In Chapter 3 we presented a number of rate laws that depended on both concentration and temperature. For the elementary reaction A + B C + D the elementary rate law is _{} We want to provide at least a qualitative understanding as to why the rate law takes this form. We will first develop the collision rate, using collision theory for hard spheres of cross section S_{r}, _{} When all collisions occur with the same relative velocity, U_{R} the number of collisions between A and B molecules, _{}, is _{} {collisions/s/molecule] Next we will consider a distribution of relative velocities and only consider those collisions that have an energy of E_{A} or greater in order to react to show _{} where _{} with _{AB} = collision radius, k_{B} = Boltzmann's constant, _{AB} = reduced mass, T = temperature and N_{Avo} = Avogadro's number. To obtain an estimate of E_{A}, we use the Polyani Equation _{} Where H_{Rx} is the heat
of reaction and _{} and _{P} are the Polyani Parameters. With these
equations for A and E_{A} we can make a first approximation to the rate law
parameters without going to the lab. 
References for Collision Theory, Transition State Theory and Molecular Dynamics
P. Atkins, Physical Chemistry, 6th ed. (New York: Freeman, 1998)
P. Atkins, Physical Chemistry, 5th ed. (New York: Freeman, 1994).
G. D. Billing and K. V. Mikkelsen, Introduction to Molecular Dynamics and Chemical Kinetics (New York: Wiley, 1996).
P.W. Atkins, The Elements of Physical Chemistry, 2nd ed. (Oxford: Oxford Press, 1996).
K. J. Laidler, Chemical Kinetics, 3rd ed. (New York: Harper Collins, 1987).
G. Odian, Principles of Polymerization, 3rd ed. (New York: Wiley 1991).
R. I. Masel, Chemical Kinetics and Catalysis, Wiley Interscience, New York, 2001.
As a short hand notation we will use the following references nomenclature
A6p701 Means Atkins, P. W. Physical Chemistry, 6th ed. (1998) page 701.
L3p208 Means Laidler, K. J., Chemical Kinetics, 3rd,ed. (1987) page 208.
This nomenclature means that if you want background on the principle, topic, postulate, or equation being discussed, go to the specified page of the text referenced.
The objectives of the development that follows are to give the reader insight as to why the rate laws depend on the concentration of the reacting species (i.e., r_{A} = kC_{A}C_{B}) and why the temperature dependence is the form of the Arrhenius law, k=Ae^{/RT}. To achieve this goal we consider the reaction of two molecules in the gas phase
A + B _{} C + D
We will model these molecules as rigid spheres.
s_{A} and s_{B}, respectively
Figure PRS.3A1 Schematic of molecules A and B
We shall define our coordinate system such that molecule B is stationary wrt molecule A so that molecule A moves towards molecule B with a relative velocity U_{R}. Molecule A moves through space to sweep out a collision volume with a collision cross section _{} illustrated by the cylinder shown below.
Figure PRS.3A2 Schematic of collision cross section
The collision radius is _{}.
_{}
If the center of a "B" molecule comes within a distance of _{} of the center of the "A" molecule they will collide.The collision cross section of rigid spheres is S_{r} _{}. As a first approximation we shall consider S_{r} constant. This constraint will be relaxed when we consider a distribution of relative velocities. The relative velocity between two gas molecules A and B is U_{R}.^{1}
(1)
where
k_{B} = Boltzmann's constant = 1.381 X 10^{23}J/K/molecule
= 1.381 kg m^{2}/s^{2}/K/molecule
m_{A} = mass of a molecule of species A (gm)
m_{B} = mass of a molecule of species B (gm)
μ_{AB} = reduced mass = _{} (g), [Let μ μ_{AB}]
M_{A} = Molecular weight of A (Daltons)
N_{Avo} = Avogadros' number 6.022 molecules/mol
R = Ideal gas constant 8.314 J/mol•K = 8.314 kg • m^{2}/s^{2}/mol/K
We note that R = N_{Avo} k_{B} and M_{A} = N_{Avo} • m_{A}, therefore we can write the ratio (k_{B}/_{AB}) as
(2)
An order of magnitude of the relative velocity at 300 K is _{}, i.e., ten times the speed of Indianapolis 500 Formula 1 car. The following collision diameter and velocities at 0°C are given in Table PRS.3A1
