the need for research evidence to be supported by implementation in school C2 (teacher perspective)

Slater’s theory treats, in detail, the changing positions of the atoms in the molecule as the molecule vibrates, and studies these different arrangements of the atoms as a function of time. All the interatomic distances, bond lengths and bond angles are calculated as a function of time for a specified vibrational energy and energy dis-tribution of the activated molecule. Since molecular vibrations vary periodically, then the bond lengths, bond angles and atomic distances will also change periodically with time. The changing atomic positions depend critically on the vibrational energy of the activated molecule, and the way in which this energy is distributed among the normal modes of vibration.

Fixing the magnitude of the energy, and the energy distribution, results in an inevitable and totally unique variation of the positions of the atoms with respect to each other, and reaction can only occur if certain atomic dimensions are achieved.

What they are depends on the particular reaction, and the theory first decides what the critical configuration shall be, and then focuses attention on a critical aspect of the critical configuration, such as a bond length, a bond angle or some combination of both. The theory sets out the physical conditions and requirements which have to be met before it is posssible for the activated molecule to reach the critical dimensions.

Figure 4.31 shows the sine waves for the three normal modes of vibration for the linear N₂O molecule. For the unimolecular decomposition N₂O!N₂þO it is assumed that reaction occurs with the breaking of the N–O bond after it reaches a certain critical extension. The diagram shows the summing of these three sine waves and the effect which that has on the extension of the N–O bond with time. The diagram shows the value of the critical extension which must be reached before the bond will break. The vibrational energy, and the distribution of this energy among the normal modes can be seen to be such that the bonddoesreach the critical value, and breaks.

The average number of times per unit time that the combination of sine waves, calculated for the summed normal modes, exceeds the critical extension can be found, and this gives a value for the first order high pressure rate constant.

Slater’s theory assumes that the normal modes behave as harmonic oscillators, which requires that there be no flow of energy between the normal modes once the molecule is suitably activated, and so the energy distribution remains fixed between collisions. But spectroscopy shows that energy can flow around a molecule, and allowing for such a flow between collisions vastly improves the theory. Like Kassel’s theory a fully quantum theory would be superior.

The equation for the high pressure rate constant predicts strict Arrhenius behaviour, and also a non-linear plot for 1/kobsversus 1/[A], both in agreement with experiment.

However, the beauty of Slater’s theory is that it gives a very clear description of the process of reaction, which is easy to visualize; that is, ‘the molecule splits up if some bond(s), bond angle(s) or a combination of both is extended too far’.

N _ O distance

critical value q₀

bending time

symmetric stretch asymmetric stretch

actual motion: sum of all three Figure 4.31 Summing of normal modes of vibration for N₂O

THE SLATER THEORY 161

Further reading

Laidler K.J.,Chemical Kinetics, 3rd edn, Harper and Row, New York, 1987.

Laidler K.J. and Meiser J.H.,Physical Chemistry, 3rd edn, Houghton Mifflin, New York, 1999.

Alberty R.A. and Silbey R.J.,Physical Chemistry, 2nd edn, Wiley, New York, 1996.

Logan S.R., Fundamentals of Chemical Kinetics, Addison-Wesley, Reading, MA, 1996.

* Nicholas J.,Chemical Kinetics, Harper and Row, New York, 1976.

* Wilkinson F., Chemical Kinetics and Reaction Mechanisms, Van Nostrand Reinhold, New York, 1980.

Robson Wright M.,Fundamental Chemical Kinetics, Horwood, Chichester, 1999.

* Laidler K.J.,Theories of Chemical Reaction Rates, McGraw-Hill, New York, 1969.

* Glasstone S., Laidler K.J. and Eyring H.,Theory of Rate Processes, McGraw-Hill, New York, 1941.

* Out of print, but should be in university libraries.

Further problems

1. Find the change in the number of degrees of freedom of each type on forming the activated complex for the following reactions:

atomþnon-linear molecule to non-linear AC

linear moleculeþnon-linear molecule to non-linear AC two linear molecules to linear AC

two linear molecules to non-linear AC non-linear molecule to non-linear AC.

