Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (13.81 MB, 563 trang )
Ch005.qxd
12/22/2006
116
10:27 AM
Page 116
5. Collisions and Molecular Dyanmics
data. Although they may lack the theoretical rigour of more fundamental theoretical formulations, these more empirical theories are important since, apart from their historical interest, they often act as pointers for the development of more quantitative theoretical models.
These can be directed towards real systems and can help us avoid using highly sophisticated
calculations to treat the wrong target. To quote the late John W. Tukey, who made major contributions to statistics, both in industrial and academic areas: “An approximate answer to the
right question is worth a great deal more than a precise answer to an approximate question”.
It is worth noting that John Tukey’s initial background was chemistry.
At a first glance, an ab initio theory would seem to be the most satisfactory way to
treat theoretically the chemical kinetics of any system, since this does not need to introduce any experimental data. However, only those systems involving 3–4 atoms in the
gas phase can be solved with chemical accuracy (Ϯ4 kJ/mol) by this route. For the vast
majority of reactions of chemical interest, we need either to introduce major simplifications into the most sophisticated theories so that they can be applied, or to opt for a simpler theory, including empirical parameters, which can be used without a posteriori
changes to estimate the rate constants. These approaches differ because in the first case,
approximations are made when the method is applied, while with the latter approach, the
approximations come within the formulation of the theory. In both cases, we hope that
we can understand and interpret any discrepancies between the observed and the calculated kinetic data. The approach we use frequently depends upon the nature of the chemical system. The biggest problem is how to interpret any disagreement between
experiment and calculation. This could result from experimental error, from wrong
approximations when we apply the theory or incorrect assumptions in the theory itself.
In the latter case, the theory is wrong and must be rejected. We must try to avoid the tendency of attributing any discrepancies between theoretical prediction and experimental
observation to approximations introduced in the application of the theory, rather than to
any weakness in the theory itself. The continued use of any obsolete theory past its “sellby date”, is scientifically incorrect as well as hindering the development of more appropriate models.
As a scientific theory is intended to simulate and help us understand the behaviour of
part of or the whole Universe, it should be kept sufficiently simple to be understandable,
and its proof should be purely operational. The relevant question should not be whether it
is true or false, since this does not have any meaning from a scientific viewpoint. Instead
we should ask: “does it work”?
Even so, the answer to this question may not be completely objective. We can discard a
theory that contains errors of logic, but a similar decision based upon comparison between
theoretical and experimental data depends on how rigorous we want the agreement to be.
Based on these ideas, we will discuss the simple collision theory, which is attractive
because of the relationship it provides between the orientation and the energy of the molecules involved in collisions and their reactivity. However, it is not actually satisfactory as
a theoretical hypothesis for polyatomic systems, since it cannot be used quantitatively for
the calculation of rates of chemical reactions. However, for triatomic systems whose
potential energy surfaces (PESs) are well defined, the calculation of trajectories using this
approach provides an excellent method for calculating the reaction rates. Thus, we will
restrict this type of calculation to simple systems. For polyatomic systems or reactions in
Ch005.qxd
12/22/2006
5.1
10:27 AM
Page 117
Simple Collision Theory
117
solution, it is totally impracticable and we will make use of other theories to be discussed
in the following chapter.
5.1
SIMPLE COLLISION THEORY
The simplest model for the rates of chemical reactions assumes that every time there is a
bimolecular collision, there is a reaction. From calculations of the frequency of collisions
in this system, we can then determine the rate constant.
To calculate the number of collisions per unit time, we need a model for the behaviour
of molecules in these systems. The simplest approach involves a system of two gases, A
and B, whose molecules behave as hard spheres, which are characterised by impenetrable
radii rA and rB. A collision between A and B occurs when their centres approach within a
distance dAB, such that
dAB = rA + rB
(5.1)
If we assume that the molecules of B are fixed and that those of A move with an average
–
2 –
velocity v A, each molecule A sweeps a volume dAB v A per unit time which contains sta2
tionary molecules of B. The area p ϭ dAB , is known as the collision cross section. If
there are NB/V molecules of type B per unit volume, the number of collisions of a molecule of type A with the stationary molecules B will be
zAB =
2
dAB vA N B
V
(5.2)
If the total number of molecules of A per unit volume is NA/V, then the total number of collisions of A with B per unit volume is given by
Z AB =
2
dAB vA N B N A
V2
(5.3)
As indicated above, we have assumed that the molecules of B are stationary to obtain the
above expression. In practice, Figure 5.1 shows that, for each pair of molecules, A and B,
involved in a collisional trajectory, we can define a relative velocity vAB, which is related
to their velocities, vA and vB, according to
(
2
2
vAB = vA + vB − 2 vA vB cos
)
1
2
(5.4)
The value of cos can vary between Ϫ1 and 1. As all values of between 0 and 360° are
equally probable, the positive and negative values of cos will cancel out, and the mean
value will be zero. Thus we obtain
2
2
2
vAB = vA + vB
(5.5)
Ch005.qxd
12/22/2006
10:27 AM
Page 118
118
5. Collisions and Molecular Dyanmics
Figure 5.1 Relative velocities for a collisional trajectory between molecules A and B.
