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Statistical Thermodynamics and Chemical Kinetics

Statistical Thermodynamics and Chemical Kinetics. Lecture 10. State Key Laboratory for Physical Chemistry of Solid Surfaces. 厦门大学固体表面物理化学国家重点实验室. Chapter 10 Transition State Theory. 10. 1 Motion on the Potential Surface

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Statistical Thermodynamics and Chemical Kinetics

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  1. Statistical Thermodynamics and Chemical Kinetics Lecture 10 State Key Laboratory for Physical Chemistry of Solid Surfaces 厦门大学固体表面物理化学国家重点实验室

  2. Chapter 10 Transition State Theory 10. 1 Motion on the Potential Surface The potential energy surfaces are based on the Born-Oppenheimer separation of nuclear and electronic motion. A justification for this separation is the disparity in the electron and nuclear masses, which results in very slow nuclear motion compared to electronic motion. With the Born-Oppenheimer separation, each electronic state of the chemical reactive system has a potential energy surface.

  3. For a chemical reaction involving N atoms there are 3N-6 vibrational degrees of freedom at the saddle point. One of these degrees of freedom corresponds to infinitesimal motion along the reaction path. The remaining 3N-7 degrees of freedom define vibrational motion orthogonal to the reaction path. To study properties of the saddle point in more detail, consider a potential energy contour diagram for a collinear chemical reaction (e.g., H + H-H). This collinear system has two orthogonal internal degrees of freedom, for which the coordinate are  =r1+ r2 and s=r2-r1. The saddle point has a symmetric configuration at which r1= r2 = r0,  =2r0 and s=0.

  4. The first and second derivatives of the potential at the saddle point are Therefore the vibrational frequency for motion along the reaction path will be imaginary (negative). The vibrational frequency for the  coordinate will be positive. In the vicinity of the saddle point, a Taylor’s series expansion of the potential is (10-1) The transition state theory is based on the postulate that the rate of transformation of systems from reactants to products is given solely by passage in the forward direction from coordinates s< 0 to s > 0.

  5. 10.2 Basic postulates and standard derivation of TST Transition state theory, introduced by Eyring and by by Evans and Polyani in 1935, provided the first theoretical attempt to determine absolute reaction rates. In this theory, a transition state separating reactants and products is used to formulate an expression for the thermal rate constant. The relationship between transition state theory and dynamical theories was first discussed by Wigner who emphasized that the theory was a model essentially based on classical mechanics. A number of assumptions are made in deriving the TST rate expression.

  6. The two most basic are the separation of electronic and nuclear motions, equivalent to the Born-Oppenheimer approximation in quantum mechanics, and the assumption that the reactant molecules are distributed among their states in accordance with the Maxwell-Boltzmann distri-bution.

  7. However, the following additional assumptions, which are unique to the theory, are also required. • Molecular systems that have crossed the TS in the direction of products cannot turn around and reform reactants. • In the TS, motion along the reaction coordinate may be separated from the other motions and treated classically as a translation. • The TSs that are becoming products are distributed among their states according to the M-B laws.

  8. It is customary to derive TST by postulating a quasi-equilibrium between TS species and reactants. This approach focuses on the third assumption listed. • Another approach follows the dynamical formulation of Wigner and uses the first assumption, which is considered the “fundamental assumption” of TST. • Both derivations are given here, and they use the conventional transition state theory model in which the transition state is located at the saddle point on the potential energy surface. Later in this chapter, variational TST, a more accurate method for choosing the TS, is discussed.

  9. The third assumption, the “quasi-equilibrium hypothesis”, is based on a simple physical interpretation. Suppose we have an elementary reaction A + B  X‡  C (10-2) where X‡is a TS. Suppose also that it is possible to define a small region at the top of the potential energy barrier such that all systems entering a small region from the left pass through to products without turning back, and similarly all products entering from the right must pass through to reactants. This small region is often referred to as the dividing surface and is orthogonal to the reaction pathway. Systems within this small distance are by definition transition states, to the right are products, to the left reactants. (See Fig. 10-2)

  10. Consequently, for a system in which reactants are in equilibrium with the products, there are two types of transitions states: those moving from reactant to products, and those moving in the opposite direction. We designate the concentration of these by Nf‡ and Nb‡, respectively. At equilibrium the rate of the forward reaction must be the same as the backward reaction; accordingly, the concen-trations of the TSs moving toward reactants and toward products must be equal. Hence, Nf‡ + Nb‡ = N‡ = K‡[A][B] (10-3)

