(i) Everything that you did in Kinetics in the first year is important and you should start by reviewing that and doing any problems that you failed to do then. The first two problems below are revision problems.

(ii) It is also important to revise the expressions for the equilibrium constant in terms of partition functions, which you did last term in Statistical Mechanics.

(iii) Review simple collision theory and its limitations. This is best done in terms of potential energy surfaces for reactions. It is also useful to examine the differences between collision and reaction cross sections, illustrating your discussion with examples,.

(iv) Transition state theory. The idea of the reaction coordinate as a separable motion. The use of the statistical mechanical expression for the equilibrium constant in terms of partition functions to derive an explicit formula for *k*(*T*).

(v) Comparison of the full derivation of transition state theory in (iv) with the thermodynamic formulation. Entropies of activation.

(vi) Comparison of transition state theory with collision theory. Application in the Arrhenius equation and to kinetic isotope effects. Deviations from Arrhenius behaviour. Quantum mechanical tunnelling.

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The rate of the elementary gas phase reaction

CH + O_{2} → products

was determined using flash photolysis techniques under conditions where the molecular oxygen concentration was in large excess over that of the CH radical.

(i) Obtain an expression (in integrated form) for the time dependence of the CH radical concentration under these conditions.

(ii) The CH radicals were monitored as a function of time using the technique of laser induced fluorescence. The relative fluorescence intensity, *I*, proportional to CH concentration, is given in the table below at different times, *t*.

t/10^{−6} s |
20 | 30 | 40 | 50 | 60 | 70 |

I |
0.230 | 0.144 | 0.088 | 0.050 | 0.033 | 0.020 |

Verify that the kinetics are first order, and determine the pseudo-first order rate constant for the reaction.

In the above experiments the molecular oxygen concentration was 8.8 x 10^{14} molecules cm^{-3}. Calculate the bimolecular rate constant for the reaction.

(iv) Derive an expression for the half-life of a bimolecular reaction when the two reactant concentrations are equal. What would be the half-life of the CH radicals if the O_{2} and CH concentrations were initially *both* equal to 5 x 10^{12} molecules cm^{-3}, but conditions were otherwise unchanged?

In a study of the reaction of methyl radicals with ethanal vapour a substance CH_{3}X, which decomposes with first order kinetics, is used as a source of methyl radicals. A mechanism that is consistent with the observed products is

CH_{3}X → CH_{3} + X(inactive)

CH_{3} + CH_{3}CHO → CH_{4} + CO + CH_{3}

CH_{3} + CH_{3} → C_{2}H_{6}

By applying the steady state approximation derive an expression for the rate of formation of methane in terms of the concentrations of ethanal and CH_{3}X. The rate of production of methane varies with the concentration of CH_{3}X in the following manner:

[CH_{3}X]/arbitrary units |
5 | 10 | 20 | 50 | 100 |

dCH_{4}/dt/arbitrary units |
7.24 | 10.49 | 14.49 | 22.91 | 32.40 |

Show that this data is consistent with the mechanism above. How would the rate of formation of methane be expected to depend on the ethanal concentration?

How does the form of the potential energy surface linking products with reactants influence the distribution of excess energy in the products of an exothermic reaction?

What is meant by (a) a collision cross section σ_{C}, and (b) a reaction cross section σ_{R}?

How, in principle, is a thermal rate constant *k* for a reaction at a specific temperature obtained from the reaction cross section.

One theory suggests that σ_{R} varies with reactant translational kinetic energy *E _{T}* according to the equation

σ_{R} = σ_{R}^{0}(*E _{T}* -

where σ_{R}^{0} is a constant and *E ^{0}* is the threshold energy for reaction. The reaction

K + HCl(v) → KCl + H

is slightly endothermic for v = 0. The following data were obtained for the reaction cross section σ_{R} at different energies:

E/kJ mol_{T}^{-1} |
8 | 12 | 20 | 30 |

σ_{R}/nm^{2} |
0.0033 | 0.0108 | 0.017 | 0.020 |

Estimate the threshold energy for reaction from the data. Vibrational excitation to v = 1 in HCl increases σ_{R} much more than the equivalent amount of energy in translational excitation. What qualitative conclusion can you draw about the shape of the potential energy surface?

(a) For the reaction A + BC → AB + C, show how transition state theory predicts that the pre-exponential factor of the rate constant should either be proportional to *T*^{−1/2} or temperature *independent* depending on the geometry of the transition state.

(b) Lindemann theory predicts an expression of the following form for the dependence of the rate of the unimolecular decomposition of substance A on concentration of an inert diluent [M].

based on applying the steady state approximation to the reaction scheme

Show that on the basis of this model *k*_{uni} falls to one half of its limiting high pressure value (*k*_{∞}) when [M] = *k*_{∞}/*k*_{1}. Explain why values of *k*_{1} determined from this equation are generally much larger than predicted from simple collision theory.

(c) Explain why the rates of many radical-radical recombination reactions in the gas phase at low pressure show third order kinetics, and why the rates decrease with increasing temperature.

(d) In the reaction of HCl with K, (ΔH = 4 kJ mol^{−1}) laser excitation of HCl to *v* = 1 (requiring 34.5 kJ mol^{−1}) increases the reaction rate by a factor of 100, whereas increasing the average thermal energy of the reactants by the same amount through heating of the gases only increases the reaction rate by a factor of 10. The majority of the excess energy of this reaction is partitioned directly into translational energy of the products. Discuss what can be inferred from this data about the potential energy surface of the reaction.