Table PRS.3A1 Molecular Diameters^{2}
Molecule 
Average Velocity, 
Molecular Diameter (Å) 
H_{2} 
1687 
2.74 
CO 
453 
3.12 
Xe 
209 
4.85 
He 
1200 
2.2 
N_{2} 
450 
3.5 
O_{2} 
420 
3.1 
H_{2}O 
560 
3.7 
C_{2}H_{6} 
437 
5.3 
C_{6}H_{6} 
270 
3.5 
CH_{4} 
593 
4.1 
NH_{3} 
518 
4.4 
H_{2}S 
412 
4.7 
CO_{2} 
361 
4.6 
N_{2}O 
361 
4.7 
NO 
437 
3.7 
Consider a molecule A moving in space. In a time t the volume V swept out by a molecule of A is
_{ }
Figure PRS.3A3 Volume swept out by molecule A in time Dt
The bends in the volume represent that even though molecule A may change directions upon collision the volume sweep out is the same. The number of collisions that will take place will be equal to the number of B molecules, V_{}, that are in the volume swept out by the "A" molecule, i.e.,
_{}
where _{} is in _{} rather than [moles/dm^{3}]
In a time t the number of collisions of this one A molecules with many B molecules is _{}. The number of collisions of this one A molecule with all the "B" molecules per unit time is
(3)
However, we have many A molecules present at a concentration _{}, (molecule/dm^{3}). Adding up the collisions of all the A molecules per unit volume, _{}, then the number of collisions _{} of all the A molecules with all B molecules per time per unit volume is
(4)
Where S_{r} is the collision cross section (Å)^{2}. Substituting for S_{r} and U_{R}
[molecules/time/volume] (5)
If we assume all collisions result in reactions then
[molecules/time/volume] (6)
Multiplying and dividing by Avogadros number, N_{Avo}, we can put our equation for the rate of reaction in terms of the number of moles/time/vol.
(7)
_{} (8)
where A is the frequency factor
(9)
(10)
EXAMPLE: Calculate the frequency factor A for the reaction
at 273K.
Additional Information
Using the values in Table PRS.3A1
Collision Radii
Hydrogen H _{H}=2.74 Å/4 = 0.75Å = 0.68 x 10^{10}m
Oxygen O_{2} _{} Å 1.55Å = 1.5 x 10^{10}m
_{}
(E)
(9)
The relative velocity is
(1)
Calculate the ratio (k_{B}/_{AB}) (Let _{AB})
_{}
_{}
Calculate the relative velocity
(A)
_{}
Calculate the frequency factor A
(B)
(C)
(D)
The value reported in Masel^{3} (p367) from Westley is
A=1.5*10^{14} (Å)^{3}/molecule/s
Close, but no cigar as Groucho Marx would say.
For many simple reaction molecules the calculated frequency factor, A_{calc} is in good agreement with
experiment. For other reactions, A_{calc}, can be an order of magnitude too high or
too low. In general, collision theory tends to over predict the frequency factor A
_{}
Terms of cubic angstroms per molecule per second the frequency factor is
_{}
There are a couple of things that are troublesome about the rate of reaction given by Equation (10), i.e.
(10)
First and most obvious is the temperature dependence. A is proportional to the square root of temperature and so therefore is r_{A}, i.e.
_{}
However we know that the temperature dependence of the rate of chemical reaction on temperature is given by the Arrhenius equation
(11)
or
(12)
This shortcoming of collision theory along with the assumption that all collisions result in reaction will be discussed next.
A. The collision theory outlined above does not account for orientation of the collision, front to back and along the lineofcenters. That is, molecules need to collide in the correct orientation for reaction to occur. Figure PRS.3A4 shows molecules colliding whose centers are off set by a distance b.
Figure PRS.3A4 Grazing collisions
B. Collision theory does not explain activation barriers. Activation barriers occur because bonds need to be stretched or distorted in order to react and these processes require energy. Molecules must overcome electronelectron repulsion in order to come close together^{4}
C. The collision theory does not explain the observed temperature dependence given by Arrhenius Equation
_{}
D. Collision theory assumes all A molecules have the same relative velocity, the average one.
(1)
However there is a distribution of velocities f(U,T). One distribution most used is the MaxwellBoltzmann distribution.
