Transition state theory provides an approach to explain the temperature and concentration dependence of the rate law. For example, for the elementary reaction
The rate law is
For simple reactions transition state can predict E and A in concert with computational chemistry. In transition state theory (TST) an activated molecule is formed during the reaction at the transition state between forming products from reactants.
The rate of reaction is equal to the product of the frequency, vI, of the activated complex crossing the barrier and the concentration of the transition state complex
The transition state molecule and the reactants are in pseudo equilibrium at the top of the energy barrier.
We will now use statistical and quantum mechanics to evaluate to arrive at the equation
where q¢ is overall the partition function per unit volume and is the product of translational, vibration, rotational and electric partition functions, i.e.,
The individual partition functions to be evaluated are:
for diatomic molecules
The Eyring Equation
R. I. Masel, Chemical Kinetics and Catalysis, Wiley Interscience, New York, 2001.
A5p403 Means Atkins, P. W. Physical Chemistry, 5th ed. (1994) page 403.
A6p701 Means Atkins, P. W. Physical Chemistry, 6th ed. (1998) page 701.
L3p208 Means Laidler, K. J., Chemical Kinetics, 3rd ed. (1987) page 208.
M1p304 Means Masel, R.I., 1st Edition (2001) page 304.
Figure PRS.3B-1 Evidence of Active Intermediate.
The active intermediate is shown in transition state at the top of the energy barrier. A class of reactions that also goes through a transition state is the SN2 reaction.
We shall first consider SN2 reactions [Substitution, Nucleophilic, 2nd order] because many of these reactions can be described by transition state theory. A Nucleophile is a substance (species) with an unshared electron. It is a species that seeks a positive center.
The nucleophile seeks the carbon atom that contains the halogen. The nucleophile always approaches from the backside, directly opposite the leaving group. As the nucleophile approaches the orbital that contains the nucleophile electron pairs, it begins to overlap the empty antibonding orbital of the carbon atom bearing the leaving group (Solomon, T.W.G, Organic Chemistry, 6/e Wiley 1996, p.233).
The Figure PRS.3B-1 shows the energy of the molecules along the reaction coordinate which measures the progress of the reaction. [See PRS.A Collision theory-D Polyani Equations]. One measure of this progress might be the distance between the CH3 group and the Cl atom.
Figure PRS.3B-2 Reaction coordinate for (a) SN2 reaction, and (b) generalized reaction
Collision Theory when discussing the Polyani Equation.
The energy barrier shown in Figure PRS.3B-2 is the shallowest barrier along the reaction coordinate. The entire energy diagram for the ABC system is shown in 3-D in Figure PRS.3B-3. To obtain Figure PRS.3B-2 from Figure PRS.3B-3 we start from the initial state (A + BC) and move through the valley up over the barrier, Eb, (which is also in a valley) over to the valley on the other side of the barrier to the final state (A + BC). If we plot the energy along the dashed line pathway through the valley of Figure PRS.3B-3 we arrive at Figure PRS.3B-2.
Figure PRS.3B-3 3-D energy surface for generalized reaction.
The rate of reaction for the general reaction (Lp90) is the rate of crossing the energy barrier
We consider the dissociation of the activated complex as a loose vibration of frequency vI, (s1). The rate of crossing the energy barrier is just the vibrational frequency, vI, times the concentration of the activated complex,
We assume the activated complex ABC# is in virtual equilibrium with the reactants A and BC so we can use the equilibrium concentration constant to relate these concentrations, i.e.,
Combining Eqns. (A) and (B) we obtain
The procedure to evaluate vI and is shown in the table T.S.1
Table PRS.3B-1 Transition State Procedure to Calculate vI and
Step 1)A. Molecular partition function. The number of ways, W, of arranging N molecules in m energy states, with ni molecules in the ei energy state is
The distribution that gives a maximum in W is the Boltzmann distribution from which we obtain the molecular partition function, q.
Step 2)B. Relating , ni and N.
The entropy or the system is given by the fundamental postulate
Next we manipulate the Boltzmann equation for N molecules distributed in m energy states using Stirling's approximation to arrive at
Step 3)C. Relate and q. Starting with the total energy of the system E==niei, relative to the ground state, substitute for the number of molecules, ni, in energy state eI, using the Boltzmann distribution in the last equation of Step 3
and then sum to arrive at
for non-interacting molecules. is the ground state energy.
