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1 H. Scott Fogler Chapter 3 1/4/06 Overview Collision Theory Professional Reference Shelf A. Collision Theory In Chapter 3, we presented a number of rate laws that depended on both concentration and temperature. For the elementary reaction the elementary rate law is A + B C + D r A = kc A C B = Ae E RT C A C B We want to provide at least a qualitative understanding of why the rate law takes this form. We will first develop the collision rate, using collision theory for hard spheres of cross section S r, AB. When all collisions occur with the same relative velocity, U R, the number of collisions between A and B molecules, Z AB, is Z AB = S r U R C A C B [collisions/s/molecule] Next, we will consider a distribution of relative velocities and only consider those collisions that have an energy of E A or greater in order to react to show where r A = Ae E A RT C A C B 8k A = B T AB ) µ AB 1 N Avo with σ AB = collision radius, k B = Boltzmann s constant, µ AB = reduced mass, T = temperature, and N Avo = Avogadro s number. To obtain an estimate of E A, we use the Polyani Equation E A = E A o + P H Rx Where ΔH Rx is the heat of reaction and E o A and γ P are the Polyani Parameters. With these equations for A and E A we can make a first approximation to the rate law parameters without going to the lab. 1

2 HOT BUTTONS I. Fundamentals of Collision Theory II. Shortcomings of Collision Theory III. Modifications of Collision Theory A. Distribution of Velocities B. Collisions That Result in Reaction 1. Model 1 Pr = 0 or 1. Model Pr = 0 or Pr = E E A )/E IV. Other Definitions of Activation Energy A. Tolman s Theorem E a = E * B. Fowler and Guggenheim C. Energy Barrier V. Estimation of Activation Energy from the Polyani Equation A. Polyani Equation B. Marcus Extension of the Polyani Equation C. Blowers-Masel Relation VI. Closure References for Collision Theory, Transition State Theory, and Molecular Dynamics P. Atkins, Physical Chemistry, 6th ed. New York: Freeman, 1998) P. Atkins, Physical Chemistry, 5th ed. New York: Freeman, 1994). G. D. Billing and K. V. Mikkelsen, Introduction to Molecular Dynamics and Chemical Kinetics New York: Wiley, 1996). P.W. Atkins, The Elements of Physical Chemistry, nd ed. Oxford: Oxford Press, 1996). K. J. Laidler, Chemical Kinetics, 3rd ed. New York: Harper Collins, 1987). G. Odian, Principles of Polymerization, 3rd ed. New York: Wiley 1991). R. I. Masel, Chemical Kinetics and Catalysis New York: Wiley Interscience, 001). As a shorthand notation, we will use the following references nomenclature: A6p701 means Atkins, P. W., Physical Chemistry, 6th ed. 1998) page 701. L3p08 means Laidler, K. J., Chemical Kinetics, 3rd,ed. 1987) page 08. This nomenclature means that if you want background on the principle, topic, postulate, or equation being discussed, go to the specified page of the referenced text. I. FUNDAMENTALS OF COLLISION THEORY The objective of this development is to give the reader insight into why the rate laws depend on the concentration of the reacting species i.e., r A = kc A C B ) and why the temperature dependence is in the form of the Arrhenius law, k=ae /RT. To achieve this goal, we consider the reaction of two molecules in the gas phase A + B! C + D We will model these molecules as rigid spheres.

3 A. Rigid Spheres Species A and B are modeled as rigid spheres of radius σ A and σ B, respectively. A B! A! B Figure R3.A-1 Schematic of molecules A and B. We shall define our coordinate system such that molecule B is stationary wrt molecule A so that molecule A moves towards molecule B with a relative velocity, U R. Molecule A moves through space to sweep out a collision volume with a collision cross section,! AB, illustrated by the cylinder shown in Figure R3.A-.!AB!AB The collision radius is! AB. UR Figure R3.A- Schematic of collision cross-section.! AB =! A +! B If the center of a B molecule comes within a distance of! AB of the center of the A molecule they will collide. The collision cross section of rigid spheres is S r =! AB. As a first approximation, we shall consider S r constant. This constraint will be relaxed when we consider a distribution of relative velocities. The relative velocity between gas molecules A and B is U R. where U R = 8k BT µ AB k B = Boltzmann s constant = J/K/molecule = kg m /s /K/molecule 1 R3.A-1) This equation is given in most physical chemistry books, e.g., see Moore, W. J. Physical Chemistry, nd Ed., Englewood Cliffs, NJ: Prentice Hall, p

