SF Chemical Kinetics.

Size: px

Start display at page:

Download "SF Chemical Kinetics."

Rosanna Mills
5 years ago
Views:

1 SF Chemical Kinetics. Lecture 5. Microscopic theory of chemical reaction kinetics. Microscopic theories of chemical reaction kinetics. basic aim is to calculate the rate constant for a chemical reaction from first principles using fundamental physics. ny microscopic leel theory of chemical reaction kinetics must result in the deriation of an expression for the rate constant that is consistent with the empirical rrhenius equation. microscopic model should furthermore proide a reasonable interpretation of the pre-exponential factor and the actiation energy E in the rrhenius equation. We will examine two microscopic models for chemical reactions : The collision theory. The actiated complex theory. The main emphasis will be on gas phase bimolecular reactions since reactions in the gas phase are the most simple reaction types. 1

2 References for Microscopic Theory of Reaction Rates. Effect of temperature on reaction rate. urrows et al Chemistry 3, Section 8.7, pp Collision Theory/ ctiated Complex Theory. urrows et al Chemistry 3, Section 8.8, pp tkins, de Paula, Physical Chemistry 9 th edition, Chapter, Reaction Dynamics. Section..1, pp tkins, de Paula, Physical Chemistry 9 th edition, Chapter, Section..4-.5, pp Collision theory of bimolecular gas phase reactions. We focus attention on gas phase reactions and assume that chemical reactiity is due to collisions between molecules. The theoretical approach is based on the kinetic theory of gases. Molecules are assumed to be hard structureless spheres. Hence the model neglects the discrete chemical structure of an indiidual molecule. This assumption is unrealistic. We also assume that no interaction between molecules until contact. Molecular spheres maintain size and shape on collision. Hence the centres cannot come closer than a distance δ gien by the sum of the molecular radii. The reaction rate will depend on two factors : the number of collisions per unit time (the collision frequency) the fraction of collisions haing an energy greater than a certain threshold energy E*.

3 Simple collision theory : quantitatie aspects. k ( g) + ( g) Products Hard sphere reactants Molecular structure and details of internal motion such as ibrations and rotations ignored. Two basic requirements dictate a collision eent. One must hae an, encounter oer a sufficiently short distance to allow reaction to occur. Colliding molecules must hae sufficient energy of the correct type to oercome 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 molecules for reaction. To ealuate the collision frequency we need a mathematical way to define whether or not a collision occurs. The collision cross section σ defines when a collision occurs. σ πδ π ( r + r ) δ r r Effectie collision diameter δ r + r rea σ The collision cross section for two molecules can be regarded to be the area within which the center of the projectile molecule must enter around the target molecule in order for a collision to occur. 3

Maxwell-oltzmann elocity distribution function J.C. Maxwell 1831-1879 F( ) 4π π Gas molecules exhibit a spread or distribution of speeds.

$small fraction of molecules moe with ery low speeds, a small fraction moe with ery high speeds, and the ast majority of molecules moe at intermediate speeds.$ The shape of the Gaussian distribution cure changes as the temperature is raised.

The shape of the Gaussian distribution cure changes as the temperature is raised.

greater proportion of the gas molecules hae high speeds at high temperature than at low temperature. The collision frequency is computed ia the kinetic theory of gases.

4 Maxwell-oltzmann elocity distribution function J.C. Maxwell F( ) 4π π Gas molecules exhibit a spread or distribution of speeds. m k F() T 3/ exp m kt The elocity distribution cure has a ery characteristic shape. small fraction of molecules moe with ery low speeds, a small fraction moe with ery high speeds, and the ast majority of molecules moe at intermediate speeds. The bell shaped cure is called a Gaussian cure and the molecular speeds in an ideal gas sample are Gaussian distributed. The shape of the Gaussian distribution cure changes as the temperature is raised. The maximum of the cure shifts to higher speeds with increasing temperature, and the cure becomes broader as the temperature increases. greater proportion of the gas molecules hae high speeds at high temperature than at low temperature. The collision frequency is computed ia the kinetic theory of gases. We define a collision number (units: m -3 s -1 ) Z. Z π δ n n Mean relatie elocity r Units: m s -1 n j number density of molecule j (units : m -3 ) δ r + r Mean relatie elocity ealuated ia kinetic theory. erage elocity of a gas molecule ( ) F d F( ) 4π π m k T 3/ Maxwell-oltzmann elocity Distribution function exp m kt M distribution of elocities enables us to statistically estimate the spread of molecular l elocities in a gas Some maths! 8kT π m Mass of molecule 4

