Professional Reference Shelf

R3.1 Collision Theory

1. Overview
2. Fundamentals of Collision Theory
3. Shortcoming of Collision Theory
4. Modifications of Collision Theory
   1. Distribution of Velocities
   2. Collisions Resulting in Reaction
      1. Model 1 Pr = 0 or 1
      2. Model 2 Pr = 0 or Pr = (E-EA)/E
5. Other Definitions of Activation Energy
   1. Tolman
   2. Fowler and Guggenheim
   3. Barrier Height and Potential Energy Surfaces
6. Polyani Equation
7. Problems
8. Closure

I. Overview ­ Collision Theory

II.       Fundamentals of Collision Theory

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_AC_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.

Figure PRS.3A-1  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.3A-2  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^-23J/K/molecule
 = 1.381 kg m²/s²/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 • m²/s²/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.3A-1

Table PRS.3A-1 Molecular Diameters²

Consider a molecule A moving in space. In a time t the volume V swept out by a molecule of A is

Figure PRS.3A-3  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/dm³]

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/dm³). 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)

EXAMPLE: Calculate the frequency factor A for the reaction

at 273K.
Additional Information
Using the values in Table PRS.3A-1
Collision Radii
      Hydrogen H           _H=2.74 Å/4 = 0.75Å = 0.68 x 10^­10m

                        Oxygen O₂             Å 1.55Å = 1.5 x 10^­10m

Solution

                  (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 Masel³ (p367) from Westley is

A=1.5*10¹⁴ (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.

III.     Shortcomings of Collision Theory

A.  The collision theory outlined above does not account for orientation of the collision, front to back and along the line-of-centers. That is, molecules need to collide in the correct orientation for reaction to occur. Figure PRS.3A-4 shows molecules colliding whose centers are off set by a distance b.

Figure PRS.3A-4  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 electron-electron repulsion in order to come close together⁴

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 Maxwell-Boltzmann distribution.

IV.    Modifications of Collision Theory

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

A. Distribution of Velocities

We will use the Maxwell-Boltzmann 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.3A-5.

Figure PRS.3A-5  Maxwell-Boltzmann 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., U²/2), 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 A-B 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  is⁵

            (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_BT and noting , we obtain

         (20)

Again recall the tilde, i.e.,  denotes that the specific reaction rate is per molecule (dm³/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.

B.  Collisions that Result in Reaction

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.3A-6  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 is⁶

             (24)

and

A plot of the reaction cross section as a function of the kinetic energy of approach,

is shown in Figure PRS.3A-7.

Figure PRS.3A-7  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.3A-8 shows a plot the distribution function given by Equation (17) after it has been converted to an energy distribution.

            We can write the Maxwell-Boltzmann 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.3A-8

Figure PRS.3A-8  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.3A-10 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 CH₃HCH₃

Otherwise if it hits anywhere else, say the end carbon,  will not be formed⁷

Consequently, collision theory predicts a rate 2 orders of magnitude too high for the formation of CH₃HCH₃.

V.    Other Definitions of Activation Energy

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 .

B.  Fowler and Guggenheim

C.  Energy Barrier

The energy barrier concept will be discussed in transition state theory, Ch3 Profession Reference Shelf B.

Figure PRS.3A-9  Reaction coordinate diagram

For simple reactions the energy, E_A, can be estimated from computational chemistry programs such as Cerius² or Spartan, as the heat of reaction between reactants and the transition state

VI.     Estimation of Activation Energy From the Polyani equation

A.  Polyani Equation

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.3A-10.

Figure PRS.3A-10  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 reaction⁸

We consider the superposition of two attraction/repulsion potentials V_BC, V_AB similar to the Lennard-Jones 6-12 potential. For the molecules BC, the Lennard-Jones potential is

                       (35)

r_BC = Distance between molecules (atoms) B and C.

            In addition to the Lennard-Jones 6-12 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.3A-11.

            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.3A-11 Potentials (Morse or Lennard-Jones)

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.3A-11. 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.3A-11. The overlapping Morse potentials are shown in Figure PRS.3A-11. We now superimpose the potentials for AB and BC to form the following figure

Figure PRS.3A-12  Overlap of potentials (Morse or Lennard-Jones)

Let S₁ be the slope of the BC trajectory between r₁ and r_BC = r_AB = r^*. Starting at E_1R and r₁, the energy of BC, E₁, at a separation distance of r_BC can be calculated from the product of the slope S₁ and the distance from r₁ at E_1R.

              (38)

For Example, the energy, E₁ (kJ/mol), of the BC molecule at a position r_BC = 5 nm relative to r₁ (2 nm) where E_1R = -50 kJ/mol and S₁ = 10 kJ/mol/nm is

Let S₂ be the slope of the AB trajectory between r₂ 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 E₂ 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, E₂, at a position r_BC = 7 nm relative to r₂ (10 nm) when S₂ = -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

(44)

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)

In reasoning similar to developing the Polyani Equation Marcus shows (see Masel pp.584-586)

C. Blowers-Masel 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.3A-13

Figure PRS.3A-13  Comparison of models with data.  From Masel.
Courtesy of Wiley

We see the greatest agreement between theory and experiment is found with the Blowers-Masel model.

VII.    Problems

1. † 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

2. † O'Hanlon, J. F., A User's Guide to Vacuum Technology, Wiley, New York (1980)

3. † Masel, R. I. Chemical Kinetics and Catalysis, p367, Wiley, New York (2001).

4. † Masel, R. I., lit. cit.

5. † Moore, lit. cit. p185, Atkins, P. W. Physical Chemistry, 5th ed. p.36 (1994).

6. †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.

7. † Masel, lit. cit. p.36

8. † after R. I. Masel Lit cit., Chemical Kinetics and Catalysis, Wiley, 2001