  11. Now, if all the products are suddenly removed from equilibrium, that is, if Nb‡ =0 , then, since the forward rate of reaction must be the same regardless of the back reaction rate, it follows that the concentration of the forward-moving transition states is the same as it would be for full equilibrium; hence, Nf‡ = K‡[A][B]/2 (10-5) This is the “quasi-equilibrium hypothesis” of transition state theory. To calculate the net rate of reaction, the rate at which transition states pass over the barrier to products is needed; i.e., (10-6)

  12. The number of transition states per unit volume having velocity between v and v+dv in one direction is presented by determining N‡. The average time dt for the N‡ transition states to cross the barrier is equal to the thickness d of the dividing surface divided by the average velocity vs at which the transition states traverse the dividing surface; that is, (10-7) Hence, (10-8) From equation 10-5, the number of transition states crossing the dividing surface in the direction of products is one-half the total number of transition states, i.e., N‡= N‡/2 (10-9)

  13. Therefore, the number of transition states crossing the barrier per unit volume in unit time is, (10-10) On the assumption that there is an equilibrium distribution of velocities, the average velocity of the transition state moving in one direction, e.g., the forward direction, is (10-11) Where ms is the reduced mass for motion through the dividing surface.

  14. Inserting this value into equation (10-10) gives (10-12) Since the transition state is in equilibrium with the reactants, it is possible to obtain N‡ from equilibrium statistical mechanics, namely, through (10-13) This statement of equilibrium does not imply that the TS is long-lived. This equilibrium constant can be expressed in terms of partition functions as (10-14)

  15. Combining equation 10-12 and 10-14, we have, for the reaction rate, (10-15) In the approaches of Eyring and Evans and Polyani, the partition function for the reaction coordinate in the transition state is considered to be a translational function. Since this motion is assumed to be separable, the translational partition function is therefore separable as (10-16) where Qs is the partition function for the reaction coordinate motion and Q‡ is the partition function for all other 3N-1 degrees of freedom in the transition state.

  16. The translational partition function for motion in one dimension in a system of length d is (10-17) This Qs is not divided by 2, since it is the partition function for all the transition states. Substituting this expression into equation 10-16 and the result into equation 10-15 gives (10-18) From chapter 1, we know that the experimental rate equation for a bimolecular reaction is (10-19)

  17. Comparison of equations 10-18 and 10-19 gives the absolute rate coefficient for A + B  products, (10-20) Note that the artificial constructs d and ms, as well as the factor 1/2, have cancelled out of the absolute rate theory expression. The ratio kBT/h, having the units of frequency and the magnitude 6.25x1012 sec-1 (for T=300K), is frequently termed the frequency factor.

  18. In order to be able to evaluate equation 10-20, we must be able to calculate the transition state partition function Q‡ , using the statistical mechanical techniques discussed before. To do that, we need to know the structural parameters of the transition state (specifically, the moment of inertial I‡ ) , as well as its 3N-7 vibrational frequencies {v‡ }. In practice, such parameters can only be estimated approximately. With detailed information about potential surface now becoming available from ab initio theory, however, we may expect to see more precise calculations of rate coefficients from transition state theory.

  19. The energy E0 is the difference in zero-point energy between the transition state and the reactants. It is useful to point out the relationship between E0 and the experimental activation energy Ea. As shown by Tolman et al. (10-21) where <Er(T)> is the average energy of molecules undergoing reaction and <E(T)> is the average energy of all reactant molecules. In transition state theory, <Er(T)> is given by the average energy of the transition states plus E0.

  20. 10.3 Quantum mechanical effects in TST A basic difficulty in generalization of classical transition state theory to quantum mechanics is that the reaction criterion cannot be formulated as the condition that a trajectory pass through a critical surface. This is connected with the fact that, in a quantum mechanical observation, the coordinates and momenta of a system con not be assigned simultaneously. Consider the Heisenberg uncertainty relations, For the reaction coordinate. If Dp is replaced by ħ/l, where l is the de Broglie wavelength, we have the relation Dq>l.

  21. Thus the uncertainty in the value of the reaction coordinate at the transition state must be larger than the wavelength associated with motion along the reaction coordinate. In other words, quantum mechanically, the transition state is not localized. The uncertainty relation DEDt  ħ may be analyzed in a similar way. For the thermal averaging in TST to have a meaning, it is necessary that the translational energy E in the reaction coordinate be much less than kBT. As a result, the lifetimeDt of the transition state must be larger than ħ/kBT. Therefore, in the quantum case, the transition state can not be considered a definite configuration of nuclei during an infinitesimal interval of time.

  22. Thus, for the calculation of the reactive flux in the quantum case, it is necessary to consider explicitly the dynamics of the trajectories’ motion in the region Dq or to follow the eolution with time of the system for time Dt. In classical TST, the potential energy is constant and the reaction coordinate motion is separated from the remaining internal motions at the localized position along the reaction coordinate which defines the transition state. Quantum mechanical delocalization of the transition state along the reaction coordinate can lead to two problems.

