(i) Experimental methods for following the rates of reactions and the various procedures used for determining the order and rate constant of a reaction. Distinction between order and molecularity and the application of kinetic measurements in understanding mechanism.

(ii) The Arrhenius equation and the determination of activation energies and the frequency factor A. Typical values of these quantities.

(iii) The steady state approximation and its applications. This is one of the most widely used ideas in kinetics and is used whenever there are highly reactive intermediates in a kinetic scheme more complex than a single elementary step. Two important examples are (a) unimolecular reactions and (b) the Michaelis-Menten mechanism of enzyme catalysis.

(iii) The collision theory of bimolecular gas phase reactions. The frequency
of collisions between molecules (*you will have to do some simple kinetic
theory here*). The concept of activation energy in collision theory.
Comparison with experiment (*find some examples*). Possible defects in
the collision theory.

(v) The thermodynamic formulation of Activated Complex Theory (Transition State Theory) (Simple derivation). The entropy of activation.

To help you understand the application of the Steady State approximation to more complex reaction schemes, two applets are shown below (Click here for information concerning applets; the relevant filenames, if you wish to download them, are respectively tutorials/kinetics/steadystate.jar and tutorials/kinetics/SteadyStateAppletJ.html for the consecutive ABC reaction applet and tutorials/kinetics/Michaelis.jar and tutorials/kinetics/MichaelisAppletJ.html for the enzyme kinetics applet). The first applet shows how the reactant, product and intermediate vary during a consecutive reaction using either steady state kinetics or the full numerical solution.

The applet above calculates the exact behaviour of A, B and C in the consecutive reaction

It is possible to obtain an analytic solution for the kinetics of the above scheme when *k*_{−1} is zero (the solution is given in Atkins: *Physical Chemistry*). However, when this condition is not fulfilled it is necessary to solve the equation numerically. The left hand side of the diagram shows this numerical solution and the right hand side shows the plot of log[A] against *t*, which would be an appropriate analysis under certain circumstances. There are two commonly used approximate solutions, the *Steady state approximation* and the *Pre-equilibrium approximation*. In the former the assumption is made that the concentration of the intermediate [B] is approximately constant with time, and in the latter A and B are assumed to be in equilibrium throughout the course of the reaction. The button on the diagram above will add the behaviour of A, B and C predicted by these to approximations.

Try the following:

(i) With the results of the steady state approximation on the screen vary the three rate constants and show that the range where the steady state approximation is a good approximation corresponds to the situation where the consumption of the intermediate B is relatively fast. Note that the steady state approximation is not valid at low concentrations and so the concentrations of the different species are not plotted at low concentrations.

(ii) With the results of the pre-equilibrium approximation on the screen vary the three rate constants and show that the range where this approximation works corresponds to the situation where rates of conversion of B to A and A to B are fast relative to the conversion of B to C, *i.e.* the opposite of (i).

As for the first scheme an accurate solution of the equation can only be obtained numerically. Although the original solution of Michaelis and Menten was based on the prequilibrium approximation here we only compare the results of the *Steady state approximation* with the exact solution. There are two steady state approximations that can be made and it is possible to have a moderately accurate solution from these over the whole range (for a paper, which you might just manage to understand and for which the research was done in Oxford, click here). However, the steady state solution shown here is only the simple one based on the steady state concentration of the ES complex being zero.

(i) With the results of the steady state approximation on the screen vary the three rate constants and show that the range where the steady state approximation is a good approximation corresponds to the situation where the consumption of the intermediate ES is relatively fast. Note that the steady state approximation is not valid at low concentrations and so the concentrations of the different species are not plotted at low concentrations.

(ii) Show that the enzyme/substrate ratio also has a serious effect on the validity of the steady state approximation. For many *in vivo* situations this ratio is close to unity.

** Where the question number is enclosed in a button, e.g. , you can obtain help or comments about the question by clicking it.**

The following data have been obtained at 900 K for the reaction:

t/s |
0 | 10 | 20 | 40 | 60 | 100 |

[C_{2}H_{4}]/mol
dm^{ -3} |
0.884 | 0.621 | 0.479 | 0.328 | 0.250 | 0.169 |

(a) Find the order of reaction and the rate constant.

(b) The rate constant for the reaction at 1100 K is 2.229 mol^{
-1} dm^{3} s^{-1}
. Calculate the activation energy for the reaction.

Sodium monoxide, NaO, is formed in the upper atmosphere by the reaction between sodium atoms and ozone:

The rate constant for this reaction has recently been measured in the laboratory
in a flow system at room temperature. Ozone and sodium atoms were mixed at
a certain point and the decay of sodium was monitored at various distances
*x* downstream by measuring the intensity of its laser induced fluorescence.
In an experiment in which the ozone concentration was 5.34 x 10^{
12} molecules cm^{-3} and very much
greater than the concentration of sodium, the following results were recorded:

x/cm |
1.5 | 3.0 | 4.5 | 6.0 |

I/arb. units |
933 | 175 | 32.6 | 6.09 |

The linear flow velocity was 1480 cm s^{-1} at a
temperature of 293 K. Determine the value of the rate constant for the reaction
from these data.