4 m A = mass of a molecule of species A gm) m B = mass of a molecule of species B gm) m µ AB = reduced mass = A m B g), [Let µ µ m A + m AB ] B M A = Molecular weight of A Daltons) N Avo = Avogadro s number 6.0 molecules/mol R = Ideal gas constant J/mol K = kg m /s /mol/k We note that R = N Avo k B and M A = N Avo m A, therefore we can write the ratio k B /µ AB ) as k B R = R3.A-) µ AB M A M B M A + M B An order of magnitude of the relative velocity at 300 K is U R! 3000 km hr, i.e., ten times the speed of an Indianapolis 500 Formula 1 car. The collision diameter and velocities at 0 C are given in Table R3.A-1. Molecule Table R3.A-1 Molecular Diameters Average Velocity, meters/second) Molecular Diameter Å) H CO Xe He 100. N O H O C H C 6 H CH NH H S CO N O NO Consider a molecule A moving in space. In a time Δt, the volume ΔV swept out by a molecule of A is Courtesy of J. F. O Hanlon, A User s Guide to Vacuum Technology New York: Wiley, 1980). 4

5 !l !V = U R!t ) AB!V A Figure R3.A-3 Volume swept out by molecule A in time Δt. The bends in the volume represent that even though molecule A may change directions upon collision the volume sweep out is the same. The number of collisions that will take place will be equal to the number of B molecules, ΔVC B, that are in the volume swept out by the A molecule: [ C B!V = No. of B molecules in!v] [ ] rather than [moles/dm3 ] where C B is in molecules dm 3 In a time Δt, the number of collisions of this one A molecule with many B molecules is U R C B AB t. The number of collisions of this one A molecule with all the B molecules per unit time is Z 1A B =! AB C B U R R3.A-3) However, we have many A molecules present at a concentration, C A, molecule/dm 3 ). Adding up the collisions of all the A molecules per unit volume, C A, then the number of collisions Z AB of all the A molecules with all B molecules per time per unit volume is S 67 r 8 Z AB = AB U R C A C B = S r U R C A C B R3.A-4) Where S r is the collision cross section Å). Substituting for S r and U R Z 8k AB = B T AB ) µ 1 C A C B [molecules/time/volume] R3.A-5) If we assume all collisions result in reactions, then 1 r A = Z 8k AB = B T AB * C A C B [molecules/time/volume] R3.A-6) µ ) Multiplying and dividing by Avogadrós number, N Avo, we can put our equation for the rate of reaction in terms of the number of moles/time/vol. r A N 1 Avo 3 r A 8k N Avo = )* B T AB )µ 1 C A N Avo C A C B N Avo C B N Avo R3.A-7) 5

6 1 8k r A = B T AB * N Avo µ ) 44 3 where A is the frequency factor 8k A = B T AB ) A µ AB C A C B [moles/time/volume] R3.A-8) 1 N Avo R3.A-9) r A = AC A C B R3.A-10) Example Calculate the frequency factor A for the reaction at 73K. Additional information: Using the values in Table R3.A-1 Collision Radii Hydrogen H H + O OH + O σ H =.74 Å/4 = 0.68Å = 0.68 x m Oxygen O O = 3.1 Å 1.55Å = 1.5 x m R = 8.31J mol K = kg m s K mol Solution A = S r U R N Avo R3.A-E-1) A = 8k AB U R N Avo = B T AB ) µ The relative velocity is U R = 8k B T 1 µ 1 N Avo R3.A-9) R3.A-1) Calculate the ratio k B /µ AB Let µ µ AB ) k B µ = R M A M B M A + M B = kg m s K mol 1g mol) 3g mol) 1g mol + 3g mol = 8.314kg m s K mol 0.97 g mol ) 1kg 1000g 6

