The Arrhenius Law: Pre-exponential Factors

Pre-exponential factor (A) is part of the Arrhenius Equation. The Arrhenius equation is temperature dependent and it was discovered by the Swedish chemist Svante Arrhenius in 1889.

Introduction

In the Arrhenius Equation: A (red) is the pre-exponential factor which is also known as a frequency factor. It represents the frequency of collisions between reactant molecules. Pre-exponential factor is also dependent on temperature because it is related to molecular collisions.

The Units of the pre-exponential factors vary depending on the order of the reaction and thus the rate constant. In a first order reaction, the units of the pre-exponential factor will be per second. Since the pre-exponential factor depends on frequency of collisions, its related to Collision theory and Transition State Theory.

Temperature Dependence of Reactions

This equation introduces the relationship between rate and A, Ea or T. Where A is the pre-exponential factor, Ea is the activation energy, and T is the temperature. The pre-exponential factor, A, is a constant that can be derived experimentally or numerically. It is also called the frequency factor and describes the number of times two molecules will collide. In empirical settings, the pre-exponential factor is simply a constant defined as A.

When dealing with the collision theory, the pre-exponential factor is defined as Z and its equation can be derived by considering the factors that affect the frequency of collision for a given molecule. Consider the most elementary bimolecular reaction:

A + A -->  Product

An underlying factor to the frequency of collisions is the space or volume in which this reaction is allowed to occur. Consider these two molecules in a cylinder:

Intuitively, it makes sense for the frequency of collisions between these two molecules to be dependent upon the dimensions of their respective containers. Using this logic, we define Z as:

Z = (Volume of the cylinder * Density of the particles) / time

Using this relationship, we can derive an equation for the collision frequency, Z, of molecule A and A:

$$Z_{AA} = 2N^2_Ad^2 \sqrt{\dfrac{\pi{k_{b}T}}{m_a}}$$

For a more complex collision, such as one between A and B:

A + B ? Product

We use the same reasoning as above to derive the following equation for the collision frequency, Z, of molecule A and B(2).

$$Z_{AB} = N_AN_Bd^2_{AB} \sqrt{\dfrac{8{k_{b}T}}{\mu}}$$

Substituting our collision factor back into the original Arrhenius equation yields:

$$k = Z_{AB}e^{\frac{-E_A}{RT}}") }} \(k = Z_{AB} = N_AN_Bd^2_{AB} \sqrt{\dfrac{8{k_{b}T}}{\mu}}e^{\frac{-E_A}{RT}$$

This equation produces a rate constant with the standard units of (M-1 s-1); however, since we are working on a molecular level, a rate constant with molecular units would be more useful. To obtain this constant, we divide the rate by NANB. This produces a rate constant with units (m3 molecule-1 s-1) and provides the following equation:

$$k = Z_{AB}e^{\frac{-E_A}{RT}}$$

Divide both sides by NANB(2)

$$\dfrac{k}{N{_A}N{_B }} = d^2_{AB} \sqrt{\dfrac{8{k_{b}T}}{\mu}}e^{\frac{-E_A}{RT}$$

ZAB becomes zAB

$$\dfrac{Z_{AB}}{N{_A}N{_B }} = z_{AB}$$

And finally,

$$k = z_{AB}e^{\frac{-E_A}{RT}}$$

Our pre-exponential factor has now been defined by the parameters of the collision theory as being:

$$d^2_{AB} \sqrt{\dfrac{8{k_{b}T}}{\mu}}$$

A and Z are practically interchangeable terms for collision frequency. Often times however, when the term is determined experimentally, A is the preferred variable and when the constant is determined mathematically, Z is the variable more often used. The derivation for Z, while mostly accurate, ignores the steric effect of molecules. For a reaction to occur, two molecules must collide in the correct orientation. Intuitively, each collision will not result in the proper orientation and thus will not yield a corresponding product.

To account for this steric effect, we must introduce the variable P, which represents the probability of two atoms colliding with the proper orientation. The Arrhenius equation now turns into:

$$k = Pze^{\frac{-E_A}{RT}$$

The probability factor, P, is very difficult to asses and thus still leaves the Arrhenius equation as imperfect.

Transition State Theory Pre-exponential Theory

The collision theory deals with gases and neglects to account for structural complexities in atoms and molecules. Therefore, the collision theory estimation for probability is not accurate for anything other than gases. The transition state theory attempts to solve for this discrepancy. It uses the foundations of thermodynamics to give a representation of the most accurate pre-exponential factor that yields the corresponding rate. The equation is derived through laws concerning Gibbs free energy, enthalpy and entropy to be:

$k = \dfrac{k_bT}{h} e^{\frac{\Delta S^o}{R}}}e^{\frac{-\Delta H^o}{RT}}}(M^{1-m})$

Pre-exponential factor table

 Factor Type A Empirical $$d^2_{AB} \sqrt{\dfrac{8{k_{b}T}}{\mu}}$$ Collision Theory $$\dfrac{k_bT}{h}$$ Transition State Theory

Relationship to Temperature

The pre-exponential factor is a function of temperature. As we see in the above table, the factor for the collision theory and the transition state theory are both responsive to temperature changes. The factor for the collision theory is proportional to the square root of T, while that of the transition state theory is proportional to T. The empirical factor is also sensitive to temperature. Intuitively, as temperature increases, molecules will begin to move faster and as molecules move faster they are more likely to collide and therefore effect the collision frequency, A.

Sources

1. Atkins, Peter, and Julio De Paula. Physical Chemistry for the Life Sciences. Alexandria, VA: Not Avail, 2006.
2. Chang, Raymond. Physical Chemistry for the Biosciences. Sausalito, CA: University Science, 2005.

Contributors

• Golshani (UCD)