We are now going to account for the fact that we have (1) a distribution of relative velocities U_{R} and (2) that not all collisions result in a reaction, only those collisions with an energy E_{A} or greater. The goal is to arrive at
_{}
We will use the MaxwellBoltzmann Distribution of Molecular Velocities (A6p.26). For a species of mass m, the Maxwell
distribution of velocities (relative velocities) is
(13)
A plot of the distribution function, f(U,T), is shown as a function of U in Figure PRS.3A5.
Figure PRS.3A5 MaxwellBoltzmann distribution of velocities
Replacing m by the reduced mass of two molecules A and B.
_{}
The term on the left hand side of Equation (13), [f(U,T)dU] is the fraction of A molecules with relative velocities between U and (U + dU). Recall from Eqn. (4) that the number of AB collisions for a reaction cross section S_{r} is
(14)
except now the collision cross section is a function of the relative velocity.
Note we have written the collision cross section S_{r} as a function of velocity U, i.e., S_{r}(U). Why does the velocity enter into reaction cross section, S_{r}' Because not all collisions are head on, and those that are not, will not react if the energy, (i.e., U^{2}/2), is not sufficiently high. Consequently, this functionality, S_{r} = S_{r}(U), is reasonable because if two molecules collide with a very very low relative velocity it is unlikely that such a small transfer of kinetic energy is likely to activate the internal vibrations of the molecule to cause the breaking of bonds. On the other hand for collisions with large relative velocities most collision will result in reaction.
We now let k(U) be the specific reaction rate for a collision and reaction of AB molecules with a velocity U.
(15)
Equation (15) will give the specific reaction rate and hence the reaction rate for only those collisions with velocity U. We need sum up the collisions of all velocities. We will use the Maxwell Boltzmann distribution for f(U,T) and integrate over all relative velocities.
(16)
Maxwell distribution function of velocities for the A/B pair of reduced mass _{} is^{5}
(17)
Combining Equations (16) and (17)
(18)
We let S_{r}=S_{r}(U) for brevity. We will now express the distribution function in terms of the translational energy _{T}.
We are now going to express the equation for _{}. Equation (18) in terms of kinetic energy rather than velocity. Relating the differential translational kinetic energy, _{t}, to the velocity U.
_{}
noting and multiplying and dividing by _{} and we obtain
_{}
and hence the reaction rate
_{}
Simplifying
_{}
(19)
_{}
Multiplying and dividing by k_{B}T and noting _{}, we obtain
(20)
Again recall the tilde, i.e., _{} denotes that the specific reaction rate is per molecule (dm^{3}/molecule/s). The only thing left to do is to specify the reaction cross section, S_{r}(E) as a function of kinetic energy E for the A/B pair of molecules.
We now modify the hard sphere collision cross section to account for the fact that not all collisions result in reaction. Now we define S_{r} to be the reaction cross section defined as
_{}
Where P_{r} is the probability of reaction. In the first model we say the probability is either 0 or 1. In the second model P_{r} varies from 0 to 1 continuously. We will now insert each of these modules into Equation (20).
B.1 Model 1
In this model we say only those hard collisions that have kinetic energy E_{A} or greater will react. Let E _{t}. That is, below this energy, E_{A}, the molecules do not have sufficient energy to react so the reaction cross section is zero, S_{r}=0. Above this kinetic energy, all the molecules that collide react and the reaction cross section is _{}
Figure PRS.3A6 Reaction cross section for Model 1
Integrating Equation (20) by parts for the conditions given by Equations (21) and (22) we obtain
(23)
_{} _{}
_{ Generally , so}
Converting _{}to a per mole basis rather than a per molecular basis we have
_{}
_{}
The good news and the bad news. This model gives the correct temperature dependence but predicted frequency factor A' is even greater than A given by Eqn. (9) (which itself is often too large) by a factor (E_{A}/RT). So we have solved one problem, the correct temperature dependence but created another problem, too large a frequency factor. Let's try Model 2.
B.2 Model 2
In this model we again assume that the colliding molecules must have an energy
E_{A} or greater to react. However, we now assume that only the kinetic energy directed along the line of centers E_{<<}, is important. So below E_{A} the reaction cross section is zero, S_{r}=0. The kinetic energy of approach of A towards B with a velocity U_{R} is E = _{AB} _{}. However, this model assumes that only the kinetic energy directly along the line of centers contributes to the reaction.