7 Calculate the relative velocity k B µ = 8571m s K U R = 8 ) 73K) 8571) m s K S r = AB 3.14 Calculate the frequency factor A A = m molecule 1 = 441m s = Å s R3.A-E-) = [ A + B ] = [ m m] = m molecule [ 441m s] [ molecule mol] R3.A-E-3) A = m 3 mol s = dm 3 mol s A = m 3 mol s 1mol Å ) molecule m A = The value reported in Masel from Wesley is A = Close, but no cigar, as Groucho Marx would say. 3 R3.A-E-4) Å) 3 molecule s R3.A-E-5) Å) 3 molecule s For many simple reaction molecules, the calculated frequency factor A calc, is in good agreement with experiment. For other reactions, A calc, can be an order of magnitude too high or too low. In general, collision theory tends to overpredict the frequency factor A 10 8 dm 3 mol s < A calc <1011 dm 3 mol s Terms of cubic angstroms per molecule per second the frequency factor is 10 1 Å 3 molecule s < A calc <10 15 Å 3 molecule s There are a couple of things that are troubling about the rate of reaction given by Equation R3.A-10), i.e. M1p367. 7

8 r A = AC A C B R3.A-10) First and most obvious is the temperature dependence. A is proportional to the square root of temperature and, therefore, is r A : r A ~ A ~ T However we know that the temperature dependence of the rate of chemical reaction on temperature is given by the Arrhenius equation or r A = Ae E RT C A C B R3.A-11) E RT k = Ae R3.A-1) Next, we will discuss this shortcoming of collision theory, along with the assumption that all collisions result in reaction. II. SHORTCOMINGS OF COLLISION THEORY A. The collision theory outlined above does not account for orientation of the collision, front-to-back and along the line-of-centers. That is, molecules need to collide in the correct orientation for reaction to occur. Figure R3.A-4 shows molecules colliding whose centers are offset by a distance b. A b b = impact parameter B Figure R3.A-4 Grazing collisions. B. Collision theory does not explain activation barriers. Activation barriers occur because bonds need to be stretched or distorted in order to react and these processes require energy. Molecules must overcome electron-electron repulsion in order to come close together C. The collision theory does not explain the observed temperature dependence given by Arrhenius equation k = Ae E RT D. Collision theory assumes all A molecules have the same relative velocity, the average one. U R = 8k 1 BT R3.A-1) µ AB However, there is a distribution of velocities fu,t). One distribution most used is the Maxwell-Boltzmann distribution. Masel, 1p 8

9 III. MODIFICATIONS OF COLLISION THEORY We are now going to account for the fact that we have 1) a distribution of relative velocities U R and ) that not all collisions only those collisions with an energy E A or greater result in a reaction--the goal is to arrive at k = Ae E A RT A. Distribution of Velocities We will use the Maxwell-Boltzmann Distribution of Molecular Velocities A6p.6). For a species of mass m, the Maxwell distribution of velocities relative velocities) is f U,T)dU = 4 m k B T 3 e )mu k B T U du R3.A-13) A plot of the distribution function, fu,t), is shown as a function of U in Figure R3.A-5. T1 T!>!T1 f U,T) T Figure R3.A-5 Maxwell-Boltzmann distribution of velocities. Replacing m by the reduced mass µ of two molecules A and B f U,T)dU = 4 µ k B T 3 U e )µu k B T U du The term on the left side of Equation R3.A-13), [fu,t)du], is the fraction of molecules with velocities between U and U + du). Recall from Equation R3.A- 4) that the number of A B collisions for a reaction cross section S r is Z AB = S r U)U C 13 A k U ) C B R3.A-14) except now the collision cross-section is a function of the relative velocity. Note we have written the collision cross section S r as a function of velocity U: S r U). Why does the velocity enter into reaction cross section, S r? Because not all collisions are head on, and those that are not will not react if the energy U /µ) is not sufficiently high. Consequently, this functionality, S r = S r U), is reasonable because if two molecules collide with a very very low relative velocity it is unlikely that such a small transfer of kinetic energy is likely to activate the internal vibrations of the molecule to cause the breaking of bonds. On the other hand, for collisions with large relative velocities most collisions will result in reaction. 9