We now relate the aerage elocity to the mean relatie elocity. If and are different molecules then The aerage relatie elocity is gien by The expression across.

r 8k T π μ + r j 8k T π m j Z π δ n n r Reduced mass mm μ m + m 8kT Z nnσ π μ Zn n Collision frequency factor 1/ For collisions between like molecules The number of collisions per unit time between a

5 We now relate the aerage elocity to the mean relatie elocity. If and are different molecules then The aerage relatie elocity is gien by The expression across. Hence the collision number between unlike molecules can be ealuated. r 8k T π μ + r j 8k T π m j Z π δ n n r Reduced mass mm μ m + m 8kT Z nnσ π μ Zn n Collision frequency factor 1/ For collisions between like molecules The number of collisions per unit time between a single molecule and other Molecules. Z n σ Total number of collisions between like molecules. We diide by to ensure That each, encounter Is not counted twice. Z r 8k T π m Z n 8kT n σ π m 1/ 1/ E * Molecular collision is effectie only if translational energy of reactants is greater than some threshold alue. Fraction of molecules with kinetic energy greater Than some minimum Threshold alue ε* F ε * ( ε ε *) exp > kt 5

The simple collision theory expression for the reaction rate R between unlike molecules.

We introduce molar ariables E* Nε * n n c c N N ogadro dn dc N constant dt dt R dc dt kc c Hence the SCT

1/ k T k N σ exp oth of these π m RT expressions are similar to the z exp RT rrhenius equation.

6 The simple collision theory expression for the reaction rate R between unlike molecules. dn ε * R Znn exp dt kt 8k T Z σ The rate expression for a bimolecular reaction between and. We introduce molar ariables E* Nε * n n c c N N ogadro dn dc N constant dt dt R dc dt kc c Hence the SCT rate expression. The rate constant for bimolecular collisions between like molecules. 1/ k T k N σ exp oth of these π m RT expressions are similar to the z exp RT rrhenius equation. dc R ZNccexp dt RT π μ The bimolecular rate constant for collisions between unlike molecules. Collision Frequency factor 1/ 1/ 8k T k N σ exp π μ RT z exp RT 6

7 We compare the results of SCT with the empirical rrhenius eqn. in order to obtain an interpretation of the actiation energy and pre-exponential factor. 1/ k T E * k N σ exp π m RT z exp RT E k obs exp RT, encounters 1/ 8k T k N σ exp π μ RT z exp RT, encounters z '' k '' N σ 8 SCT predicts π m that the pre-exponential factor should depend on temperature. The threshold energy and the actiation energy can also be d ln k compared. dt obs ctiation energy exhibits a weak T dependence. T rrhenius E RT obs z ' 8k ' N σ π μ T d ln k E * + RT dt RT E E * RT + Pre-exponential factor SCT E E * SCT : a summary. The major problem with SCT is that the threshold energy E* is ery difficult to ealuate from first principles. The predictions of the collision theory can be critically ealuated by comparing the experimental pre-exponential factor with that computed using SCT. We define the steric factor P as the ratio between P the experimental and calculated factors. exp calc We can incorporate P into the SCT expression for the rate constant. For many gas phase reactions k Pz exp RT P is considerably less than unity. Typically SCT will predict that calc will be in k Pz the region Lmol -1 s -1 exp regardless of RT the chemical nature of the reactants and products. What has gone wrong? The SCT assumption of hard sphere collision neglects the important fact that molecules possess an internal structure. It also neglects the fact that the relatie orientation of the colliding molecules will be important in determining whether a collision will lead to reaction. We need a better theory that takes molecular structure into account. The actiated complex theory does just that. 7

to compute P from molecular parameters 1/ No way to compute E* from first principles. k T k N Theory quantitatie predictie.