Assuming that the molecular velocities are described by a Maxwell–Boltzmann distribution, the mean velocity of the molecules of a gas A is given by eq. (1.11). Consequently,
the relative mean molecular velocity of molecules of types A and B will be
vAB =
8 k BT
(5.6)
where the reduced mass is ϭ mAmB/(mA ϩ mB). Introducing this value for the relative
velocity in eq. (5.3), we obtain
Z AB =
2
dAB vAB N B N A
V2
(5.7)
If we consider a gas that contains only molecules of type A, the total number of collisions
would be
Z AA =
2
2
1 4rA vAA N A
2
V2
(5.8)
where the factor ½ arises because we cannot count the same molecule twice. Since from
eq. (5.5), the relative mean velocity of the molecules of a gas can be related to the mean
–
–
velocity of the molecules, v AAϭ √2 v A, eq. (5.8) can be rewritten as
Z AA =
2
2
2 2 rA vA N A
V2
(5.9)
The expressions for the number of collisions per unit time, also known as the collision densities, contain two factors involving the number of molecules per unit volume. These can be
expressed in terms of pressures or molar concentrations, giving the rate constant for a bimolecular reaction between molecules of A and B in the experimentally more meaningful form
kAB =
v
R AB
2
= dAB
8 k BT
(5.10)
Ch005.qxd
12/22/2006
5.1
10:27 AM
Page 119
Simple Collision Theory
119
We can test this collision theory approximation by comparing the value calculated for the
elementary reaction
NO2 + CO ⎯⎯ NO + CO2
→
(5.I)
at 870 K with the experimental value, kexp ϭ 1.5ϫ102 dm3 molϪ1 secϪ1. From the van der
Waals radii of N and O atoms, the reactant bond lengths, and assuming that these behave
as hard spheres, we can estimate that dAB ϭ 4ϫ10Ϫ10 m. The reduced mass
can be calculated from the relative molecular masses of nitrogen dioxide and carbon
monoxide
=
46 × 28
1.661 ×10 −27 = 2.89 ×10 −26 kg
46 + 28
(5.11)
where the units conversion factor in the unified atomic mass unit, from which we obtain the
relative mean velocity, using the relationship 1 J ϭ 1 kg m2 secϪ2,
vNO2 ,CO =
8 × 1.38 × 10 −23 × 870
= 1.03 × 103 m sec −1
3.14 × 2.89 × 10 −26
(5.12)
such that we calculate the rate constant to be kAB ϭ 5.17ϫ10Ϫ16 m3 moleculesϪ1 secϪ1 ϭ
3.11ϫ1011 dm3 molϪ1secϪ1. We can see from the comparison between the experimental
and calculated values that the latter are a factor of at least 109 greater than the former.
In this simple collision theory, there are two implicit approximations which lead to calculated values that are considerably greater than the experimental ones: (i) not all the collisions occur with a favourable orientation for reaction; (ii) we need a certain amount of
energy for the conversion of the reactants into products, and not all the collisions have
enough energy to produce this chemical transformation.
The first of these approximations can be corrected for by introducing a geometric factor, g Յ 1. This can be calculated using the molecular geometries of the reactants and products. In the example of reaction (5.I), it is necessary that in the collision the NO bond is
extended and its oxygen atom directly approaches the carbon atom of CO for the reaction
to occur (Figure 5.2). When the N᎐O and C᎐O bonds are collinear, the chemical transformation results from the stretching of these bonds, since these stretching vibrations are of
high frequency and energy. In this model, the geometric factor g is given by the product
of the orientation factor for the CO molecule, ½, and that of the NO2 molecule, which, as
indicated in Figure 5.2, will be approximately 2/17. Therefore, for this reaction, g ϭ 1/17.
