The pre-exponential factor is a measure of the probability that two (or more) molecules involved in a reaction collide. It is worth reviewing the kinetic theory of gases to get a better understanding of what it is. As for the activation energy, it can be seen as the barrier of energy that has to be overcome so the reaction can occur.

Both parameters are very important and can certainly be used in reaction engineering, process modelling and optimization process.

A typical chemical reaction has the form:

A+B -> C

And the reaction rate will be:

r = k0*exp(-E/RT) * [A][B]

So the product of the concentrations "[A][B]" gives you an idea of how frequently the two molecules will collide. The exponential term "exp(-E/RT)" is the probability that the molecules will be at an energy level greater than the activation energy - notice how the exponential term tends to one as temperature (and hence kinetic energy) increases. The pre-exponential is an indication of how quickly the reaction happens, or how frequently. The probabilities and frequencies combine to give a reaction rate.

A more basic reaction is:

A->B

r = k0*exp(-E/RT) * [A]

Here, no collision is necessary. "k0" again relates the frequency at which A attempts to change into B, and the exponential term gives the probability that A has enough energy.

These rate equations are still empirical though, but are extremely important in basic chemical reaction kinetics (e.g. those well described by the kinetic theory of gasses). Similar equations can be derived from statistical mechanics for simple molecules. The more complex the chemical reaction (e.g. biochemical or polymeric), the less adequate the expression becomes.

For simple molecules the significance of the quantity can be obtained directly from the velocity distribution function.

The correct expression for an Arrhenius rate is

K=K0*T^b*exp(-E/KT).

Any reaction follow a reaction path, in the sense that the reactants move along a trajectory feeling a force different in any position. In general the interaction is modeled by a potential surface, depending on many parameters, between them the distance between the centers of mass of the two molecules. This potential energy surface presents often a barrier that must be passed to have the reaction. Considering a maxwell distributions, molecules that can pass the barrier are those of energy higher that the barrier. therefore the activation energy is the height of the barrier. In this case we are neglecting the contribution of tunneling and the shape of the barrier which determine the number of particles with energy higher than the barrier that really makes the reaction. This number is given by K0. However the percent of particles with sufficient energy that make the reaction can depend on the energy, that's why we introduce a pre-exponential factor function of the temperature. This approach is commonly used in calculating rates, evaluating firstly the cross sections.

As a example you can give a look the the work of Esposito on QCT (Quasi Classical Trajectory Method). However the Arrhenius formula is not general, and in some cases fails, and modified version of the rates can be encountered.

The terms in the Arrhenius equation aren't just related to probabilities, but to rates as well. For instance, protein folding takes time, it does not happen instantaneously. The time it takes to attempt or complete the reaction is incorporated into the pre-exponential constant k0 - it is a function of the reaction itself. Potential energy surfaces indicate equilibrium positions and energy barriers do contribute to reaction rates, but in general, the value k0 cannot be determined from energy considerations only.

Following the thermodynamic interpretation of transition state theory is possible to show that the pre-exponential factor is related to entropy of activation. Another way to see the k0 factor is relate it to the orientation of reactants during a colision. Only colisions with the correct orientation will lead to chemical reactions and the k0 factor represents this effect and other (such as tunneling as mentioned above). The thermal rate coefficients will be related to colisions that occurs with the proper orientation and sufficient energy (activation energy) to form the products.

Any combustion model needs these parameters to perform a simulation. Quality air models needs those parameters too. Just two examples.

Just one comment about unimolecular reactions:

Any thermal unimolecular reactions such as A => B occurs by colisions. Colisions with vessel's walls, bath gases or self colisions. The accepted mechanism (Lindemann mechanism) for unimolecular reactions is

A + M => A* + M (activation)

A* + M => A + M (de-activation)

A* => B

The reaction in the high pressure limit will follow a first order reaction ; v=k[A]

However, in low pressure the collision step will be very important and the rate law will present an order different of 1.

Only photochemical reactions may proceed without colisions.

@Tobias

dear,

I was talking about the traditional explanation of the Arrhenius law, for bimolecular reactions, the same low can be valid also for other situations, for different reasons. I do have very little experience about protein folding. In my case a potential energy surface is not characterized only the equilibrium position and by its maximum, but also by the shape of the surface. If you have a square barrier, classically it cannot be passed, no matter how high is the energy. When Leonardo talk about relative molecule orientation is making an example of the influence of the potential shape (the orientation is one parameter of the potential surface). Therefore the number of particles of a given energy that react is a function of the number of trajectories of the reactants that ended in the product. There are classical methods that are based on this approach to calculate rate coefficients.

There is also a discussion open about the validity of the Arrhenius low.

Dear Colona,

You can reasoning the protein folding in terms of potential energy surface. It is impossible for ours methods determine the exact PES for a protein, but the folding will be ruled by the shape of PES and barriers. I like the explanation of a chemical reaction in term of trajectories and agrees your previous comments.