8 Summary of SCT., encounters 1/ 8k T k N σ exp π μ RT z exp RT, encounters Transport property Energy criterion k Pz exp RT k Pz exp RT Steric factor (Orientation requirement) Weaknesses: No way to compute P from molecular parameters 1/ No way to compute E* from first principles. k T k N Theory quantitatie predictie. σ exp not tat or π m RT Strengths: Qualitatiely consistent with obseration z exp (rrhenius equation). RT Proides plausible connection between microscopic molecular properties and macroscopic reaction rates. Proides useful guide to upper limits for rate constant k. Henry Eyring Deeloped (in 1935) the Transition State Theory (TST) or ctiated Complex Theory (CT) of Chemical Kinetics. 8

9 Potential energy surface Can be constructed from experimental measurements or from Molecular Orbital calculations, semi-empirical methods, Various trajectories through the potential energy surface 9

10 Potential energy hypersurface for chemical reaction between atom and diatomic molecule. * [ C] C + C + Reading reaction progress on PE hypersurface. E E ΔU 1

11 Transition state theory (TST) or actiated complex theory (CT). In a reaction step as the reactant molecules and come together they distort and begin to share, exchange or discard atoms. They form a loose structure of high h potential ti energy called the actiated complex that is poised to pass on to products or collapse back to reactants C + D. The peak energy occurs at the transition state. The energy difference from the ground state is the actiation energy E a of the reaction step. The potential energy falls as the atoms 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. Energy ctiated complex Transition state E a E a + Reaction coordinate C + D Transition state theory The theory attempts to explain the size of the rate constant k r and its temperature dependence from the actual progress of the reaction (reaction coordinate). The progress along the reaction coordinate can be considered in terms of the approach and then reaction of an H atom to an F molecule When far apart the potential energy is the sum of the alues for H and F When close enough their orbitals start to oerlap bond starts to form between H and the closer F atom H F F The F F bond starts to lengthen s H becomes closer still the H F bond becomes shorter and stronger and the F F F bond becomes longer and weaker The atoms enter the region of the actiated complex When the three atoms reach the point of maximum potential energy (the transition state) a further infinitesimal compression of the H F bond and stretch of the F F bond takes the complex through the transition state. 11

12 Thermodynamic approach Suppose that the actiated complex is in equilibrium with the reactants with an equilibrium constant designated K and decomposes to products with rate constant k K k [ ] + actiated complex products where K [][] Therefore rate of formation of products k [ ] k K [][] Compare this expression to the rate law: rate of formation of products k r [][] Hence the rate constant k r k K The Gibbs energy for the process is gien by G RTln (K ) and so K exp( G/RT) Hence rate constant k ( r k exp (( H T S)/RT). ) Hence k r k exp( S/R) exp( H/RT) This expression has the same form as the rrhenius expression. The actiation energy E a relates to H Pre-exponential factor k exp( S/R) The steric factor P can be related to the change in disorder at the transition state Statistical thermodynamic approach The actiated complex can form products if it passes though the transition state ΔE The equilibrium constant K can be deried from statistical mechanics K exp(- ) q is the partition function for each species qq RT E (kj mol -1 )is the difference in internal energy between, and at T Suppose that a ery loose ibration-like motion of the actiated complex with frequency along the reaction coordinate tips it through the transition state. The reaction rate is depends on the frequency of that motion. Rate [ ] It can be shown that the rate constant k r is gien by the Eyring equation the contribution from the critical ibrational motion has been resoled out from quantities K and q cancels out from the equation k oltzmann constant h Planck s constant k r kt h Hence k r K q kt q h q q - ΔE exp( RT ) 1

Statistical thermodynamic approach Can determine partition functions q and q from spectroscopic measurements but transition state has only a transient existence (picoseconds) and so cannot be studied

