Adsorption - Pre Exponential frequency factor

In summary, the pre-exponential factor in the Arrhenius equation represents the frequency at which reactants collide in order for a reaction to occur. It can be calculated using statistical mechanics and the partition function. By calculating the partition functions for the reactants and the activated complex, we can find the frequency factor and use it to calculate the rate constants for the reaction.
  • #1
dikmikkel
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Homework Statement


I have an adsorption reaction [itex] A+* \leftrightharpoons A^*[/itex]
Then I am to finde the rate constants and these should be given by an Arhennius relation, e.g. from left to right in the reaction:
[itex] k_+ = f_+\exp(-\Delta E_a/k_bT)[/itex] where the delta energy is the activation energy and f+ is the frequency factor

My question is, how do you find that pre-exponential factor? And this should be from theory and i'd rather use statistical mechanics(partition functions)



The Attempt at a Solution


I have seen it written as easy as: [itex] f_+ = \dfrac{k_bT}{h}[/itex]
but I suspect I am missing some partition functions or so?
This is maybe chemistry but I need a physisits opinion and stringent use of units.
 
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  • #2


Hello there,

I would approach this problem by first understanding the physical meaning of the pre-exponential factor in the Arrhenius equation. The pre-exponential factor, also known as the frequency factor, represents the frequency at which the reactants collide in order for the reaction to occur. This factor takes into account the orientation of the molecules and their energy levels.

In order to calculate this factor, we can use statistical mechanics and the partition function. The partition function takes into account all possible energy states of a system at a given temperature. In this case, we can use the partition function to calculate the frequency of collisions between the reactants.

To do this, we first need to calculate the partition function for the reactants and the activated complex (A+*). Then, we can use the ratio of these two partition functions to calculate the frequency factor.

The partition function for a single molecule is given by:

Z = \sum_i g_iexp(-E_i/k_bT)

where g_i is the degeneracy of the energy level i and E_i is the energy of that level.

For the reactants, we can calculate the partition function by summing over all possible energy levels of A and * separately and then multiplying them together.

Z_r = \sum_A g_Aexp(-E_A/k_bT) \times \sum_* g_*exp(-E_*/k_bT)

Similarly, for the activated complex, we can calculate the partition function by summing over all possible energy levels of A+ and * separately and then multiplying them together.

Z_ac = \sum_{A+} g_{A+}exp(-E_{A+}/k_bT) \times \sum_* g_*exp(-E_*/k_bT)

The frequency factor can then be calculated as:

f_+ = \dfrac{Z_ac}{Z_r}

Using this approach, we can find the pre-exponential factor from theory and use it to calculate the rate constants for the reaction. I hope this helps. Let me know if you have any further questions.
 

1. What is the pre-exponential frequency factor in adsorption?

The pre-exponential frequency factor, also known as the pre-exponential factor or pre-exponential term, is a constant in the Arrhenius equation that describes the rate of adsorption. It represents the frequency of collisions between the adsorbate molecules and the surface of the adsorbent.

2. How does the pre-exponential frequency factor affect the rate of adsorption?

The pre-exponential frequency factor is directly proportional to the rate of adsorption. This means that a higher value of the pre-exponential factor will result in a faster rate of adsorption, while a lower value will lead to a slower rate of adsorption.

3. What factors can influence the value of the pre-exponential frequency factor?

The value of the pre-exponential frequency factor can be influenced by various factors such as temperature, pressure, and the nature of the adsorbate and adsorbent. Changes in these factors can alter the frequency of collisions between the molecules and the surface, thereby affecting the value of the pre-exponential factor.

4. How is the pre-exponential frequency factor determined experimentally?

The pre-exponential frequency factor can be determined experimentally by measuring the rate of adsorption at different temperatures and using the Arrhenius equation to calculate the value of the pre-exponential factor. This involves plotting a graph of the natural logarithm of the rate constant against the reciprocal of temperature, and the slope of the line will be equal to the pre-exponential frequency factor.

5. Can the pre-exponential frequency factor be used to predict the rate of adsorption at different temperatures?

Yes, the pre-exponential frequency factor can be used to predict the rate of adsorption at different temperatures using the Arrhenius equation. This equation can be used to calculate the activation energy of adsorption, which can then be used to determine the rate of adsorption at any given temperature.

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