Another thing that Tobias explained and I think in a different way is that the time to complete the folding is incorporated in the pre-exponential factor. The folding may be viewed as a conjunct of conformational changes, such as paralel first order reactions. The time to occur the folding will be ruled by the complete rate law of process: the rate coefficient that involves the pre-exponential factor and activation energy, plus the reactant (protein) order (that I think is first order). If the protein thermal energy is increased in time, the probability of a different tridimensional structure be obtained may be higher than the expected structure.

In Kinetic analysis, often times low activation energy at high temperature means high rate constant and hence speed up the reaction.

Whereas, pre-exponential factor expresses the fraction of reactant molecules that possess enough kinetic energy to react. This fraction can run from zero to nearly unity, depending on the magnitudes of Ea and of the temperature. In other words, pre-exponential factor is the fraction of molecules that would react if either the activation energy were zero, or if the kinetic energy of all molecules exceeded Ea.

Thank you all for your answers, I'm learning from the comments. I agree that the Arrhenius equation is based on empirical observations and has limited validity. I also definitely agree that protein folding (and similar reactions with complex molecules) occur via a series of reactions to reach it's final state. However, it isn't often practical, or even possible, to know the exact series of reactions, much less the PES describing it. If the entire reaction pathway, including the various transition states, are known, then the trajectory of the chemical reaction will certainly follow the path of steepest descent to a minimum on the Gibbs free energy surface, right? (please correct me if I'm wrong, I'm working this out for myself.) The problem is that we usually don't have the entire reaction path. The Arrhenius equation is then our empirical substitute to enable us to make rough estimations.

Intuitively, I still think there should be more involved in the reaction rate than just the potential energy surface and the entropy change. While these factors indisputably tell you where the reaction ends up, there is more involved in determining how quickly the reaction goes. Think about a soccer ball rolling down a hill. The potential energy of the ball is completely determined by the shape of the hill, but it will take the ball a lot longer to get to the bottom if it's covered in honey, right? How does energy or entropy account for that?

Thanks for the discussion. I'm enjoying the challenge of trying to understand these concepts!

Dear Tobias

I think there is something wrong in your considerations.

To describe the kinetics of an elementary chemical reaction by rate coefficients, you have to make the assumption that the interaction between the reactant is a collision, i.e. it is instantaneous and happen in a geometrical point. This approximation is usually valid because the interaction time is in the scale of fs while reaction characteristic time is around ns. If this assumption is not valid, you cannot use rate coefficients at all.

This means that there is no relation between the interaction time and rate coefficients.

Thanks everybody for your input- you enriched me with your theoretical treatment using thermodynamic interpretation of transition state theory. I still need to know, how we can use activation energy values for any real life applications- say reaction engineering.

Dana,

Any simualtion of reactors, or combustion simulations (see the kintecus web site), catalysis and air quality models needs the the rate coefficient or activation energy plus pre-exponential factor. I know a group that are improving a air quality model, including a kinetic model in a mass transport model.

The comments mentioned above are valid only for homogeneous gas phase reaction under equilibrium. In case of a heterogeneous solid state reaction, one can not simply consider the activation energy as the energy barrier to carry out a certain reaction. In fact, the currently kinetic evaluation methods are mainly based on Arrhenius function, which was established only for homogeneous gas phase chemical reaction.However, the thing is that the distribution of energy in solids is not represented by the Boltzmann equation. The redistribution of vibrational energy within a crystalline phase differs from the mechanism of collisions between freely moving molecules, as envisaged in homogeneous reaction rate (Arrhenius) theory. In this case, the pre-exponential factor can not be solely considered as frequency of collision and you have to consider the inherent chemical/physical background of your reaction, to figure out the decomposition mechanism. Consider all possible physical processes (melting, evaporation or sublimation et. al.) that included in your investigated reaction, because they could significantly affect the apparent activation energy and pre-exponential factor. In contrast, reactivities of species within the solid material that undergoes chemical change in heterogeneous reactions may vary with position and time. Moreover, the term ‘concentration’ has a different significance in describing solid-state reactants. Part of the statements are rebuilt from the literature (Journal of Thermal Analysis and Calorimetry, Vol. 86 (2006) 1, 267–286). I hope my comment would not confuse you.

@Qi-Long Yan

Your comment is not clear. Do you want to say that in solids the Arrhenius law is not valid or you say that in spite of the Arrhenius theory cannot be applied for the reasons you listed, the Arrhenius law is still valid?

Another thing that is not clear is that the energy transfer in solid follow different mechanisms. I am not an expert, but as far as I know, in the second quantization theory, the solid vibrations can be modeled by the phonon gas, that can be approximated by an ideal gas of fermions or bosons according to the nature of the atoms in the crystal. You have also energy transfer by electrons and holes that behaves like a Fermi gas.

In both cases, if the temperature is high enough, both Fermi and Bose distributions can be approximated by a Boltzmann.

I should say that also in gas phase the Arrhenius law is not always valid. In my work I rarely use it.