The geometric factor may become much less than unity with increasing complexity of
the reaction in the gas phase. In addition, with reactions in solution, we need to consider
effects of solvent cage, which allow many collisions to occur during an encounter between
the two reactants, such that the effect of the geometric factor becomes more difficult to calculate. However, as we will see, in these systems, too, it is less relevant.
The energetic factor is of fundamental importance in the determination of reaction
probabilities and, hence, rate constants. Once more, we start from the Maxwell–Boltzmann
distribution (eq. 1.10) to calculate this factor. In practice, in collision theory, we normally
Ch005.qxd
12/22/2006
10:27 AM
Page 120
120
5. Collisions and Molecular Dyanmics
Figure 5.2 Sketch of the origin of the geometric factor in a collision. The favourable orientations
for the attack of the carbon atom of CO are shaded.
use the distribution function of molecular velocities in two dimensions rather than in
three.
⎛ − mv 2 ⎞
dN
m
=
exp ⎜
vdv
N 0 k BT
⎝ 2 k BT ⎟
⎠
(5.13)
Although there is no strong theoretical basis for this decrease in dimensionality, it simplifies the system, and we can rationalise it by considering that at the instant of collision
between the two molecules, the velocity vectors have a common point such that they lie
within a plane. Thus, the components of the velocities within the two dimensions that define
this plane are sufficient to describe an effective collision. The distribution of molecular
energies, based on the above equation, is
dN (
N0
)=
⎛ − ⎞
1
exp ⎜
d
k BT
⎝ k BT ⎟
⎠
(5.14)
which can be integrated to obtain the fraction of collisions with energy equal to or greater
than the critical value, c, for the reaction to occur.
∫
ε > εc
dN (
N0
)=∞
1
⎛ − ⎞
εc
B
B
∫ k T exp ⎜ k T ⎟ d
⎝
⎠
⎛− ⎞
= exp ⎜ c ⎟
⎝ k BT ⎠
(5.15)
If we include this energetic term and the geometric factor g in the expression for the rate
constant, we obtain the final expression for the rate constant given by the collision theory
Ch005.qxd
12/22/2006
5.1
10:27 AM
Page 121
Simple Collision Theory
121
that considers only effective collisions.
2
kcol = g dAB
8 k BT
⎛− ⎞
exp ⎜ c ⎟
⎝ k BT ⎠
(5.16)
This expression bears an obvious resemblance to that of the Arrhenius law. In fact the preexponential factor in the latter expression can be identified as
2
A = g dAB
8 RT
(5.17)
The activation energy, expressed in molar terms, is given by
Ea =
d ln ( kcol )
d (1 RT )
= Ec + 1 RT
2
(5.18)
The expressions (5.16)–(5.18) were first applied to bimolecular gas phase reactions by
M. Trautz (1916) and W.C. McC. Lewis (1924). The terms in d and in eq. (5.17) generally compensate each other, such that, if we exclude the geometric factor g, in many cases
log A ϭ10.5Ϯ0.5, with A given in dm3 molϪ1 secϪ1. Steric hindrance, caused by the presence of bulky groups in the positions adjacent to the point of attack, is seen mainly through
an increase in the activation energies of reactions and not through a decrease in the frequency of effective collision. Thus, the simple collision theory does not lead to significant
correlations between the changes in structures and the calculated values for A. Some people have refined this theory by defining an entropy of activation for the molecular collisions, leading to
2
A = g dAB
8 RT
(
exp ⌬S ∗ R
)
(5.19)
The introduction of this extra term in the expression for the pre-exponential factor may
explain values much less than those determined by the frequency of collisions. However,
as in most situations it is not easy to calculate this entropy of activation, this new term is
only of limited use.
The numerical value of the collision theory value rate constant, previously calculated for
the NO2 ϩ CO reaction (3.11ϫ1011 dm3 molϪ1 secϪ1) corresponds to the typical values calculated for the pre-exponential factors of gas-phase bimolecular reactions. Dividing this value
by the molar volume of an ideal gas at standard temperature and pressure (22.421 dm3 molϪ1),
we obtain a relaxation rate, which is approximately (p )Ϫ1 ϭ 1010 atmϪ1 secϪ1 at 273 K. This
implies that the mean free time between collisions of molecules of an ideal gas is 0.1 nsec,
and that the corresponding collision frequency is collϪ1
1010 secϪ1. Some caution must
be made in using these values, since molecules interact with a range of velocities and
instantaneous inter-molecular distances, such that there is no single collision frequency, but
Ch005.qxd
12/22/2006
10:27 AM
Page 122
122
5. Collisions and Molecular Dyanmics
there is a distribution of time intervals between collisions. This distribution can be given in
terms of a Poisson function in n
P (n) = e −
n
(5.20)
n!
where the mean value of n is ϽnϾ ϭ ϭ T/ coll, and T is the number of collisions per unit
time.
