Collision rate termination reaction hydrogen oxygen explosion/combustion

[Tigrisje]

Hello,

I'm making a question on the explosion of hydrogen/oxygen. All the things you need to know are the following. The formation of H-radicals in the H2-O2 mixture are responsible for ignition of the explosion. This means the H-radicals need to be formed faster than they are destructed to let the mixture explode. One of the 'termination' reaction in which H-radicals are removed is through collisions with the vessel wall.

H + wall --> removal with rate constant k_1

The question now is: "Textbooks state that the rate constant k1 can be written as:
k_1 = k_d / P with P the total pressure and k_d another rate constant. Assuming that the vessel is spherical with radius R, and d is the diameter of a hydrogen atom, and m its mass. Derive now the following equation:

k1 = (12/ R^2) * ((kb*T ) / (Pi*d^2 * P * Srqt(2) ))* Sqrt( 8*kb*T / Pi*m)

How do I do this? It has something to do with determining the collion rate to the wall and the last Sqrt-term will be from the Boltzmann velocity, but I just can't see where all the terms come from.

 

[nate808]

Hello,

Thank you for your question. The derivation of the equation for k1 involves several steps, so I will try to explain each one in detail for better understanding.

First, we need to understand the concept of rate constant and how it relates to the collision rate between molecules. The rate constant, k1, for the reaction H + wall --> removal is defined as the proportionality constant between the rate of the reaction and the concentration of H radicals. In other words, it represents the probability of a collision between a H radical and the vessel wall resulting in its removal. This probability is influenced by factors such as the total pressure, P, of the mixture and the size of the vessel.

Next, we need to consider the size of the hydrogen atom and the vessel in which the reaction takes place. Since the vessel is assumed to be spherical with a radius R, the surface area of the vessel is 4πR^2. Now, we need to determine the number of collisions that occur between the H radicals and the vessel wall in a given time. This is known as the collision rate, denoted by Z, and is given by the equation:

Z = (N/V) * (4πR^2 * v),

where N/V is the number of H radicals per unit volume and v is the average velocity of the H radicals. The first term, N/V, can be represented as the concentration of H radicals, [H], which is equal to P/RT according to the ideal gas law. The average velocity, v, can be calculated using the Maxwell-Boltzmann distribution, which gives the probability of finding a molecule with a particular velocity at a given temperature. The equation for v is given by:

v = (2kbT/πm)^1/2,

where kb is the Boltzmann constant, T is the temperature, and m is the mass of the H atom.

Now, we can substitute these values into the equation for Z to get:

Z = (P/RT) * (4πR^2 * (2kbT/πm)^1/2).

Next, we need to determine the collision frequency, denoted by ν, which is the number of collisions per unit time. This can be calculated by multiplying the collision rate, Z, by the cross-sectional area of the hydrogen atom, πd^2/4. This gives us:

ν = (πd^2/4)