  23. First, if the potential is not flat in the region Dq, so that the system is not freely moving, it is incorrect to treat the reaction coordinate as being a classical translation motion. Rather, the potential will usually have a concave-down shape, and quantum mechanical tunneling will be occur through the potential. The second problem is more critical. If there is curvature along the reaction coordinate in the region Dq, the reaction coordinate is not separable from the remaining internal degrees of freedom. Thus, the rate constant expression can not be factored into a frequency kBT/h for the reaction coordinate and a partition function for the remaining degrees of freedom.

  24. Also, since the curvature couples the reaction coordinate with the remaining modes, tunneling can not be treated as a one dimensional reaction coordinate barrier-penetration problem. Instead, there will be a multitude of tunneling paths which involve all the coordinates. To correct the tunneling, we have to consider the problem of crossing the barrier quantum mechanically rather classically as done in previous sections. According to quantum mechanics, there is a probability that a particle finds its way from reactants to products, and that probability varies continuously with energy; for E<E0 there is a finite nonzero probability, and for E > E0, it approaches unity.

  25. The probabilities can be calculated for any barrier from the Schrödinger equation (10-23) We have assumed a one-dimensional barrier, which is allowable since, with the separable approximation, there is reaction only long one coordinate, that is, the reaction pathway. A solution of equation 10-23 is obtained for a suitable choice of potential V(s) to describe the barrier. The resulting expression for the probability of tunneling through the barrier is given by (10-24)

  26. where G(E), called the permeability function, depends on the exact shape of the barrier along the reaction coordinate, and s1and s2are turning points, i.e., the coordinates of the reaction path for which V(s)=E. The tunneling correction is obtained by integrating over a M-B distribution, i.e., (10-25) Even for simple barrier shapes, a closed solution for the permeability function is difficult to obtain; and, in any case, the integration over all energies must be carried out numerically.

  27. However, one barrier shape for which an exact expression for G(E) is known and is physically meaningful is the symmetrical Eckart barrier. The potential for this barrier has the form V(s) = E0sech2(ps/l) (10-26) The permeability function for this barrier is (10-27) where And where l is the width of the barrier.

  28. For small barrier heights and widths, the tunneling correction approaches an asymptotic form given by (10-28) Where sis the imaginary frequency of the transition state at the top of the barrier. The experimental evidence for tunneling comes from Arrhenius plots of lnkversus 1/T. If the Arrhenius show significant nolinearity, then tunneling may be the cause. The curvature in the Arrhenius plots of lnk vs. 1/T tends to be concave upward when tunneling is important, because the calculated reaction rate deviation is positive and increases for large values of 1/T.

  29. 10.4 Thermodynamic formulation of TST The transition state derived rate constant can be reformulated in thermodynamic terms, since it is sometimes more useful to work with the rate constant in this form than with partition functions. The rate constant expression given by equation 10-20, viz., can be rewritten as (10-29) where (10-30)

  30. which is the equilibrium constant for formation of the transition state. If the equilibrium constant is expressed in terms of the molar Gibbs standard free energy using the van’t Hoff relation (10-31) Then equation 10-20 can be written as (10-32) (10-33) (10-34)

  31. Equation 10-34 is similar to the Arrhenius equation (10-35) and the thermodynamic parameters can be related to the Arrhenius parameters. The Arrhenius activation energy is defined by (10-36) Taking the logarithm of equation 10-29 and differentiating with respect to T gives (10-37) (10-38)

  32. Inserting 10-38 into equation 10-37 and comparing the result with 10-36 gives (10-39) The relationship between the standard thermodynamic energy and enthalpy for a constant-pressure process is (10-40) since H=E + PV. The quantity DV0‡ is known as the standard volume of activation. With equation 10-40, equation 10-39 becomes (10-41) For a unimolecular reaction, there is no change in the number of molecules, so the volume maintains. Therefore, (10-42)

  33. Inserting 10-42 into equation 10-34 leads to (10-43) Hence, (10-44) For gas phase reactions other than unimolecular, the relationships between the Arrhenius parameters and thermodynamica terms different from the relationships just given. If the ideal gars relation is assumed, i.e., (10-45) then, from equation 10-41, one obtains (10-46)

  34. And the Arrhenius pre-exponential factor is (10-47)

  35. 10.5 Applications of TST • 10.5.1 Evaluating partition functions Determining rate constants from the canonical level of TST is an invaluable aid in both elucidating mechanisms and evaluating rate constants. Calculating the rate constant for a reaction requires calculating the partition functions. The total partition function associated with the internal motion for each molecule is given as follows, (10-48) To calculate the individual partition functions, one needs to know moments of inertia, vibrational frequencies, and electronic states. Such information can be obtained by quantum chemical calculations.