10 We now let k U) be the specific reaction rate for a collision and reaction of A-B molecules with a velocity U. [ ] R3.A-15) k U) = S r U)U m 3 molecule s Equation R3.A-15) will give the specific reaction rate and hence the reaction rate for only those collisions with velocity U. We need to sum up the collisions of all velocities. We will use the Maxwell-Boltzmann distribution for fu,t) and integrate over all relative velocities. ) = k k T U)fU,T)dU = f U,T) S r U 0 0 ) UdU R3.A-16) Maxwell distribution function of velocities for the A/B pair of reduced mass µ AB is ) = 4 f U,T Combining Equations 16) and 17) ) = S r k T µ k B T µ U 4 0 * k B T) 3 U 3 U ) µu k e B T R3.A-17) + µu k e B T du R3.A-18) For brevity, we let S r =S r U), we will now express the distribution function in terms of the translational energy ε T. We are now going to express the equation for k T) in terms of kinetic energy rather than velocity. Relating the differential translational kinetic energy, ε, to the velocity U: Multiplying and dividing by µ and hence, the reaction rate ) = 4 k T Simplifying µ k B T µ = 4 k B T! t = µu and µ, we obtain d t = µ UdU 3 ) * S 0 r µ 3 µu e + ) * S µ 0 r + t e µu 1 k B T 1, + t k B T µ µudu 1 3 d+ t dµu 1 3 d, t p185, A5p36 10

11 ) = k T 8 µ k B T ) 3 1 * + t e ) S 0 r, + t k B T d+ t [ m 3 s molecule] R3.A-19) ) = k T 8 µ k B T ) 3 1 * + S r ) t ) t e,) t k B T d) t Multiplying and dividing by k B T and noting t k B T) = E RT), we obtain ) = 8k BT k T µ 1 * ) 0 0 ) S r E +E RT Ee RT, de /. 1 R3.A-0) - RT0 Again, recall the tilde, e.g., k T), denotes that the specific reaction rate is per molecule dm 3 /molecule/s). The only thing left to do is to specify the reaction cross-section, S r E), as a function of kinetic energy E for the A/B pair of molecules. B. Collisions that Result in Reaction We now modify the hard sphere collision cross section to account for the fact that not all collisions result in reaction. Now we define S r to be the reaction cross section defined as S r = P r! AB where P r is the probability of reaction. In the first model we say the probability is either 0 or 1. In the second model P r varies from 0 to 1 continuously. We will now insert each of these modules into Equation R3.A-0). B.1 Model 1 In this model, we say only those hard collisions that have kinetic energy E A or greater will react. Let E ε t. That is, below this energy, E A, the molecules do not have sufficient energy to react so the reaction cross section is zero, S r =0. Above this kinetic energy all the molecules that collide react and the reaction crosssection is S r = AB P r = 0 P r =1 S r [ E,T] = 0 for E < E A S r [ E,T] = AB for E E A R3.A-1) R3.A-) 11

12 Figure R3.A-6 Reaction cross section for Model 1. Integrating Equation R3.A-0) by parts for the conditions given by Equations R3.A-1) and R3.A-) we obtain k = 8 k B T 1 ) 1+ E, A µ * + RT-. e/e A RT 0 AB R3.A-3) Click Back 1) ) = k T = U R AB 8 µ k B T ) E A RT ) e*e A RT Derive Click Back 1) * + T )+ T e,+ T k B T d+ T ) S o r S r = 0 < * ) = k T 8 µ k B T S r = AB ) 3 1 ) AB * +, * ** t e -* T k B T d* T 8 = AB µ k B T ) 3 ) ) 1 k B T) + *, T e -* T k B T d* T E* k B T k B T 8k = B T AB ) µ 1 * E + e,e RT de E A RT RT X = T k B T, dx = d T k B T, X = E RT, T k B T = E RT 1

13 Click Back 1 cont d) udv = uv vdu ) = AB k T 8k B T ) µ 1 * + { X e E A RT 1,X dx 3 =,Xe,X- / u dv. * / E A RT + *, e,x dx E A RT 8k = B T AB ) µ k T 1 E A 8k = B T AB ) µ + RT e *E A RT + e *E A RT. - 0, / 1 * E A E A RT > 1 ) = N Avo k T) = AB, + RT / e 0E A RT. 8k B T ) µ = A E A RT S r = P r AB 1 N Avo E A RT Over predicts the frequency factor Generally, E A >>1, so RT E A k = U R AB 1 4 RT 4 3 A e E A RT Converting k to a per mole basis rather than a per molecular basis we have 1 A = E A *8k ) B T AB N Avo RT µ 1444 AB k = A e E A RT = E A RT AeE A RT A = E A RT A We have good news and bad news. This model gives the correct temperature dependence but predicted frequency factor A is even greater than A given by Equation R3.A-9) which itself is often too large) by a factor E A /RT). So we have solved one problem, the correct temperature dependence, but created another problem, too large a frequency factor. Let s try Model. 13

