision nuber (units: -3 s - ) Z. Z d n n r Mean relatie elocity Units: s - n j = nuber density of olecule j (units : -3 ) Mean relatie elocity ealuated ia inetic theory. d r r erage elocity of a gas olecule F F( ) 4 d T 3/ Maxwell-oltzann elocity Distribution function T M distribution of elocities enables us to statistically estiate the spread of olecular elocities in a gas Soe aths! 8T Mass of olecule 4
We now relate the aerage elocity to the ean relatie elocity. If and are different olecules then The aerage relatie elocity is gien by The ression across. r Hence the collision nuber between unlie olecules can be ealuated. r 8T j 8T j Z d n n r Reduced ass 8T Z nns Zn n Collision frequency factor / For collisions between lie olecules The nuber of collisions per unit tie between a single olecule and other Molecules. Total nuber of collisions between lie olecules. We diide by to ensure That each, encounter Is not counted twice. Z Z n s r 8 T Z n 8T n s / / E * Molecular collision is effectie only if translational energy of reactants is greater than soe threshold alue. Fraction of olecules with inetic energy greater Than soe iniu Threshold alue e* F e * e e * T 5
The siple collision theory ression for the reaction rate R between unlie olecules. dn e * R Znn dt T 8 T Z s The rate ression for a biolecular reaction between and. dc R cc dt We introduce olar ariables ogadro constant E* N e * n c N dn dt dc N dt n c N The rate constant for biolecular collisions between lie olecules. oth of these ressions are siilar to the rrhenius equation. Hence the SCT rate ression. dc E * R ZN cc dt RT / T E * N s RT E * z RT The biolecular rate constant for collisions between unlie olecules. Collision Frequency factor / / 8 T E * N s RT E * z RT 6
We copare the results of SCT with the epirical rrhenius eqn. in order to obtain an interpretation of the actiation energy and pre-onential factor. / T E * N s RT E * z RT obs E RT, encounters / 8 T E * N s RT E * z RT, encounters obs z '' '' N s 8 SCT predicts that the pre-onential factor should depend on teperature. The threshold energy and the actiation energy can also be copared. ctiation energy exhibits a wea T dependence. T d ln dt rrhenius E RT obs z ' 8 ' N s T d ln E * RT dt RT E E * RT Pre-onential factor SCT E E * SCT : a suary. The ajor proble with SCT is that the threshold energy E* is ery difficult to ealuate fro first principles. The predictions of the collision theory can be critically ealuated by coparing the eriental pre-onential factor with that coputed using SCT. We define the steric factor P as the ratio between P the eriental and calculated factors. calc We can incorporate P into the SCT ression for the rate constant. E * For any gas phase reactions Pz RT P is considerably less than unity. Typically SCT will predict that calc will be in E * Pz the region - Lol - s - regardless of RT the cheical nature of the reactants and products. What has gone wrong? The SCT assuption of hard sphere collision neglects the iportant fact that olecules possess an internal structure. It also neglects the fact that the relatie orientation of the colliding olecules will be iportant in deterining whether a collision will lead to reaction. We need a better theory that taes olecular structure into account. The actiated coplex theory does just that. 7
Suary of SCT., encounters / 8 T E * N s RT E * z RT, encounters / T E * N s RT E * z RT Transport property Pz Pz Steric factor (Orientation requireent) Energy criterion E * RT E * RT Weanesses: No way to copute P fro olecular paraeters No way to copute E* fro first principles. Theory not quantitatie or predictie. Strengths: Qualitatiely consistent with obseration (rrhenius equation). Proides plausible connection between icroscopic olecular properties and acroscopic reaction rates. Proides useful guide to upper liits for rate constant. Henry Eyring 9-98 Deeloped (in 935) the Transition State Theory (TST) or ctiated Coplex Theory (CT) of Cheical Kinetics. 8
Potential energy surface Can be constructed fro eriental easureents or fro Molecular Orbital calculations, sei-epirical ethods, Various trajectories through the potential energy surface 9
Potential energy hypersurface for cheical reaction between ato and diatoic olecule. C * C C Reading reaction progress on PE hypersurface. E E U
Energy Transition state theory (TST) or actiated coplex theory (CT). In a reaction step as the reactant olecules and coe together they distort and begin to share, exchange or discard atos. They for a loose structure of high potential energy called the actiated coplex that is poised to pass on to products or collapse bac to reactants C + D. The pea energy occurs at the transition state. The energy difference fro the ground state is the actiation energy E a of the reaction step. The potential energy falls as the atos rearrange in the cluster and finally reaches the alue for the products Note that the reerse reaction step also has an actiation energy, in this case higher than for the forward step. ctiated coplex Transition state E a Ea + Reaction coordinate C + D Transition state theory The theory attepts to lain the size of the rate constant r and its teperature dependence fro the actual progress of the reaction (reaction coordinate). The progress along the reaction coordinate can be considered in ters of the approach and then reaction of an H ato to an F olecule When far apart the potential energy is the su of the alues for H and F When close enough their orbitals start to oerlap bond starts to for between H and the closer F ato H F F The F F bond starts to lengthen s H becoes closer still the H F bond becoes shorter and stronger and the F F bond becoes longer and weaer The atos enter the region of the actiated coplex When the three atos reach the point of axiu potential energy (the transition state) a further infinitesial copression of the H F bond and stretch of the F F bond taes the coplex through the transition state.