(Click Back 2)
Here, as E increases above E_{A} the number of collisions that result in reaction increases. The probability for a reaction to occur is^{6}
(24)
and
A plot of the reaction cross section as a function of the kinetic energy of approach,
_{}
is shown in Figure PRS.3A7.
_{}
Figure PRS.3A7 Reaction cross section for Models 1 and 2
Recalling Equation (20)
(26)
Substituting for S_{r} in Model 2
(27)
Integrating gives
_{} _{}
Multiplying by Avogodro's number
_{}
(28)
Which is similar to the equation for hard sphere collisions except for the term _{}

(29)
This equation gives the correct Arrhenius dependence and the correct order of magnitude for A.
(11)
Effect of Temperature on Fraction of Molecules Having Sufficient Energy to React
We now will manipulate and plot the distribution function to obtain a qualitative understanding of how temperature increases the number of reacting molecules. Figure PRS.3A8 shows a plot the distribution function given by Equation (17) after it has been converted to an energy distribution.
We can write the MaxwellBoltzmann distribution of velocities
_{}
in terms of energy by letting _{} to obtain
_{} _{}
Fraction of collisions that have E_{A} or above
_{}
This integral is shown by the shaded area on Figure PRS.3A8
Figure PRS.3A8 Boltzmann distribution of energies
As we just saw only those collisions that have an energy E_{A} or greater result in reaction. We see form Figure PRS.3A10 that the higher the temperature the greater number of collision result in reaction.
However, this Equations (9) and (11) cannot be used to calculate A for a number reactions because of steric factors and the molecular orientation upon collision need to be considered. For example, consider a collision in which the oxygen atom, O, hits the middle carbon in the reaction to form the free radical on the middle carbon atom CH_{3}_{}HCH_{3}
_{}
Otherwise if it hits anywhere else, say the end carbon, _{} will not be formed^{7}
_{}
Consequently, collision theory predicts a rate 2 orders of magnitude too high for the formation of CH_{3}_{}HCH_{3}.
We will only state other definition is passing, except for the energy barrier concept which will be discussed in transition state theory.
A. Tolman's Theorem E_{a} = E^{*} _{}
_{}
The average transitional energy of a reactant molecule is _{}.
_{}
The energy barrier concept will be discussed in transition state theory, Ch3 Profession Reference Shelf B.
_{}
Figure PRS.3A9 Reaction coordinate diagram
For simple reactions the energy, E_{A}, can be estimated from computational chemistry programs such as Cerius^{2} or Spartan, as the heat of reaction between reactants and the transition state
_{}
The Polyani equation correlates activation energy with heat of reaction. This correlation
(33)
works well for families of reactions. For the reactions
_{}
where R = OH, H, CH3 the relationship is shown below in Figure PRS.3A10.
Figure PRS.3A10
Experimental correlation of E_{A} and H_{Rx} from
Masel (Lit cit). Courtesy of Wiley.
For this family of reactions
(34)
For example, when the exothermic heat of reaction is
_{}
The corresponding activation energy is
_{}
To develop the Polyani equation we consider the elementary exchange reaction^{8}
_{}
We consider the superposition of two attraction/repulsion potentials V_{BC}, V_{AB} similar to the LennardJones 612 potential. For the molecules BC, the LennardJones potential is
(35)
r_{BC} = Distance between molecules (atoms) B and C.
In addition to the LennardJones 612 model, another model often used is the Morse potential which has a similar shape
(36)
When the molecules are far apart the potential V (i.e., Energy) is zero. As they move closer together they become attracted to one another and the potential energy reaches a minimum. As they are brought closer together the BC molecules begin to repel each other and the potential increases. Recall that the attractive force between the B and C molecules is
(37)
The attractive forces between the BC molecules are shown in Figure PRS.3A11.
A potential similar to atoms B and C can be drawn for the atoms A and B. The F shown on the figures represents the attractive force between the molecules as they move in the direction shown by the arrows. That is the attractive force increase as we move towards the well (r_{o}) from both directions, r_{AB}>r_{o} and r_{AB}<r_{o}.