14 B. Model In this model, we again assume that the colliding molecules must have an energy E A or greater to react. However, we now assume that only the kinetic energy directed along the line of centers E << is important. So below E A the reaction cross section is zero, S r =0. The kinetic energy of approach of A toward B with a velocity U R is E = µ AB U R. However, this model assumes that only ) the kinetic energy directly along the line of centers contributes to the reaction. Click Back ) Here, as E increases above E A the number of collisions that result in reaction increases. The probability for a reaction to occur is and Click Back ) P r = E E A E for E > E A R3.A-4) S r E,T) = 0 for E E A E E S r = A ) AB E for E > E A R3.A-5) R3.A-6) A U R b U LC B The impact parameter, b, is the off-set distance of the centers as they approach one another. The velocity component along the lines of centers, U LC, can be obtained by resolving the approach velocity into components. At the point of collision, the center of B is within the distance σ AB. U R b! AB Courtesy of J. I. Steinfeld, J. S. Francisco, and W. L. Hayes, Chemical Kinetics and Dynamics, Englewood Cliffs NJ: Prentice Hall, 1989, p.50); Mp

15 Click Back cont d) The energy along the line of centers can be developed by a simple geometry argument sin = b 1) AB The component of velocity along the line of centers The kinetic energy along the line of centers is U LC = U R cos θ ) E LC = U LC = U R cos = E cos 3) µ AB µ AB E LC = E[ 1 sin ] = E 1 b * 4) AB ) The minimum energy along the line of centers necessary for a reaction to take place, E A, corresponds to a critical value of the impact parameter, b crit. In fact, this is a way of defining the impact parameter and corresponding reaction cross section S r = b crit 5) Substituting for E A and b crit in Equation 4) Solving for b crit E A = E 1 b crit ) 6) b cr AB = AB 1 E A ) 7) E The reaction cross section for energies of approach, E > E A, is S r = b crit = AB The complete reaction cross section for all energies E is S r = 0 S r = AB 1 E A * 8) E ) E E A 1 E ) A + E > E A E * A plot of the reaction cross section as a function of the kinetic energy of approach E = µ AB U R 9) 10) 15

16 is shown in Figure R3.A-7. AB Model S r Model 1 Recalling Equation 0) E A E Figure R3.A-7 Reaction cross section for Models 1 and. ) = 8k B T k T Substituting for S r in Model Integrating gives Click Back 3) k T k T ) = 8k B T µ k T µ 1 1 ) = AB * ES r E)e )E RT de + R3.A-0) 0 RT ) + ) AB E * E A )e *E RT de, R3.A-7) E A RT 8k B T ) µ E E A P r = 0 S r = 0 E > E A P r =1 * ) = 8 = AB µ k B T 8 µ k B T ) 3 ) 3 ) ) 1 ) AB ) 1 e *E A RT S r = AB 1 E * A ), E +, 1* +* / e *+ kt d+ E A kt) 3 +, * e -* RT + d* - ** e **, kt ** -* kt Derive Click Back 3) d* kt 16

17 Click Back 3 cont d) k B T = E RT 8k = B T AB ) µ = k T Multiplying both sides by N Avo ) = AB Multiplying by Avogodro s number 1 E A RT + 1* E A ) e RT 8k B T ) µ k T) = Ae E A RT ) = k T k T)N Avo 1 e *E A RT *E RT Bingo! r A = AB U R N Avo e E A RT C A C B R3.A-8) This is similar to the equation for hard sphere collisions except for the term e E A RT ) = AB k T 8k B T ) µ 1 N Avo e *E A RT R3.A-9) This equation gives the correct Arrhenius dependence and the correct order of magnitude for A. r A = Ae E RT C A C B R3.A-11) Effect of Temperature on Fraction of Molecules Having Sufficient Energy to React Now we will manipulate and plot the distribution function to obtain a qualitative understanding of how temperature increases the number of reacting molecules. Figure R3.A-8 shows a plot of the distribution function given by Equation R3.A-17) after it has been converted to an energy distribution. We can write the Maxwell-Boltzmann distribution of velocities 3 µ f U,T)dU = 4 U ) µu k e B T du k B T in terms of energy by letting = µu to obtain 17