The simple collision theory treatment shows the basic elements for a bimolecular rate
constant: we need a collision to occur, a certain critical energy has to be exceeded and the
particles involved in the collision have to have the correct relative orientations.
5.2
COLLISION CROSS SECTION
Instead of assuming hard sphere behaviour, we can refine the collision theory by using
cross sections calculated on the basis of microscopic interaction potentials of species A and
B during their collision, R. Before we develop the problem for reactive collisions, we will
start by treating the simpler situation of an elastic collision between two bodies subjected
to a central force, which only leads to scattering.
The collision of these two bodies can be seen by a system of coordinates based on the
centre of mass (Figure 5.3). The equations of motion will be determined by the conservation of initial energy, E, and of angular momentum, L. As the direction of L is conserved,
the trajectory has to remain in a plane and the equations of motion can be written, in polar
coordinates
L = vb = r 2
dφ
dt
2
2
1 2 1 ⎛ dr ⎞ 1 2 ⎛ dφ ⎞
E=
v =
+
r ⎜ ⎟ + V (r )
2
2 ⎜ dt ⎟ 2
⎝ ⎠
⎝ dt ⎠
(5.21)
2
1 ⎛ dr ⎞
L2
=
+ V (r )
⎜ dt ⎟ +
2 ⎝ ⎠ 2 r2
where v is the relative incident velocity and b is the impact parameter. Solving these equations, we get
dφ =
L2
dt
r2
(5.22)
and
12
⎧2 ⎡
L2 ⎤ ⎫
⎪
⎪
dr = − ⎨ ⎢ E − V ( r ) −
⎥ ⎬ dt
2 r 2 ⎦⎭
⎪
⎪
⎩ ⎣
(5.23)
Ch005.qxd
12/22/2006
5.2
10:27 AM
Page 123
Collision Cross Section
123
Figure 5.3 Classical trajectory for velocity v and impact parameter b, with deflection angle , following a system of coordinates based on the centre of mass. The classical turning point is rc.
or
⎧2 ⎡
L2 ⎤ ⎫
⎪
⎪
dt = − ⎨ ⎢ E − V (r ) −
2 ⎥⎬
2 r ⎦⎭
⎪
⎪
⎩ ⎣
−1 2
dr
(5.24)
Now, we can write
t ⎛ L ⎞
φ ( t ) = ∫ ⎜ 2 ⎟ dt
−∞ ⎝ r ⎠
= −∫
L r2
r
∞
12
⎧2 ⎡
L2 ⎤ ⎫
⎪
⎪
E − V (r ) −
⎬
⎨ ⎢
2 r 2 ⎥⎪
⎪ ⎣
⎦⎭
⎩
dr
(5.25)
and use the relationship L ϭ vb ϭ b(2 E)½ to obtain
φ ( t ) = ±b ∫
dr
r
∞
⎡ b V (r ) ⎤
r ⎢1 − 2 −
⎥
E ⎦
⎣ r
2
12
2
(5.26)
The sign must be chosen to agree with that of the radial velocity: that is positive for the
species leaving and negative for those approaching.
For any collision, there will always be a maximum approach distance rc for which r(t)
has its minimum value. In this classical turning point all the initial kinetic energy is
converted into potential energy. It is convenient to define t ϭ 0 for r(t) ϭ rc and φ(0)ϭ φc.