The thermodynamic formulation of transition state theory for a gas phase bimolecular reaction may be written

(a) Discuss briefly the origin of the factor *k*_{B}*T*/*h* appearing in this expression.

(b)(i) The activation energy of a reaction is defined

Use this equation, together with equation (1), and the van't Hoff equation

to show that

*E*^{ ‡} = Δ*H*^{ ‡O} + 2*RT*

(ii) Hence show that the Arrhenius A-factor for a gas phase bimolecular reaction can be written as

where *RT*/*P*^{O} is typically expressed in units of dm^{3} mol^{−1}.

(iii) Briefly explain what is meant by the terms Δ*S* ^{‡O} and Δ*H*^{ ‡O} appearing in the above equations.

(c)(i) Simple collision theory yields the following expression for the Arrhenius A-factor, expressed in dm^{3} mol^{−1} s^{−1}

where <*c*_{rel}> = (8*k*_{B}*T*/πμ)^{1/2} is the mean relative speed of the reactants (in m s^{1}), σ is the collision cross-section (in m^{2}), and *P* is the steric factor. Taking *P* = 1, and making reasonable estimates for a typical collision cross-section and relative speed
at 300 K, estimate *A*_{sct}.

(ii) Use your estimate, together with equation (2), to determine a simple collision theory value for Δ*S*^{ ‡O} at 300 K.

(iii) Account for the sign of Δ*S*^{ ‡O} obtained in (c)(ii). Explain why Δ*S*^{ ‡O} values derived from experimental kinetic data for bimolecular reactions in the gas phase are often more negative than the simple collision estimate, but are rarely less negative.

Theoretical expressions for the rate constant of a bimolecular elementary reaction step include the Eyring equation from transition state theory:

and the collision theory result:

(a) Explain the significance of the terms *P*, κ, *K*^{‡}, σ and μ in the above expressions.

(b) For what type of reaction are the predictions of the two theories identical, with *P* and κ equal to unity? Obtain the specific form of the Eyring equation, in terms of the partition functions, for this type of reaction.

(c) Express the rate constant for the radical reaction

in terms of the molecular partition functions. Hence deduce how the pre-exponential factor for the radical reaction should depend on temperature. (Assume that all vibrational partition functions are unity)

(d) When the experimental activation energy (50 kJ mol^{−1}) and the gas-kinetic collision cross sections are used, a *P* factor of 0.01 is found necessary to reconcile the collision theory result with results for the radical reaction. Explain why this factor is significantly less than one.

(a) Discuss the physical interpretation of the symbols σ, <*c*> and *E*_{0} in the simple collision theory expression for the rate coefficient of a bimolecular reaction,

(b) The steric factor *P* can be calculated using transition state theory, which gives the rate coefficient of a bimolecular reaction A + B → products as

State the physical assumptions that lead to this expression, and provide one example of a situation where these assumptions might be expected to break down.

(d) Given that q_{‡}, q_{A} and q_{B} are molecular partition functions per unit volume, the translational contributions to which can be written as (e.g.)

show that in the hypothetical case of a reaction between two hard spheres the transition state theory rate coefficient reduces to the simple collision theory result with *P* = 1, *E*_{0} = Δ*E*_{0}^{‡}, and

(e) Hence deduce that the steric factor for a reaction between two molecules A and B that proceeds via a linear transition state is given by

where the translational contributions are absent from the primed partition functions q_{‡}^{/}, q_{A}^{/} and q_{B}^{/} and the rotational contribution is also absent from q_{‡}^{/}. Noting that q_{vib} ≈ 1 per vibrational degree of freedom and q_{rot} ≈ 10 per rotational degree of freedom at room temperature, estimate the steric factor for (i) a reaction between an atom and a diatomic molecule, and (ii) a reaction between two diatomic molecules, assuming that the transition state is linear in both cases. Comment on the results you obtain.

(a) Consider the following gas phase reactions:

Estimate the ratio of the rate constants *k*_{2}/*k*_{1} at 300 K using the vibrational wavenumbers given below. Assume that the effect of isotopic substitution is limited to changes in the zero point energies of reactants and transition states, and that the transition states are linear.

- | H_{2} |
Transition state 1 | Transition state 2 |

stretch (cm^{−1} |
4395 | 1762 | 1773 |

bend (degeneracy = 2)(cm^{−1}) |
- | 694 | 870 |

(b) For the reaction of an atom X with a diatomic molecule Y_{2}

proceeding through a linear transition state, Transition State Theory gives

where *q*_{Xint} and *q*_{Y2int} are the partition functions for the internal modes (rotational, vibrational, electronic) of the reactants and *q*_{t} is related to the masses of the reactants by

Outline the origin of the expression for *k*, comment on the assumptions involved, and explain the meanings of the symbols *q*_{int}^{‡} and ε_{0}.

(c) Show that the expression for *k* in (c) leads to the following equation, stating and justifying the assumptions and approximations used.

The 1,2 subscripts refer to the two reactions in (a) and *I*^{‡} denotes the moment of inertia of the transition state.

(d) Hence obtain a revised estimate of *k*_{2}/*k*_{1} at 300 K. (*I*_{1}^{‡} = 4.02 x 10^{−47} kg m^{2}, *I*_{2}^{‡} = 3.95 x 10^{−47} kg m^{2}; the relative atomic masses of H and D are 1 and 2 respectively).

(e) The experimental value of *k*_{2}/*k*_{1} is 14 at 300 K. Why is this value larger than predicted by Transition State Theory?