Step 4)D. Canonical partition function for interacting molecules. We need to consider interacting molecules and to do this we have to use Canonical partition function
with the probability of finding a system with energy Ei is
These relationships are developed with the same procedure as that used for the molecular partition function. For indistinguishable molecules, the canonical and molecular partition functions are related by
using the above equation we can arrive at
Step 5)E. Thermodynamic relationship to relate , and qi, the molecular partition function
We begin by combining the Maxwell relationship, i.e.,
with the last equation in Step 4 where the tilde (e.g. ) represents the symbols are in units of kcal or kJ without the tilde is in units per mole, e.g., kJ/mol. We first use the last equation for S in Step 4 to substitute in the Maxwell Eqn. We next use the relationship between Q and q, i.e.,
To relate to q, the molecular partition function. For N indistinguishable molecules of an ideal gas
Step 6)F. Relate G to the molar partition function qm. We define qm as
and then substitute in the last equation in Step 5.
Note: The tilde's have been removed.
where n = Number of moles, and NAvo = Avogadro's number and where G and Uo are on a per mole basis, e.g. (kJ/mole).
Step 7)G. Relate the dimensionless equilibrium constant K and the molar partition function qmi
For the reaction
the change in the Gibb's free energies is related to K by
Combining the last equation in Step 6 and the above equations
Step 8)H. Relate the partition function on a per unit volume basis, q¢, and the equilibrium constant K
Where Vm is the molar volume (dm3/mol). Substituting for qmi in the equation for K in Step 7 we obtain
Step 9)I. Recall the relationship between K and KC from Appendix C
Equate the equilibrium constant K given in the last equation of Step 8 to the thermodynamic K for an ideal gas, () to obtain KC in terms the partition functions, i.e.,
For the transition state A B C#, with d = 1,
we also know
Equating the two equations and solving for
The prime, e.g., q', denotes the partition functions are per unit volume.
Step 10) J. The loose vibration.
The rate of reaction is the frequency, vI, of crossing the barrier times the concentration of the activated complex
This frequency of crossing is referred to as a loose (imaginary) vibration. Expand the vibrational partition function to factor out the partition function for the crossing frequency.
Note that # has moved from a superscript to a subscript to denote the imaginary frequency of crossing the barrier has been factored out of both the vibrational, and overall partition functions, q#, of the activated complex.
Combine with rate equation, noting that vI cancels out, we obtain
where A is the frequency factor.
Step 11) K. Evaluate the partition functions
Evaluate the molecular partition functions.
Using the Schrödenger Equation
we can solve for the partition function for a particle in a box, a harmonic oscillator and a rigid rotator to obtain the following partition functions
The end result is to evaluate the rate constant and the activation energy in the equation
We can use computational software packages such as Cerius2 or Spartan to calculate the partition functions of the transition state and to get the vibrational frequencies of the reactant and product molecules. To calculate the activation energy one can either use the barrier height as EA or use the Polyani Equation.
In this section we will develop and discuss the molecular partition function for N molecules with a fixed total energy E in which molecules can occupy different energy states, ei.
Total Energy of System
Total number of molecules, N,
= Number of molecules with energy ei
The total Energy E is
The number of ways, W, arranging N molecules among m energy states (e1, e2 . . . em) is
For example, if we have N=20 molecules shared in four energy levels (e1, e2, e3, e4) as shown below
there are 1.75x109 ways to arrange the 20 molecules among the four energy levels shown. There are better ways to put the 20 molecules in the four energy states to arrive at a number of arrangements greater than 1.75x109. What are they?
For a constant total energy, E, there will be a maximum in W, the number of possible arrangements and this arrangement will overwhelm the rest. Consequently, the system will most always be found in that arrangement. Differentiating Eqn. (3) and setting dW = 0, we find the distribution that gives this maximum [see A6p571]. The fraction of molecules in energy state ei is
The molecular partition function q, measures how the molecules are distributed (i.e. partitioned) over the available energy states.
Equation (7) is the Boltzmann Distribution. It is the most probable distribution of N molecules among all energy states ei from i=0 to i= subject to the constraints that the total number of molecules N and the total energy, E are constant.
This energy E = ni ei is relative to the lowest energy, U0, (the ground state) the value at T=0. To this internal energy, E, we must add the energy at zero degrees Kelvin, U0 (A6p579) to obtain the total internal energy
The tildes, , represent that this is the total energy not the energy per mole.
Comments on the Partition Function q
The molecular partition function gives an indication of the average number of states that are thermally accessible to a molecule at the temperature of the system. At low temperatures only the ground state is accessible. Consider what happens as we go to the extremes of temperature.
and we see the partition function goes to infinity as all energy states are accessible.
and we see that none of the states are accessible with one exception, namely degeneracy in the ground state, i.e.,
W is the number of ways of realizing a distribution for N particles distributed on ei levels for a total energy E
E = 1n1 + 2n2 + 3n3 + . . .
