Autocatalytic reaction and rate of reaction

[squaremeplz]

Homework Statement

Hi, I am trying to set up a mathematical model for an autocatalytic (self heating) reaction. However, I can't fully grasp the chemistry.

Homework Equations

A + X = B + 2X,
X + Y = B + 2Y,
A + Y = B.

The Attempt at a Solution

So one example I found is the all known reaction of hydrogen and oxygen, assuming the following conditions:

Let free atoms of hydrogen or oxygen arise as primers in a mixture of O2 and H2 a chain reaction arises:

O_2 and H_2 = OH + O,
O + H_2 = OH + H,
… (Basic Chain)
Next, the hydroxyl radicals OH are formed and these take part in a subsequent reaction
OH + H2 = H20 + H

As a result, the concentration of free hydrogen atoms rises in a geometric progression. Free atoms and radicals are short-lived active intermediate reaction products. They quickly recombine on the walls of the vessel or bulk. The reaction of Chain termination is usually written as:

H + wall = ½ H2
H + O2 + M = HO2 + M

Where M is an arbitrary third particle. As seen from the above reaction, the chain termination in the bulk does not lead to a stable molecule but o the peroxide-type radical HO_2, which still contains superfluous chemical energy and is a long-lved active intermediate. This radical can participate in slow chain propagating reactions:

HO2 + H2 = H2O2 + H
HO2 + H2 = H2O + OH

This type of reaction exhibits a slow increase in the rate of the overall process with time. In order to fully describe the kinetics of the reaction, we must introduce yet another degradation of the perodixed radical on the surface of the walls of the vessel (slow chain termination)

HO_2 + wall -> destruction

And autocatalytic process of chain propagation:

HO2 + H2O = H2O2 + OH
HO2 + H2P2 = H2O + O2 + OH

The last equation explains the phenomenon of autocatalysis in the final reaction product, i.e. Water. On one hand, the rate of the autocatalytic reaction increases markedly with pressure, and this may lead to thermal explosions, as we will investigate in detail.

In order for my model to work, theoretically, the reaction needs to undergo self heating leading towards increased reaction rate, eventually the heat produced will achieve a critical value at which point an explosion occurs.

Can i use the reaction above, and if so, how would I set up the arrhenius source constant,
k = z*e^(-E/RT) where is z is the collision between atoms, and e^(-E/RT) the probability of a collision leading towards a reaction.

Any help is greatly appreciated.

 

[Borek]

I am not sure if you HAVE to start with chemistry.

I did something similar eons ago - I just assumed A -> B, k given by Arrhenius equation, amount of heat evolved proportional to amount of products, and mixture temperature calculated using simple q=mcΔT. As far as I remember I got a differential equation that yielded integral that I wasn't able to calculate, but it wasn't hard to integrate numerically.

That was after some organic lab - I don't remember what I was doing, but it required ice bath and at some moment mixture got too hot... I had to clean whole fume hood

--

 

[Blueshift5]

Hello, it seems like you are trying to understand and model the kinetics of an autocatalytic reaction. An autocatalytic reaction is a type of reaction where one of the products acts as a catalyst for the reaction, leading to a self-sustaining chain reaction. In your example, the peroxide-type radical HO2 acts as the catalyst for the reaction, leading to a slow increase in the rate of the overall process with time. This type of reaction can exhibit self-heating and can potentially lead to thermal explosions if the heat produced exceeds a critical value.

To set up a mathematical model for this type of reaction, you can use the Arrhenius equation that you mentioned. The Arrhenius equation relates the rate constant (k) to the temperature (T) and the activation energy (E) of the reaction. In this case, the activation energy would include the energy required for the collision between atoms (z) and the probability of a collision leading towards a reaction (e^(-E/RT)). You can use experimental data to determine the values of z and E for your specific reaction and then plug them into the Arrhenius equation to calculate the rate constant (k) at different temperatures.

Additionally, it is important to consider the effects of pressure on the reaction, as you mentioned in your post. Changes in pressure can affect the rate of the reaction and can also lead to changes in the critical temperature for thermal explosions. Therefore, it may be necessary to also incorporate pressure into your mathematical model.

I hope this helps in setting up your model and understanding the chemistry of an autocatalytic reaction. It is a complex process, but with careful consideration of the factors involved, you can create a successful mathematical model. Good luck with your research!