A compound X can decompose by two simultaneous independent 1st order reactions

At 298 K the rate constant *k*_{1} = 0.122
min^{-1} and [X] vary with time as follows:

t/min |
0 | 2 | 4 | 8 | 12 |

[X]/mol dm^{-3} |
0.774 | 0.519 | 0.349 | 0.156 | 0.070 |

(a) Determine the value of the rate constant *k*_{2}
.

(b) The temperature of the reaction is increased. Comment on the
following two assertions, which may be incorrect or incomplete.

(i) *k*_{1} and *k*_{2}
will change as the temperature rises, but their ratio will remain constant.

(ii) There is a temperature, other than 0 K, at which the rates of both
reactions will be the same.

**4.** (a) Show that the half life for reactant decay in a first order
reaction is independent of the starting concentration of the reactant.

(b) Determine the activation energy from the following data for the first order decomposition of azomethane,

T = 571.6 K |
- | - | - | - | - |

t/s |
0 | 600 | 1200 | 1980 | 2760 |

P(H_{3}CNNCH_{3}
)/Torr |
430.8 | 371.8 | 313.6 | 251.9 | 205.2 |

T = 593.6 K |
- | - | - | - | - |

t/s |
0 | 180 | 360 | 540 | 720 |

P(H_{3}CNNCH_{3}
)/Torr |
212.3 | 161.7 | 130.3 | 102.0 | 80.6 |

One mechanism for the H_{2} + I_{
2} reaction is

where the first two steps are equilibria. According to this mechanism the rate law is

rate = *kK*_{1}*K*_{
2}[H_{2}][I_{2}
]

as observed. Verify this result.

Using the definition of activation energy
and also the integrated van't Hoff isochore, obtain an expression for the activation
energy according to this mechanism in terms of the enthalpies of reaction
Δ*H*_{1} and Δ*H*_{2}
for the first two elementary steps and the activation energy *E*^{
‡} in the third step.

The recombination of methyl radicals in nitrogen gas proceeds via the mechanism

where C_{2}H_{6}^{*} is an energised reaction intermediate.

(a) Use the steady state approximation to show that the rate of ethane production may be written

(b) Identify the conditions under which the overall rate of C_{2}H_{6} production is second order and those under which it is third order, and give expressions for the second and third order rate constants in terms of *k*_{1}, *k*_{-1} and *k*_{2}.

(c) Suggest why the recombination rate constant in the second order limit shows little temperature dependence, decreasing slightly over the temperature range 300-1000 K.

(a) Thermal gas phase unimolecular decompositions are thought to be initiated by a collisional bimolecular process and to be followed by a reactive step

Using the steady state approximation show that the overall rate constant is

Under what conditions will the overall kinetics be first order?

(a) Some enzyme catalysed reactions are well described by the simple reaction scheme:

where E, S, ES and P represent respectively enzyme, substrate, enzyme-substrate
complex and product.

Using the steady state approximation derive an expression for the steady state concentration of ES in terms of [E] and [S]. Making the substitution [E]_{0} = [ES] + [E], where [E]_{0} is the overall enzyme concentration, derive an expression for [ES] in terms of [E]_{0} and [S], and hence show that that ν, the rate of the reaction (formation of product), is given by (the Michaelis-Menten equation)

and obtain an equation relating the Michaelis constant, *K*_{M}, to
*k*_{1}, *k*_{-1}
and *k*_{2}.

(b) The following initial rates of reaction were measured for 1.0 x 10^{
-9} g of an enzyme (*M*_{r} = 29600) dissolved
in 0.01 dm^{3} of water.

[S]/10^{-5}M |
0.1 | 0.3 | 0.5 | 1.0 | 3.0 | 5.0 |

ν/10^{-9}M s^{
-1} |
0.183 | 0.417 | 0.567 | 0.750 | 0.967 | 1.017 |

Verify that these data are consistent with the above reaction scheme and
determine *K*_{M} and *k*_{2}
.

(c) What fraction of the enzyme is complexed to substrate when [S] = 5
x 10^{-5} M?

The important `oxygen-only' reactions in the stratosphere can be written

where *h*ν represents absorption of a photon of light and M represents any species that might collide and remove energy.

(i) Given that, under conditions of constant illumination, it is possible to write the rates of O atom production in steps (1) and (3) in the form 2*k*_{1}[O_{2}] and *k*_{3}[O_{3}], where *k*_{1} and *k*_{3} have units of s^{-1}, write down the rate of production of O and O_{3}.

(ii) Use the steady state approximation to show that

[O][O_{3}] = *k*_{1}[O_{2}]/*k*_{4}

With the help of this result and the rate equation for O_{3} obtained in (i), derive the following steady state expression for the ozone concentration

where

(iii) Show that the presence of any species that scavenges atomic oxygen will lead to a reduction in the stratospheric ozone concentration relative to the above scheme.

**10.** 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?

The thermodynamic formulation of transition state theory for a unimolecular reaction step in the gas phase gives the following expression for the rate constant (see derivation)

Compare this equation with the Arrhenius equation and comment on its form.
Show how the prefactor *A* in the Arrhenius equation is related to the entropy
of activation.

The limiting values of A at high pressures and at 1000 K for the two reactions below are as follows:

Calculate the entropy of activation for the above reactions. Sketch possible
transition states for each of these reactions and suggest reasons for the
large difference in Δ*S*^{‡} for the two reactions.