The Basic Postulate is
Next we relate and q through W
Stirling's approximation for the ln of factorials is
For our system this approximation becomes
Recall substituting Eqn. (4) in Eqn. (13) we find
Further rearrangement gives
combining Eqns. (10) and (14)
Recall that the fraction of molecules in the ith energy state is
Taking the natural log of Eqn. (5)
Substituting for in Eqn. (13)
Recall from Eqn. (9) for = E = Uo, where Uo o, where Uo is the ground state energy in kcal.
This result is for non interacting molecules. We now must extend/generalize our conclusion to include systems of interacting molecules. The molecular partition function, q, is based on the assumption the molecules are independent and don't interact. To account for interacting molecules distributed in different energy states we must consider the Canonical partition function, Q.
Canonical ensemble (collection) (A6p583)
We now will consider interacting molecules and to do this we must use the Canonical ensemble which is a collection of systems at the same temperature T, volume V, and number of molecules N. These systems can exchange energy with each other.
= Energy of ensemble i
= Total energy of all the systems = a constant
= Number of members of the ensemble with energy EI
= Total number of ensembles
Let Pi be the probability of occurrence that a member of the ensemble has an energy Ei. The fraction of members of the ensemble with energy Ei can be derived in a manner similar to the molecular partition function.
Q is the Canonical Partition Function.
We now rate the Canonical partition function the molecular partition function (A6p858). The energy of ensemble i, Ei, is the sum of the energies of each of the molecules in the ensembles
Expanding the i=1 and i=2 terms
Each molecule, e.g. molecule (1), is likely to occupy all the states available to it. Consequently, instead of summing over the states i of the system we can sum over the states i of molecule 1, molecule 2, etc.
This result (Eqn. (17)) is for distinguishable molecules.
However, for indistinguishable molecules it doesn't matter which molecule is in which state, i.e., whether molecule (1) is in state (a) or (b) or (c)
(1) (a) (1) (b) (1) (c)
(2) (b) (2) (c) (2) (a) E = a + b + c
(3) (c) (3) (a) (3) (b) (etc)
Consequently we have to divide by N!
The molecular partition function is just the product of the partition functions for translational (qT), vibrational (qV), rotational (qR) and electronic energy (qE) partition functions.
This molecular partition function, q, describes molecules that are not interacting. For interacting particles we have to use the canonical ensemble. We can do a similar analysis on the canonical ensemble [collection] to obtain [A5p684]
Combining Equations (17) and (20) thus
We now are going to use the various thermodynamic relationships to relate the molecular partition function to change in free energy DG. Then we can finally relate the molecular partition function to the equilibrium constant K. From thermodynamic relationships we know that the Gibbs Free Energy, , can be written as
For an ideal gas with n total moles
Again we note the dimensions of are energy (e.g. kcal or kJ) and not energy/mol (e.g. kcal/mol). Combining Equations (20) and (24) for and we obtain
Recalling the relationship of Q to the molecular partition function
We use Avogadro's number to relate the number of molecules N and moles n, i.e., N = nNavo, along with the Stirling approximation to obtain
We divide by the number of moles, n to get
Substituting for (q/N) in Eqn. (28)
To put our thermodynamic variables on a per mole basis (i.e., the Gibbs free energy and the internal energy) we divide by n, the number of moles.
where G and Uo are on a per mole basis and are in units such as (kJ/mol) or (kcal/mol).
Applying Eqn. (31) to species i
For the reaction
the change in Gibbs free energy is
Combining Eqns. (31) and (32)
From thermodynamic and Appendix C we know
Dividing by RT and taking the antilog
The molecular partition function q is just the product of the electronic, qE, translational, qT, vibrational, qV, and rotational, qR partition functions
Equations for each of these partition functions will be given later. We now want to put the molecular partition function on a per unit volume basis. We will do this by putting the translational partition function on a per unit volume basis. This result comes naturally when we write the equation for qT
By putting on a per unit volume basis we put the product on a per unit volume basis. The prime again denotes the fact that the transitional partition function, and hence the overall molecular partition function, is on per unit volume.
The molar volume is
where f° is the fugacity of the standard state of a gas and is equal to 1 atm.
(See Appendix TS2 page 29 of Transition State Theory Notes for derivation)
The equilibrium constant and free energy are related by
The standard state is fi0 = 1 atm. The fugacity is given by fA = gAPA [See Appendix C of text]
For an ideal gas
Equating Eqns. (40) and (41) and canceling on both sides
Now back to our transition state reaction
where is the molecular partition function per unit volume for the activated complex.
