Page 183 - Essentials of physical chemistry

Basic Chemical Kinetics 145

Let us check this with the data at 36 min.

b(a x) (0:02040)(0:02910)
ln ln
a(b 2x) (0:03680)(0:00518)
¼ 0:593134 L=mol min
(2a b)t ¼ k 2 ¼ [2(0:03680) 0:02040 mol=L](36 min)
The average of these two values is 0.587 L=mol min and once again there perhaps should be more
confidence in the value from the longer time period of 52 min but the two values are close enough to
support the assignment of a second-order rate process.
There is still a caution in assigning the order of a rate equation. Although, we have taken the
precaution of calculating the rate constant using the assumed order over several data points in the
examples above, it will be necessary in general to test that assumption over a wide range of time
intervals if the rate equation is not a ‘‘tight fit’’ with variability in the calculated rate constant. The
key comparison will be to compare the error range for alternate models to determine the ‘‘best’’
order. In the examples above, we did see variation in the values of the calculated rate constant but in
some cases it may be due to experimental error. The examples we have shown are fairly clear cut
determinations of the order but the case of the pseudo-first order hydrolysis was included to show
how a true second-order reaction can seem like a first-order reaction and in that case the drift in the
order is also complicated by the problem that the ice quenching of the reaction samples is not
complete, so there is an added analytical problem in the data.

ARRHENIUS ACTIVATION ENERGY
In the example above with the pseudo-first order hydrolysis reaction, the procedure tried to ‘‘quench’’
the reaction by adding the aliquot sample to ice water. This implies there is a temperature effect on the
reaction rate.The simplest treatment of this effect was bySvante A. Arrhenius(1859–1927), aSwedish
physicist who formulated the first explanation of the temperature dependence of reaction rate con-
stants. Most organic chemistry textsshow asinglepotential energy barrier to areaction alonga reaction
coordinate. Note that a ‘‘reaction coordinate’’ is a distance along a path of progressive geometrical
distortionofthereactingspecieswhichresults intheformationofproduct.The‘‘extentofreaction’’isa
quantitative mole concept while the ‘‘reaction coordinate’’ is a geometrical concept. A little thought
leads one to the realization that there must be more than one reaction coordinate since most organic
reactions lead to more than one product molecule, but each reaction coordinate will have a barrier that
needs to be overcome before the overall DH 0 is realized. Although we do not know how Arrhenius
rxn
thought of this, the formula incorporates aspects of the Boltzmann principle. In order for a molecule to
reach the energy to go over the barrier there would be a Boltzmann probability of having that energy.

E*
k ¼ A e RT
Since the exponent is a unitless number, the value of A has the units of the rate constant k. Here, A is
the ‘‘Arrhenius constant’’ and is believed to be related to the number of binary collisions Z 11 we
encountered in the kinetic theory of gases. In the experience of this author, a few calculations for
reactions of small molecules using collisions augmented with a ‘‘steric factor’’ do give qualitative
agreement with experiment for gas phase reactions. However, A becomes merely a large number
when fitted to data for reactions in solution. Even so, we can take the natural log of the equation and
gain an appreciation for how the rate constants change with temperature and that leads to an
experimental value for what is called the ‘‘activation energy, E*.’’ The activation energy is related
to the amount of energy required to pass over the barrier in the reaction coordinate. Because of the
simplicity of the Arrhenius formula and the ability to fit experimental data with only two parameters
(A, E*), the Arrhenius formula is extensively used in computer programs which model gas phase
reactions that turn out to be far more complicated than one might think. For instance, it takes well