18 Derive Click back) 3 µ f U,T)dU = 4 U ) µu k e B T du k B T f t,t)d = ) 3 1 e 1 k B T k B T d t where fε,t) dε is the fraction of molecules with kinetic energies between ε and ε+dε). We could further multiply and divide by k B T. Recalling k B T = E RT f,t)de = 1 ) k B T f E,T)dE = E 1 RT f E,T) = E 1 RT 1 1 e *E RT d k B T e )E RT de RT 1 e )E RT RT f!, ) Fraction of molecules with energy! or greater!! By letting X = E, we could have put the distribution in dimensionless form RT f X,T)dX = 1 X1 e X dx fx,t) dx is the fraction of molecules that have energy ratios between E RT and E + de RT f E,T)dE = E 1 RT Fraction of collisions that have E A or above 1 e )E RT de RT Derive = * ) E A 4 RT) E 1 e +E RT de

19 This integral is shown by the shaded area on Figure R3.A-8. Figure R3.A-8 Boltzmann distribution of energies. As we just saw, only those collisions that have an energy E A or greater result in reaction. We see from Figure R3.A-10 that the higher the temperature the greater number of collision result in reaction. However, this Equations R3.A-9) and R3.A-11) cannot be used to calculate A for a number pf reactions because of steric factors and because the molecular orientation upon collision need to be considered. For example, consider a collision in which the oxygen atom, O, hits the middle carbon in the reaction to form the free radical on the middle carbon atom CH 3 C HCH 3 CH O CH 3 CH 3 C H CH 3 + OH CH Otherwise, if it hits anywhere else say the end carbon) CH 3 C H CH 3 will not be formed O CH 3 CH CH 3 CH 3 C HCH 3 CH 3 CH CH + OH Consequently, collision theory predicts a rate orders of magnitude too high for the formation of CH 3 C HCH 3. IV. OTHER DEFINITIONS OF ACTIVATION ENERGY We will only state other definitions in passing, except for the energy barrier concept, which will be discussed in transition state theory. M1p

20 ) A. Tolman s Theorem E a = E * E *! e * Average Energy of Molecules Average Energy! a = Undergoing of Colloiding + 1 Molecules kt Reaction The average transitional energy of a reactant molecule is 3 kt. B. Fowler and Guggenheim C. Energy Barrier Average Energy of Molecules Average Energy! a = Undergoing of Reactant Reaction Molecules The energy barrier concept is discussed in transition state theory, Ch3, Profession Reference Shelf B. A+ BC AB+ C A B C E E A A, BC AB, C Products Figure R3.A-9 Reaction coordinate diagram. For simple reactions, the energy, E A, can be estimated from computational chemistry programs such as Cerius or Spartan, as the heat of reaction between reactants and the transition state E A = H ABC H o o A H BC V. ESTIMATION OF ACTIVATION ENERGY FROM THE POLYANI EQUATION A. Polyani Equation The Polyani equation correlates activation energy with heat of reaction. This correlation E A = P H Rx + c R3.A-33) works well for families of reactions. For the reactions R + H R RH + R where R = OH, H, CH3, the relationship is shown in Figure R3.A-10. 0

21 Figure R3.A-10 Experimental correlation of E A and ΔH Rx. Courtesy of R. I. Masel, Chemical Kinetics and Catalysis New York: Wiley Interscience, 001). For this family of reactions E A =1 kcal mol H R For example, when the exothermic heat of reaction is The corresponding activation energy is H Rx = 10 kcal mol E a = 7cal mol R3.A-34) To develop the Polyani equation, we consider the elementary exchange reaction A+ BC AB+ C We consider the superposition of two attraction/repulsion potentials, V BC and V AB, similar to the Lennard-Jones 6-1 potential. For the molecules BC, the Lennard-Jones potential is * 1 r V BC = 4 0 LJ ) r 6 -, 0 / R3.A-35) +, r BC r BC. / where r BC = distance between molecules atoms) B and C. In addition to the Lennard-Jones 6-1 model, another model often used is the Morse potential, which has a similar shape [ ] R3.A-36) V BC = D e! r BC!r 0 )! e! r BC!r 0 ) When the molecules are far apart the potential V i.e., Energy) is zero. As they move closer together, they become attracted to one another and the potential energy reaches a minimum. As they are brought closer together, the BC molecules After R. I. Masel Loc cit). 1