Rearranging Eqn. (43), we solve for the concentration of the activated complex
We consider the dissociation of ABC# as a loose vibration with frequency vI in that the transition state molecule ABC# dissociates when it crosses the barrier. Therefore the rate of dissociation is just the vibrational frequency at which it dissociates times the concentration of ABC# (Lp96)
Substituting Eqn. (45) into Eqn. (46)
Where q¢ is the partition function per unit volume. Where vI is the "imaginary" dissociation frequency of crossing the barrier
The vibrational partition, , function is the product of the partition function for all vibrations
Factoring out qvI for the frequency of crossing the barrier
Note we have moved the # from a superscript to subscript to denote that is the vibrational partition function less the imaginary mode vI. is the vibrational partition function for all modes of vibration, including the imaginary dissociation frequency (Lp96).
Substituting for in Eqn. (45) and canceling vI
where is the partition function per unit volume with the partition function for the vibration frequency for crossing removed.
q = molecular partition function
q' = molecular partition function per unit volume
q = q' V
molar partition function
q'# = Partition function (per unit volume) of activated complex that includes partition function of the vibration frequency vI, the frequency of crossing
q'# = Partition function (per unit volume) of the activated complex but does not include the partition function of the loose vibration for crossing the barrier
What are the Equations for q'T, qV, qE, and qR
Schrödinger Wave Equation
We will use the Schrödinger wave equation to obtain the molecular partition functions. The energy of the molecule can be obtained from solutions to the Schrödinger wave equation (A5p370)
This equation describes the wave function, y, for a particle (molecule) of mass m and energy E traveling in a potential energy surface V(x,y,z). "h" is Planck's constant. The one dimensional form is
The probability of finding a particle in a region between x and x+dx is
is the probability density (A5p373)
et, eV, and eR) used in the partition function q. The equation is solved for three special cases
1. Translational energy, et. Particle in a Box.
2. Vibrational energy, eV. Harmonic Oscillator.
3. Rotational Energy, eR. Rigid Rotator.
4. Electronic Energy, eE.
The electronic partition functions qE, is most always close to one.
Table PRS.3B-2Overview of q¢
1 atomic mass unit º 1 amu = 1.67x1027 kg, h = 6.626x1034kg•m2/s,
kB = 1.38x1023kg•m/s2/K/molecule
1. Transitional Partition Function
Substituting for kB, 1 amu and Plank's constant h
2. Vibrational Partition Function qv
Expanding in a Taylor series
3. Electronic Partition Function
4. Rotational Partition Function qR
For linear molecules
where ABC are the rotational constants (Eqn. 57) for a nonlinear molecule about the three axes at right angles to one another
For a linear molecule
For non-linear molecules
Sy = symmetry number of different but equivalent arrangements that can be made by rotating the molecule. (See Laidler p. 99)
For the water and hydrogen molecule Sy = 2
For HCl Sy = 1
Estimate A from Transition State Theory
Let's do an order of magnitude calculation to find the frequency factor A.
Let's first calculate the quantity
6 x 1032 m3
6 x 1032
6 x 1032
6 x 1032 m3
600 x 1032 m3
6000 x 1032 m3
which compares with A predicted by collision theory
For the reaction
The rate law is
Now let's compare this withn transition state theory.
The rate of reaction is the rate at which the activated complex crosses the barrier.
Factoring out partition function for loose vibrational frequency, vI, from the vibrational partition function, gives
then from Eqn. (43) we can obtain
which is referred to as the Eyring Equation.
The overall dimensionless terms of mole fraction xi and the activity coefficients gi
S# will be negative because we are going form 0 less ordered system of A, BC moving independently as reactants to a more ordered system of A, B and C being connected in the transition state. The entropy can be thought of as the number of configurations/orientations available for reactions. That is
will be positive because the energy of the transition state is greater than that of the reactant state.
Case I Liquid
For liquid CT = a constant = CT0 (Recall for water that Cw = 55.5 mol/dm3
Here we see the temperature dependence as
Case II Gases
Here we see the temperature dependence as
As with liquids S# is negative and is positive.
A and HRx
Now lets compare the temperature dependent terms. The heat of reaction will be positive because the activated state is at a higher energy level than the reactants. See figure PRS.3B-2.
Comparing Eqns (1) and (16), the activation energy the Erying Equation is
with the frequency factor
For the general reaction
Molar partition function