13 Statistical thermodynamic approach Can determine partition functions q and q from spectroscopic measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry) Need to postulate t a structure tu for the actiated t complex and determine a theoretical alue for q. Complete calculations are only possible for simple cases, e.g., H + H H + H In more complex cases may use mixture of calculated and experimental parameters Potential energy surface: 3-D plot of the energy of all possible arrangements of the atoms in an actiated complex. Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition state. For the simplest case of the reaction of two structureless particles (e.g., atoms) with no ibrational energy reacting to form a simple diatomic cluster the expression for k r deried from statistical thermodynamics resembles that deried from collision theory. Collision theory works.for spherical molecules with no structure Example of a potential energy surface Hydrogen atom exchange reaction H + H H C H H + H C toms 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 (mountains) of higher energy and descends down along the other alley. Paths and go oer much more difficult routes through regions of high energy Can inestigate this type of reaction by collision of molecular/ atomic beams with defined energy state. Determine which energy states (translational and ibrational) lead to the most rapid reaction. Mol H H Mol H H C Diagram: 13

14 dantages of transition state theory Proides a complete 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-exponential factor with units t -1 that derie from frequency or elocity the meaning of the actiation energy E a Rather complex fundamental theory can be expressed in an easily understood pictorial diagram of the transition state - plot of energy s the reaction coordinate The pre-exponential factor can be deried a priori from statistical mechanics in simple cases The steric factor P can be understood as related to the change in order of the system 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 system on the transition state t (e.g., solent effects). Disadantage Not easy to estimate fundamental properties of the transition state except for ery simple reactions theoretical estimates of and E a may be in the right ball-park but still need experimental alues Quantitatie ctiated Complex Theory (CT) (g) + (g) X * ν * Products * K c imolecular reaction Equilibrium constant for reactant/transition state transformation. K c x cc R ν x Reaction molecularity m : number of molecules which come together to form the actiated complex. m 1 : unimolecular reaction m : bimolecular reaction. x K c ab frequency of decomposition of actiated complex asic assumption : actiated complex X * is treated as a thermodynamic quantity in equilibrium with the reactants. Reaction rate R depends on the frequency of decomposition and concentration of actiated complexes. R ν K c ab rate constant k bimolecular reaction R kc c ν Kc This fundamental expression must now be ealuated. 14

15 ideal gas or for reactions in solution : K C K ( c Δn* ) Can be proed rigorously From Statistical Mechanics. We assume that the TS decomposes with a frequency gien by: k T ν h h Planck s constant 6.63x1-34 Js k oltzmann constant 1.38x1-3 JK -1 For T 98 K ν 6 1 s 1 1 This is of the correct order of magnitude for a molecular ibration frequency. thermodynamic equilibrium constant K k ν K C ν c K C Not all transition states go and form products. K c 1 moll -1 Δn Change in # moles for reactant /TS transformation 1 1 k has units Lmol -1 s -1 K k κ ν KC κ ν c kt κ K hc κ transmission coefficient < κ < 1 We relate K* to the Gibbs energy of actiation ΔG* ΔG RT ln K kt ΔG k κ exp hc RT K ΔG exp RT Eyring equation : fundamental CT result. Δ k T S ΔH k κ exp exp hc R RT ΔG ΔH enthalpy of actiation TΔS entropy of actiation E k exp RT We obtain a useful interpretation for actiation energy and pre-exponential factor. 15

16 Relating CT parameters and rrhenius Parameters. E E ΔH RT ΔU ΔU d ln k dt + RT + PΔV Δ k T S ΔH k κ exp exp hc R RT internal energy of actiation pre-exponential factor k T ΔS k κ exp + hc R E exp RT olume of actiation bimolecular gas condensed phases m bimolecular reaction phase reaction ΔV PΔV Δn RT ( 1 m) RT RT ΔH E mrt ideal gases reaction molecularity m 1, condensed phases, unimolecular gas phase reactions m, bimolecular gas phase reactions CT interpretation of rrhenius Equation. k m k T κ h ΔS exp m + exp m1 ( c ) R RT E m molecularity ΔS * explained in terms of changes in translational, l rotational and ibrational degrees of freedom on going from reactants to TS. kt κ m h( c ) 1 Pre-exponential factor related to entropy of actiation (difference in entropy between reactants and actiated complex ΔS exp m + R ln ln k E S R 1 T kt PZ κ m h( c ) collision theory 1 ΔS exp m + R steric factor P 1 P < 1 P > 1 ΔS ΔS ΔS negatie positie TS more ordered than reactants TS less ordered than reactants 16

SF Chemical Kinetics.

SF Chemical Kinetics. 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