22 begin to repel each other and the potential increases. Recall that the attractive force between the B and C molecules is F BC = dv BC dx R3.A-37) The attractive forces between the B C molecules are shown in Figure R3.A-11. A potential similar to atoms B and C can be drawn for the atoms A and B. The F shown in figures a and b represents the attractive force between the molecules as they move in the distances shown by the arrows. That is the attractive force increase as we move toward the well r o ) from both directions, r AB >r o and r AB <r o. BC!! AB V BC V AB kj molecule F BC! LJ F BC kj molecule F AB r 0 r BC r AB a) Figure R3.A-11 Potentials Morse or Lennard-Jones). r BC r AB r 0 b) One can also view the reaction coordinate as a variation of the BC distance for a fixed AC distance, l l l l A B C A BC AB C r AB r BC For a fixed AC distance as B moves away from C, the distance of separation of B from C, r BC increases as N moves closer to A. See point in Figure R3.A-11. As r BC increases, r AB decreases and the AB energy first decreases and then increases as the AB molecules become close. Likewise, as B moves away from A and toward C, similar energy relationships are found. E.g., as B moves toward C from A, the energy first decreases due to attraction and then increases due to repulsion of the AB molecules as they come close together at point in Figure R3.A-11. The overlapping Morse potentials are shown in Figure R3.A-11. We now superimpose the potentials for AB and BC to form Figure R3.A-1.

23 Reference BC S S 1 AB Energy E a E 1R E P!H Rx =E P E 1P r BC r 1 r* BC = r AB * r* r Reaction Coordinate r AB Figure R3.A-1 Overlap of potentials Morse or Lennard-Jones). Let S 1 be the slope of the BC line between r 1 and r BC =r AB. Starting at E 1R at r 1, the energy E 1 at a separation distance of r BC from r 1 can be calculated from the product of the slope S 1 and the distance from E 1R. The energy, E 1, of the BC molecule at any position r BC relative to r 1 is E 1 = E 1R +S 1 r BC r 1 ) R3.A-38) e.g., E 1 = 50kJ + 10kJ nm [ 5 ]nm = 0kJ ) Let S be the slope of the AB line between r and r BC =r AB. Similarly for AB, starting on the product side at E P, the energy E at any position r AB relative to r is At the height of the barrier Substituting for E 1 and E E = E P +S r AB r ) R3.A-39) * e.g., E = 80kJ + 0kJ -, ) 7 10)nm = 0kJ/ + nm. E 1 = E at r BC = r AB R3.A-40) Rearranging E 1R +S 1 r BC r 1 S 1 r BC r 1 ) = E P +S r AB ) = E P E 1R ) +S r AB r H Rx r ) R3.A-41) ) R3.A-4) 3

24 yields r BC = r AB = r S 1 r r 1 ) = H Rx +S r r ) Solving for r* and substituting back into the Equation E a = E 1 E 1R = S 1 r r 1 E a = E a! + P H Rx ) R3.A-43) Derive click back) see p.19) R3.A-44) Equation 8) gives the Polyani correlation relating activation energy and heat of reaction. Values of E a o and γ P for different reactions can be found in Table 5.4 on page 54 of Masel. One of the more common correlations for exothermic and endothermic reactions is given on page 73 of Laidler. 1) Exothermic Reactions E a = H Rx If H Rx = 100kJ mol, then E a = ) 100 kj mol ) E a = 3.1kJ mol ) kj mol R3.A-45) ) Endothermic Reactions E a = H Rx ) kj mol R3.A-46) E a = ) 100 kj mol) =19 kj mol = 9 kcal mol Also see R. Masel Chemical Kinetics and catalysis, p603 to calculate the activation energy from the heat of reaction. Szabo proposed an extension of Polyani equation E a = D j Breaking) D j Forming) R3.A-47) D i = Dissociation Energies Derive Click back) S 1 r r 1 ) = H Rx +S r r ) A) Rearranging S 1 S )r = H Rx +S 1 r 1 S r B) Solving for r* r = H Rx ) + S 1 r 1 S r S 1 S S 1 S C) 4