The complete classical trajectory can be determined by integrating eq. (5.26) from infinity to rc and then from rc forward. However, it is unnecessary to follow the trajectory
Ch005.qxd
12/22/2006
10:27 AM
Page 124
124
5. Collisions and Molecular Dyanmics
with this detail when we are only observing what happens after the collision. In this case
it is sufficient to know the total deflection, Θ. As shown in Figure 5.3, this deflection is
given by
Θ ( E , b) = − 2φc
= − 2b ∫
∞
rc
dr
⎡ b V (r ) ⎤
r ⎢1 − 2 −
⎥
E ⎦
⎣ r
12
2
(5.27)
2
For us to analyse the behaviour of the angle of deflection as a function of the energy and
of the impact parameter, we have to define the potential. In the simplest case of the potential of hard spheres
V (r ) = 0, r > dAB
(5.28)
V (r ) = ∞, r ≤ dAB
it is obvious that independent of the energy of collision rc ϭ dAB. Under these conditions,
the integration of eq. (5.27) leads to
Θ ( E , b ) = − 2b ∫
= − 2b ∫
∞
dAB
∞
dAB
dr
⎡ b2
⎤
r 2 ⎢1 − 2 − 0 ⎥
r
⎣
⎦
dr
1
2
r r 2 − b2
r =∞
⎧1
⎛ b⎞ ⎫
= − 2b ⎨ cos−1 ⎜ ⎟ ⎬
⎝ r ⎠ ⎭r = d
b
⎩
AB
(5.29)
⎡
⎛ b ⎞⎤
⎛ b⎞
= π − 2 ⎢cos−1 ⎜ ⎟ − cos−1 ⎜
⎥
⎝ ∞⎠
⎝ dAB ⎟ ⎦
⎠
⎣
⎛ b ⎞
= − + 2 cos−1 ⎜
⎝ dAB ⎟
⎠
⎛ b ⎞
= 2 cos−1 ⎜
⎝ dAB ⎟
⎠
For impact parameters greater than the hard sphere diameter dAB, eq. (5.29) requires the
reciprocal of the cosine of a number greater than one, which is not possible. For this case,
Θ(b) ϭ 0. For impact parameters less than dAB, the value of Θ(b) varies between 0 and
(Figure 5.4).
Ch005.qxd
12/22/2006
5.2
10:27 AM
Page 125
Collision Cross Section
125
Figure 5.4 Deflection function Θ(b) for the hard sphere potential of diameter dAB.
Figure 5.5 Lennard–Jones potential, showing the minimum at rm ϭ21/6/
LJ
with an energy V(rm)ϭϪ .
A potential energy function that represents the interaction between two bodies in a more
realistic form is the Lennard–Jones potential
⎡⎛ LJ ⎞ 12 ⎛ LJ ⎞ 6 ⎤
V ( r ) = 4 ⎢⎜
⎟ −⎜ r ⎟ ⎥
⎝
⎠ ⎥
⎢⎝ r ⎠
⎣
⎦
(5.30)
shown in Figure 5.5, where the attractive term in r represents the dispersion forces as well
as dipole–dipole and dipole-induced dipole interactions. The purely empirical repulsive
part was chosen from adjustment to a wide range of atomic potentials. The deflection function of this potential can be calculated using eq. (5.27). The arithmetical treatment of this
Ch005.qxd
12/22/2006
126
10:27 AM
Page 126
5. Collisions and Molecular Dyanmics
Figure 5.6 Deflection function Θ(E,b) for the Lennard–Jones potential. The minima correspond to
the “rainbow” angle discussed in the text.
solution is very complicated. In Figure 5.6, we show only the results in terms of the
reduced impact parameter b* ϭ b/ LJ for a range of reduced energies E* ϭ E/ . This figure shows three regions with a different dispersive behaviour.
For collisions that are almost head-on (b → 0), the deflection function is very similar to
that of collision between hard spheres (Figure 5.4), because the impinging particle passes
through the attractive part of the potential and is repelled by the wall given by rϪ12. This
behaviour of specular reflection can be found for scattering with almost all potentials.
For very large-impact parameters, the dispersion is also similar to that for the hard
sphere, that is, the impinging particle is hardly scattered in its trajectory. This is due to the
rapid decay of the attractive part of the potential to zero. We should note that this limit is
reached more rapidly when the energy of the incident particle increases.
In the intermediate region of b* and E* we find a new behaviour. The deflection angles
can take considerably negative values, even less than Ϫ . This results from the strong
interaction of the incident particle with the attractive part of the Lennard–Jones potential,
as is shown in Figure 5.7.
The above discussion focussed on the trajectory of a single event, specified by an initial energy and an impact parameter. From the practical point of view, although it is possible to accurately define the initial energy in a molecular beam experiment, it is
impossible to isolate a single impact parameter. Thus, the experimental results are presented in terms of collision cross sections, as shown in Figure 5.8. In the case of elastic
scattering, we can distinguish between the total cross section (E) and the differential
cross section d /d . This latter parameter is defined as the intensity of scattering per solid