2. How canH^6¼ and S^6¼ be found for a gas phase reaction? Why is it that a V^6¼ value cannot be found for gas phase reactions?

3. Use transition state theory to calculate approximateS^6¼values for the following reactions:

BrþH2 !HBrþH CH₃þH₂ !CH₄þH F2þClO₂!FClO2þF C₄H₆þC₂H₄ !cycloC₆H₁₀ Comment on the values obtained.

The following are rough values for the contributions to the entropy for translation: 50 J mol¹K¹per degree of freedom

rotation: 40 J mol¹K¹per degree of freedom

vibration: 10 J mol¹K¹per degree of freedom.

4. Taking the collision diameter to be 400 pm, calculate the collision number,Z, for collisions between the molecules CH2CH–CHCH2 and CH2CH–CHO at 500 K, and from this find the pre-exponential factor,A.

The following are rough values for the contributions to the entropy from translation: 40 J mol¹K¹per degree of freedom

rotation: 25 J mol¹K¹per degree of freedom

vibration: 10 J mol¹K¹per degree of freedom.

Using this, make a rough estimate of the transition state theory pre-exponential factor.

(The experimentalAfactor¼1:610⁶ mol¹dm³s¹.)

5. Use the following data for a unimolecular decomposition to determinek1andk2

which appear in the simple Lindemann mechanism; assume thatk1has a value of 5:010¹⁰mol¹dm³s¹. From this determine the mean lifetime of the activated molecule. Comment on the results.

(Proceed by plotting 1/k^1st_obs versus 1/[A].)

10⁶½A

mol dm³ 10 15 20 40

k_obs^1st

s¹ 0:333 0:391 0:426 0:495

The following values extend the data to higher concentrations:

10⁶½A

mol dm³ 100 200 400

k_obs^1st

s¹ 0:625 0:781 0:952

Comment on the plot of 1/kobsversus 1/[A] which is obtained if all the values from both tables are included.

FURTHER PROBLEMS 163

6. Predict the temperature dependence ofA for the bimolecular reactions given in question 3.

BrþH₂!HBrþH CH₃þH2!CH4þH F₂þClO₂ !FClO₂þF C4H6þC2H4 !cyclo-C6H10

(a) the translational partition function for each degree of freedom is proportional to T^1/2;

(b) the rotational partition function for each degree of freedom is proportional toT^1/2; (c) the vibrational partition function for each degree of freedom is independent of temperature at low temperatures, but is proportional toT¹at high temperatures.

7. The following are values of entropies of activation for some unimolecular reactions, together with values ofS for the overall reaction:

Reaction S^6¼

J mol¹ K¹

S J mol¹ K¹

cyclo-propane!propane 39 29:5

cyclo-butane!2C₂H₄ 42 146:9

CH₂ O

CH₂ CH₃CHO

21 21:8

CH₃NC!CH₃CN 4 3:3

cyclo-C4F8!2C2F4 49 172:9

Compare the S^6¼ values with the corresponding S, and comment on the comparisons. Find theS^6¼values for the reverse reactions, and comment on the values obtained.

The values ofS given in the table are based on a standardconcentration, as distinct from the normal values based on a standardpressure.This has been done because the values ofS^6¼ are based on a standardconcentration.

5 Potential Energy Surfaces

The calculation of the PE surface is basically quantum mechanical. Accurate surfaces are used to show how the topography of the surface affects the ‘reaction unit’ as it changes configuration across the surface. Predictions can be made, and these can be tested by molecular beams, spectroscopic techniques and chemiluminescence.

Aims

By the end of this chapter you should be able to

distinguish between symmetrical, early and late barriers

correlate the relative distances between atoms in the ‘reaction entity’ at the critical configuration with the type of potential energy surface

list the properties of early barriers and correlate them with molecular beam results list the properties of late barriers and correlate them with molecular beam results

and

predict the dominant type of energy present in products and which type of energy promotes reaction.