25 Recalling Equation R3.A-43) E a = E 1 E 1R = S 1 r r 1 ) R3.A-43) Substituting Equation C) for r* in Equation R3.A-43) Rearranging H E a = S Rx 1 + S 1r 1 S r )S 1 r 1 E) S 1 S S 1 S S E a = 1 * S )H Rx +S 1 r 1 S r - 1, r 1 / F) S 1 S + S 1 S. S = 1 * S )H Rx +S 1 r 1 S r S 1 r 1 +S r 1-1, / G) S 1 S + S 1 S. Finally, collecting terms we have S E a = 1 *H Rx + S 1S [ r 1 r ] S S S S ) P Note: S 1 is positive, S is negative, and r > r 1 ; therefore, E a! is positive, as is γ P. E a = E a! + p H Rx E a + H) B. Marcus Extension of the Polyani Equation In reasoning similar to developing the Polyani Equation, Marcus shows see Masel, pp ) E A = 1+ H Rx 4E A0 E A0 C. Blowers-Masel Relation The Polyani Equation will predict negative activation energies for highly exothermic reactions. Blowers and Masel developed a relationship that compares quite well with experiments throughout the entire range of heat of reaction for the family of reactions as shown in Figure R3.A-13. R + H R RH + R 5

26 VI. CLOSURE Figure R3.A-13 Comparison of models with data. Courtesy of R. I. Masel, Chemical Kinetics and Catalysis New York: Wiley Interscience, 001). We see the greatest agreement between theory and experiment is found with the Blowers-Masel model. We have now developed a quantitative and qualitative understanding of the concentration and temperature dependence of the rate law for reactions such as with A + B C + D r A = k C A C B We have also developed first estimates for the frequency factor, A, from collision theory 1 8k A = S r U R N Avo = B T AB ) N Avo µ AB and the activation energy, E A, from the Polyani Equation. E A = E A o + P H Rx These calculated values for A and EA can be substituted in the rate law to determine rates of reaction. r A = kc A C B = Ae E A RT 6

27 VI. OTHER STUFF Potential Energy Surfaces and the barrier height ε hb Figure R3.A-14 Reaction coordinates. The average transitional energy of a molecule undergoing collision is k B T. The molecules with a higher energy are more likely to collide. A. How to Calculate Barrier Height 1) Ab Initio calculations. No adjustment of parameters or use of experimental data. Solve Schrödinger s Equation E b = Barrier Height ε hb E b = 40 kj/mol for the H + H exchange reaction ) Semiempirical Uses experimental measurements spectroscopic). Adjustments are made to get agreement. 3) London-Eyring-Polanyi LEP) Method LEP) Surface Use spectroscopic measurement in conjunction with the Morse potential equation E = D e! r!r 0 )! e! r!r 0 ) [ ] D = dissociation energy r o = equilibrium internuclear distance β = constant Model B3 Stored Energy This third method considers vibrational and translation energy in addition to translational energy. Even though this approach is oversimplified, it is satisfying because it gives a qualitative feel for the Arrhenius temperature dependence. In this approach, we say the fraction of molecules that will react are those that have acquired an energy E A. 7

28 Maxwell-Boltzmann Statistics for n Degrees of Freedom Here, energy can be stored in the molecule by different modes. For n degrees of freedom transitional, vibrational, and rotational) L3p76) kt f,t) = 1 n 1 ) e R3.A-30) k )! B T k B T df = f,t)d = fraction of molecules with energy between ε and ε + dε) For methane, there are 3 transitional, 3 rotational, and 9 vibrational degrees of freedom, n = 15. Because Equation 31) is awkward, the two dimension equation is often used, with n = and the distribution function is n f,t) = e k B T k B T R3.A-31) Activation Energy From the Maxwell-Boltzmann distribution of velocities we obtain the distribution of energies of the molecules as shown below. Here, fdε is the fraction, df, of molecules between ε and ε + dε. df = f,t)d We now say only those molecules that have an energy, E A or greater, will react. Integrating between the limits E* i.e., E A ) and infinity, we find the fraction of molecules colliding with energy E A or greater F = 1 k B T d e k BT * Multiplying and dividing by k B T and noting again Integrating df = e E RT de E A F = e E A RT k B T = E RT R3.A-3) We now say that this is the fraction of collisions that have energy, E A or greater, and can react. Multiplying this fraction times the number of collisions, e.g., Equation R3.A-10), gives r A = Ae E A RT C A C B which gives the Arrhenius Equation k = A e E A RT * kt = A e 8