  36. 10.5.1.1 Electronic partition function. (10-49) where gi is the degeneracy and Ei is the energy above the lowest state of the system. In most reactions, few electronic energy levels other than the ground state must be considered. For reactions involving doublet and triplet systems, the degeneracy factor is the corresponding spin degeneracy which would contribute a factor of two or more to the calculated rate constant. • 10.5.1.2 Translational partition function. (10-50) (10-51)

  37. 10.5.1.3 Vibrational partition function. A polyatomic molecule has s=3N-6 vibrational modes if it is nonlinear and 3N-5 modes if it is linear; the vibrational partition function for a polyatomic molecule is (10-52) • 10.5.1.4 Rotational partition function. For a polyatomic molecule with moments of inertia Ia, Ib, and Icabout its principal axes. The rotational partition function is (10-53)

  38. 10.5.2 Symmetry and Statistical Factors If molecules involved in a reaction have elements of symmetry, then in calculating the rate expression, the symmetry must be accounted for in the partition functions. For molecules which have rotational system, it is customary to divide the rotational partition function for each molecule by its appropriate symmetry number , defined as the number of equivalent arrangements that can be obtained by rotating the molecule. For example, consider H2. The number of identical atoms in the molecule is 2, and by rotating we get two equivalent arrangements: H1-H2 and H2-H1. Therefore, the symmetry number is 2. For planar NO3 radical, the symmetry number is 6.

  39. In taking into account the symmetry numbers in the rotational partition function, the standard procedure is to divide by the symmetry number. This procedure is correct when calculating rate constants, but it has some limitations. A good example which illustrates this is the reaction, H + H2 [H · · · H-H]‡ H2 + H. The symmetry number is 2 for H2, and so is the symmetry number of the complex; consequently, the rate constant is (10-54) If we compare the rate obtained for the reaction D + H2 [D · · · H-H]‡  DH + H.

  40. we see that the symmetry number for the complex is 1; therefore, (10-54) The conclusion is that the second reaction is favored over the first by a factor of 2. However, since both reactions are extracting hydrogen atoms from H2, the rates can not differ by a factor of 2; this factor should appear in both equations! The best way to resolve this dilemma is by using a statistical factor, defined as the number of different transition states that can be formed if all identical atoms are labeled to distinguish them from one another.

  41. Thus the above reactions both have a statistical factor of 2. In general, with this definition, we can omit symmetry numbers from the partition functions and multiply the rate expression by the statistical factor L‡, i.e., (10-55) Great care must be used in the choice of a transition state with the correct degree of symmetry.

  42. 10.5.3 Application: The F + H2 Reaction The reaction H2 + F H + HF has been of particular interest in chemical kinetics because it is the rate-limiting step in the chain reaction H2 + F2 2HF, which plays an important role in the kinetics of the HF chemical laser. It is also of special theoretical interest, because it is one of the simplest examples of an exothermic chemical reaction. The expression for the rate constant can be written as These rations can be evaluated separately.

  43. Properties of the reactants and transition state.

  44. We note that the electronic degeneracy for the fluorine atom in its ground electronic state (2P3/2) is 4, and that for the linear FH2 complex(2P) is the same. We neglect contributions from the (2P1/2) spin-orbit state of F at 404 cm-1. The degeneracy for H2 in its ground state (1Sg+) is 1. Consequently, the electronic partition function contribution is unity. The translational partition function ratio is And inserting the values given in the table gives

  45. The rotational and vibrational partition functions are dimensionless, and contain no contribution from the mono-atomic F species. The rotational partition function ratio is In evaluating the vibrational partition function ratio we obtain the expression Thus

  46. The statistical factor L contributes a factor of 2. Thus we obtain for the rate constant The experimental data are best represented by This is very reasonable agreement, for such a simple model.

  47. 10.6 Variational TST Classical TST gives the exact rate constant if the net rate of reaction equals the rate at which trajectories pass through the transition state. These two rates are equal, however, only if trajectories do not recross the transition state: any recrossing makes the reactive flux smaller than the flux through the transition state. Thus the classical transition state theory rate constant may be viewed as an upper bound to the correct classical rate constant. The effect of recrossing on the reaction rate can be taking into account by means of the variational transition state theory. In the canonical approach, the minimum in the canonical transition state theory constant given by equation 10-20 is found along the reaction path

  48. (10-56) Since the canonical rate constant is related to the free energy according to equation 10-32, i.e. the canonical variational transition state will be located at the maximum in the free energy along the reaction path. To apply canonical variational TST, the transition state’s partition function (or free energy) must be calculated as a function of the reaction path. These calculations require values for the classical potential energy, zero-point energy, vibrational frequencies, and moments of inertia as a function of the reaction path.

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