Therodynaic approach Suppose that the actiated coplex is in equilibriu with the reactants with an equilibriu constant designated K and decoposes to products with rate constant K + actiated coplex products where K Therefore rate of foration of products = [ ] = K [][] [ ] = [][] Copare this ression to the rate law: rate of foration of products = r [][] Hence the rate constant r = K The Gibbs energy for the process is gien by Δ G = RTln (K ) and so K = ( Δ G/RT) Hence rate constant r = ( (Δ H TΔ S)/RT). Hence r = (Δ S/R) ( Δ H/RT) This ression has the sae for as the rrhenius ression. The actiation energy E a relates to Δ H Pre-onential factor = (Δ S/R) The steric factor P can be related to the change in disorder at the transition state Statistical therodynaic approach The actiated coplex can for products if it passes though the transition state The equilibriu constant K can be deried fro statistical echanics q is the partition function for each species ΔE (J ol - )is the difference in internal energy between, and at T= Suppose that a ery loose ibration-lie otion of the actiated coplex with frequency along the reaction coordinate tips it through the transition state. The reaction rate is depends on the frequency of that otion. Rate = [ ] It can be shown that the rate constant r is gien by the Eyring equation the contribution fro the critical ibrational otion has been resoled out fro quantities K and q cancels out fro the equation = oltzann constant h = Planc s constant r K T = h Hence r q = q K = q T q h q ΔE (- ) RT q - ΔE ( RT )
Statistical therodynaic approach Can deterine partition functions q and q fro spectroscopic easureents but transition state has only a transient existence (picoseconds) and so cannot be studied by noral techniques (into the area of fetocheistry) Need to postulate a structure for the actiated coplex and deterine a theoretical alue for q. Coplete calculations are only possible for siple cases, e.g., H + H H + H In ore coplex cases ay use ixture of calculated and eriental paraeters Potential energy surface: 3-D plot of the energy of all possible arrangeents of the atos in an actiated coplex. Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition state. For the siplest case of the reaction of two structureless particles (e.g., atos) with no ibrational energy reacting to for a siple diatoic cluster the ression for r deried fro statistical therodynaics resebles that deried fro collision theory. Collision theory wors.for spherical olecules with no structure Exaple of a potential energy surface Hydrogen ato exchange reaction H + H H C H H + H C tos constrained to be in a straight line (collinear) H H H C Path C goes up along the alley and oer the col (pass or saddle point) between regions (ountains) of higher energy and descends down along the other alley. Paths and go oer uch ore difficult routes through regions of high energy Can inestigate this type of reaction by collision of olecular/ atoic beas with defined energy state. Deterine which energy states (translational and ibrational) lead to the ost rapid reaction. Mol H H Mol H H C Diagra:www.oup.co.u/powerpoint/bt/atins 3
dantages of transition state theory Proides a coplete description of the nature of the reaction including the changes in structure and the distribution of energy through the transition state the origin of the pre-onential factor with units t - that derie fro frequency or elocity the eaning of the actiation energy E a Rather coplex fundaental theory can be ressed in an easily understood pictorial diagra of the transition state - plot of energy s the reaction coordinate The pre-onential factor can be deried a priori fro statistical echanics in siple cases The steric factor P can be understood as related to the change in order of the syste and hence the entropy change at the transition state Can be applied to reactions in gases or liquids llows for the influence of other properties of the syste on the transition state (e.g., solent effects). Disadantage Not easy to estiate fundaental properties of the transition state except for ery siple reactions theoretical estiates of and E a ay be in the right ball-par but still need eriental alues Relating CT paraeters and rrhenius Paraeters. E E H RT U U condensed phases d ln dt RT PV internal energy of actiation olue of actiation T S H hc R RT pre-onential factor = biolecular reaction T S hc R E RT biolecular gas phase reaction V PV n RT RT RT H E RT ideal gases reaction olecularity =, condensed phases, uniolecular gas phase reactions =, biolecular gas phase reactions 4
5 RT E R S c h T R S c h T R E S ln T Pre-onential factor related to entropy of actiation (difference in entropy between reactants and actiated coplex R S c h T PZ collision theory positie P negatie S S P S P steric factor TS less ordered than reactants TS ore ordered than reactants S * lained in ters of changes in translational, rotational and ibrational degrees of freedo on going fro reactants to TS. = olecularity ln CT interpretation of rrhenius Equation.