The Azimuth Project
Experiments in bimolecular reactions

A bimolecular reaction

For the general terms and definitions used in this project see

Here we will consider an example of a bimolecular reaction. This is example 6.3 in the paper

From Equation (12) page 13 we have the following chemical reaction network.

k 1:2X 1+X 22X 3 k_1 : 2 X_1 + X_2 \rightarrow 2 X_3
k 2:2X 32X 1+X 2 k_2: 2 X_3 \rightarrow 2 X_1 + X_2

We quote from the reference.

The stochastic Petri net

bimolecular reaction

The rate equation

ddtX 1=2k 1X 1 2X 2+2k 2X 3 2 \frac{d}{d t} X_1 = -2 k_1 X_1^2 X_2 + 2 k_2 X_3^2
ddtX 2=k 1X 1 2X 2+k 2X 3 2 \frac{d}{d t} X_2 = - k_1 X_1^2 X_2 + k_2 X_3^2
ddtX 3=2k 1X 1 2X 2+2k 2X 3 2 \frac{d}{d t} X_3 = 2 k_1 X_1^2 X_2 + -2 k_2 X_3^2

The master equation

H=k 1(a 3 a 3 a 1a 1a 2a 1 a 1 a 1a 1a 2 a 2)+k 2(a 1 a 1 a 2 a 3 a 3 a 3a 3) H = k_1(a_3^\dagger a_3^\dagger a_1 a_1 a_2 - a_1^\dagger a_1^\dagger a_1 a_1 a_2^\dagger a_2) + k_2 (a_1^\dagger a_1^\dagger a_2^\dagger -a_3^\dagger a_3^\dagger a_3 a_3)

From this we arrive at the following

z 3 2;0=k 1c 1 2c 2k 2c 3 2 z_3^2; 0 = k_1 c_1^2 c_2 - k_2 c_3^2
z 1 2z 2;0=k 2c 3 2k 1c 1 2c 2 z_1^2 z_2; 0 = k_2 c_3^2 - k_1 c_1^2 c_2

category: experiments