Introduction

Kinetic studies of chemical reactions are primary to confirm a postulated reaction mechanism. In this treatment of chemical kinetics, known as deterministic approach, will yields information on the average concentration of molecules. This treatment is not applicable to cases where large fluctuations from mean concentration occur. On the other hand, this method can outcome a concentration-time curves for complex reations, by monitoring random picks of molecules represented by the digits in a computer. 

Monte Carlo Method 

Considering the first order reaction A ? B, the n particles of reactant A are mixed with s particles of solvent S and labelled, so that they are represented in the computer by the digits 1 and 0. If each A molecule in a first order reaction has a constant probability P of conversion, then the number of A species reacting in time ?t is nP?t. In the computer, every each pick of a 1 or 0 has a constant P’. The number of 1’s being converted to 0’s in ?t’ is nP’?t’. The rate law constant k can be associated with the quantity P, the probability that particle A reacts in unit time. Taking the rate constant k which is associated to P’, we weight it with a Boltzmann factor exp (-Ea/RT), where Ea is the activation energy, R the gas constant and T the temperature in Kelvins. The Boltzmann factor is the fraction of molecules having necessary energy for reaction. The rate law constant obtain by Monte Carlo Method (MC) is therefore given by an Arrhenius expression. 

k = f exp (-Ea/RT) 

where,  f = frequency factor 

The user are required to input the activation energy Ea and temperature T for the reaction into the computer. This is because according the the Arrhenius expression above, this allows the investigation of how Ea and T effect the kinetics. If S particle (digit 0) is chosen, no transition occurs and another random number of digit 1 or 0 (cycle) is selected.  

If the A molecules (digit 1) is chosen, another new cycle occurs. This time a random number between 0 and 1 is being generated. The A particles is succesfully converted to product B  
only if the random number between 0 and 1 is smaller than the Boltzmann factor. 

This means, 

Exp (-Ea/RT) > generated random number between 0 and 1 

If A particle is sucessfully converted to product B, its count decreases by one and the product is increased by one unit. The following example explains this :- 
  

Example 1 : 

Take that there is 10,000 cycles per minute 

Initial :            A       ----->        B 
                  10,000                   0 

After particle being sucessfuly converted to product B : 

                A            --------->            B 
      (10,000 – 1 )                          (0 + 1) 
  

  
If the Boltzmann factor is less than a random number between 0 and 1, another new cycle is started. Note that every cycle no matter wheather 1 or 0 is chosen it is considered 1 count of cycle. The collection of particles is sampled every 200, 500 or 1000 cycles to give the number of A molecules as a function of the number N of cycles. In each sampling, we obtain comparable results and one sampling time frame is used consistently for given kinetics runs as a function of temperature. In each reaction, a Boltzman factor will assigned a probability constant. The goal is to get the rate law constant at several temperature for several activation energy that are put into the computer by the user. At a selected temperature by making the activation energy choices, the velocity of reaction is controlled. The individual rate constant obtained by the MC method is proportional to experimental rate constant. 
  

Kinetics 

First order-first order reversible kinetics. 

         k1 
A    <====>  B 
         k2 
  

The user inputs, a selected temperature, activation energies Ea1 and Ea2 for the forward and backward reaction respectively. Each potential reaction or conversion (particle A convert to product B) must meet the the following : 

Exp (-Ea/RT) > generated random number between 0 and 1 

The number of A and B particles at cyclic intervals, are printed out and stored in the ‘clipboard’. A visual presentation of the data obtain is then produced by a graphic program. 

Any time during a run, the number of A and B particles are equal to n, 

A0 + B0 = A + B = Aeq + Beq = n 

Where A0 and B0 are initial number of A and B species and Aeq and Beq are the equilibrium (average) number of A and B particles respectively. 
  
The rate equation for A is : 

d[A]/dt = -k1 [A] + k2[B] 

At equilibrium : 

K1 [A]eq = k2 [B]eq 
  

Then rate equation for A can be tranformed algebraically to : 

d[A]/dt = -(k1 + k2)([A] – [A]eq) 
  

and integration gives : 

ln[([A] – [A]eq/[A]0 – [A]eq)] = -(k1 + k2)t 
  

This may be written in terms of numbers of particles instead of concentration as : 

ln (A – Aeq) = -(k1 + k2)t + ln(A0 – Aeq) 
  

ln (A – Aeq) is plotted against N, the number of cycles and the slope of the line is a measure of the sum of the rate constant (k1 + k2) in units of reciprocal cycles. To obtain Aeq, the data are examined determine the number of cycles elapsed before equilibrium is reached and values of A is average using a spreadsheet to yield Aeq and its standard deviation ÓA. Usually initial values for B (B0) is zero, approximately 4x104 cycles are elapsed to reach equilibrium. 
Since, 

Beq = n – Aeq 

Thus, 

Keq = Beq/Aeq = (n – Aeq)/Aeq = k1/k2 

Where Keq = equilibrium constant 

This equation is used to separate the individual rate constant from (k1 + k2). 

The relative standard deviation of A is calculated as follows : 

ÓA/Aeq = (Beq/Aeqn)1/2