Dear Prof. Colonna, You are an expert on that issue. I got some information from one book entitled “Thermal Decomposition of Solids and Melts” by Boris V. L’vov. There have been some debates on decomposition kinetics, which are usually ignored by many researchers. Three different approaches to the investigation of this problem have been established by now. It has been stated in this book that the most popular approach is based on the concept of Arrhenius that chemical reactions involve only some of the molecules, the so-called “active” particles, their actual fraction being determined by the energy barrier of the reaction and the temperature. The second approach makes use of the concepts of Hertz, Knudsen and Langmuir on the vaporization of solids and differences in the kinetics of their decomposition in vacuum under equilibrium (in Knudsen’s effusion cells) and non-equilibrium (as it is believed to be) vaporization from a free surface (by Langmuir). The third approach, which has branched off the second approach fairly recently, rests upon the assumed equilibrium mechanism of congruent dissociative vaporization (CDV) as a primary stage in the thermal decomposition of any (solid or melted) substance with subsequent condensation of decomposition products with low volatility. There is a parallelism in the establishment of the first two approaches.

Dear Qi-Long

The method you are discussing are all based on heuristic approaches, as I can guess.

In my previous comments I try to explain the meaning of Arrhenius coefficients from a microscopic point of view and using qualitative description. The Arrhenius law is very useful, but often misused. Let's discuss first the gas phase. If you have binary reactions or first order reactions, if the variation of temperature is not too high, the Arrhenius law is valid and the meaning of coefficients it is related to the height of the barrier and on the probability of successful collisions. However, also in gas phase these assumption are not always valid. First of all, increasing the temperature, some other path can be opened, with higher barrier, but with higher probability path, changing the value of the rate coefficients. Looking at the cross sections, you can have a double peak the second much higher than the first one. Another possible deviation can be due to a different contribution from excited states, mainly vibration. If the temperature is high enough, and the contribution of excited states is important you have a different behavior of the global rates. But all these conditions can be considered a combination of Arrhenius. In case of non equilibrium however, also multi-Arrhenius behavior cannot be supposed.

Now in the solid state I cannot really tell you, because I did not investigate the evaporation process (except in laser ablation, but this is another story).

I do not know in detail the theory you are listing, but I guess they start from quasi-equilibrium theory, under the condition valid for the Onsager reciprocal relations.

There are few papers on molecular dynamics simulation of the evaporation process, which are quite complex to run. In a solid or in a liquid you should consider that the interaction are multi body. The phonon gas can be approximated by an ideal gas, but what evaporate are real atoms and molecules and not phonons. You have also to distinguish between different type of evaporation processes: we can have solids that in the phase change maintain the same identity like ice and water or they suffer decomposition or readjustment. As a simple example I can consider CO2 that when dissolved in water is present as carbonic acid while in gas phase you have carbon dioxide.

What I want to say here is that I agree with you that the Arrhenius law is valid only in some conditions and that the theory cannot be extended straightforward to different situations like sublimation or evaporation.

Dear Prof. Colonna, thanks very much for your informative and impressive comments. I learned a lot from you.

@Kausik

The Arrhenius is used in real life to determine how a reaction rate changes with temperature. You would typically do a series of experiments to try and determine the activation energy and the pre-exponential coefficient, or get these values from the literature. Once you have them, you can use the reaction rate in an appropriate differential equation to calculate the yield in a reactor. For example:

In a batch CSTR, with A->B, we might have:

rA = -k0*exp( -EA/(R*T) ) * cA

rA = reaction rate

k0 = pre-exponential coefficient

EA = activation energy

R = universal gas constant

T = temperature

cA = concentration of A

The differential equation for a batch reaction in a CSTR:

dcA/dt = rA = -k0*exp( -EA/(R*T) ) * cA

By solving the differential equation, you can determine the change in "cA" with time, at a variety of different temperatures. You can make it even more complicated by including the fact that the chemical reaction may release or consume energy, thereby changing the temperature with time.

So, while the theoretical INTERPRETATION of the Arrhenius equation can be very complicated, it's USE is fairly simple: estimating reaction rates ate various temperatures. In this regard, they are essential in reactor modelling, process control, etc.

I learnt a lot from the discussions of Gianpiero ,Qi-Long, Tobius and Leonardo. Many thanks to you all.

The kinetic analysis produces the value of the preexponential factor and activation energy only in the case if the Arrhenius temperature function is applied. In case of a complex condensed-state reaction, the Arrhenius equation is the worst choice of the temperature function. Please, read the papers:

JTAC 82-2005-651

JTAC 88-2007-709

Go through the document

Method How to determine order and activation energy: Kinetics of li...

Hope you get the answer.

https://www.researchgate.net/post/Can_I_use_Arrhenius_Relationship_to_determine_Accelerated_factor_when_test_temperature_is_lower_than_use_temperature

Dear Tobias Louw, your comment makes sense, especially to a Chemical Engineer, my question is can the same analysis method be applied in the design of heterogeneous reactors, for example, a fixed bed reactor that has a solid fuel decomposing to form char and vapor products.

Hi Tobias Louw

Sure, the Ea and A i.e. the activation energy and per-exponetial are highly significant in the modeling of reaction kinetics. With the presence of those set of parameter a reactor can modeled and simulated, especially when you intend to consider a further study in reaction engineering, process modeling and optimization (for processes that involve chemical change i.e. reactions). I agree with the other respondents, Bothwell Nyoni that it is highly significant to a chemical engineer.