(a) (b)
Figure PRS.3A11 Potentials (Morse or LennardJones)
One can also view the reaction coordinate as a variation of the BC distance for a fixed AC distance, l
_{}
For a fixed AC distance as B moves away from C the distance of separation of B from C, r_{BC} increases as N moves closer to A. See point (2) in Figure PRS.3A11. As r_{BC} increases r_{AB} decreases and the AB energy first decreases then increases as the AB molecules become close. Likewise as B moves away from A and towards C similar energy relationships are found. E.g., as B moves towards C from A, the energy first decreases due to attraction then increases due to repulsion of the AB molecules as they come close together at point (2) in Figure PRS.3A11. The overlapping Morse potentials are shown in Figure PRS.3A11. We now superimpose the potentials for AB and BC to form the following figure
Figure PRS.3A12 Overlap of potentials (Morse or LennardJones)
Let S_{1} be the slope of the BC trajectory between r_{1} and r_{BC} = r_{AB} = r^{*}. Starting at E_{1R} and r_{1}, the energy of BC, E_{1}, at a separation distance of r_{BC} can be calculated from the product of the slope S_{1} and the distance from r_{1} at E_{1R}.
(38)
For Example, the energy, E_{1} (kJ/mol), of the BC molecule at a position r_{BC} = 5 nm relative to r_{1} (2 nm) where E_{1R} = 50 kJ/mol and S_{1} = 10 kJ/mol/nm is
_{}
Let S_{2} be the slope of the AB trajectory between r_{2} and r_{BC}=r_{AB} = r^{*}. Similarly for AB, starting on the product side at E_{2P}, as the distance r_{AB} increases the distance r_{BC} decreases and the energy E_{2} at a distance r_{BC} is
(39)
We note when we move up the AB trajectory from E_{2P} (80 kJ/mol) toward E^{*}, the distance between B and C, r_{BC}, decreases and the slope is negative. The AB energy, E_{2}, at a position r_{BC} = 7 nm relative to r_{2} (10 nm) when S_{2} = 20 kJ/mol/nm is
_{}
At the height of the barrier, B is equal distant from A and C, i.e.
(40)
Substituting for _{} and _{}
(41)
Rearranging
(42)
(43)
Solving for r* and substituting back into the Equation
yields
Equation (8) gives the Polyani correlation relating activation energy and heat of reaction. Values of _{} and g_{P} for different reactions can be found in Table 5.4 on page 254 of Masel. One of the more common correlations for exothermic and endothermic reactions are given on page 73 of Laidler.
(1) Exothermic Reactions
(45)
If _{}
then
_{}
(2) Endothermic Reactions
(46)
If _{}
then
_{}
Also see R. Masel Chemical Kinetics and Catalysis, page 603 to calculate the activation energy from the heat of reaction.
Szabo proposed an extension of Polyani equation
(47)
_{}
B. Marcus Extension of the Polyani Equation
In reasoning similar to developing the Polyani Equation Marcus shows (see Masel pp.584586)
_{}
C. BlowersMasel Relation
The Polyani Equation will predict negative activation energies for highly exothermic reactions Blowers and Masel
developed a relationship that compares quite well with experiment throughout the entire range of heat of reaction for
the family of reactions
_{}
as shown in Figure PRS.3A13
Figure PRS.3A13
Comparison of models with data. From Masel.
Courtesy of Wiley
We see the greatest agreement between theory and experiment is found with the BlowersMasel model.
We have now developed a quantitative and qualitative understanding of the concentration and temperature dependence of the rate law for reactions such as
A + B C + D
with
r_{A} = k C_{A}C_{B}
We have also developed first estimates for the frequency factor, A from collision theory
_{}
and the activation energy, E_{A} from the Polyani Equation
_{}
these calculated values for A and EA can be substituted in the rate law to determine rates of reaction.
_{}
† This equation is given in most physical chemistry books, e.g., see Moore, W. J. Physical Chemistry, 2^{nd} Ed. P.187, Prentice Hall, Englewood Cliffs, NJ
† O'Hanlon, J. F., A User's Guide to Vacuum Technology, Wiley, New York (1980)
† Masel, R. I. Chemical Kinetics and Catalysis, p367, Wiley, New York (2001).
† Masel, R. I., lit. cit.
† Moore, lit. cit. p185, Atkins, P. W. Physical Chemistry, 5th ed. p.36 (1994).
†Steinfeld, J. I., J. S. Francisco, and W. L. Hayes, Chemical Kinetics and Dynamics, (p250) Prentice Hall, Englewood Cliffs NJ (1989).
Masel lit. cit. p483.
† Masel, lit. cit. p.36
† after R. I. Masel Lit cit., Chemical Kinetics and Catalysis, Wiley, 2001
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