General Introduction

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What is Monte Carlo?

The Monte Carlo method is an important simulation technique in science. It is often used where the underlying fundamental processes are random in nature and indeed it derives its name from the district of Monte Carlo in Monaco - a place well known for its games of chance.

Monte Carlo simulations are particularly useful when it is inconvenient or impossible to carry out a physical experiment and instead a mathematical experiment that closely simulates the physical one is studied. For example, it would be prohibitively expensive in a study of nuclear reactor design to actually build a reactor and study the neutron processes physically. Since the fundamental particles of nuclear physics obey probabilistic laws, their actual behaviour can be simulated by carrying out mathematical experiments using random numbers, whose behaviour repesents the behaviour of the neutrons. Simulation experiments using random numbers to represent physical quantities have been studied for decades in such diverse areas as population genetics, chemical engineering operation research and chemical kinetics [1], [2].

Rather than try and explain theoretically, or through mathematics, how Monte Carlo works you will gain an understanding of the method by performing a Monte Carlo simulation manually and later using the computer.

Application to Kinetics

In the experiments to be performed in the next section the Monte Carlo method will be used to provide data which models the course of a chemical reaction and the rate of appearance and disappearance of products and reactants. Three reactions are to be studied:

1. A first order reaction of type:
   $\mathrm{A} \longrightarrow \mathrm{Products}$

2. A second order reaction of type:
   $\mathrm{A + B} \longrightarrow \mathrm{Products}$

3. A first-order consecutive reaction commencing initially with pure A only ( k₁ and k₂ are the rate constants for the two steps.
   $\mathrm{A} \stackrel{k1}{\longrightarrow} \mathrm{B} \stackrel{k2}{\longrightarrow} \mathrm{C}$

This division of reactions into first and second order is because from a theoretical point of view they proceed via different kinetic mechanisms. To describe the kinetics in a precise mathematical way we represent the reactions with their respective differential rate laws.

For first order reaction (1) the differential rate law is given by:

$$-\frac{d[A]}{dt} = k[A]$$

where [A] is the concentration of A at time t and k is the specific rate constant.

The rate of reaction is proportional to the concentration at any given time. Note that rate is always a positive value. Since concentration of A decreases with time the rate is given by:

$$-\frac{d[A]}{dt}$$

For a second order reaction (2), the rate is:

$$-\frac{d[A]}{dt} = k[A][B]$$

where k is now the second order specific rate constant.

However, experimentally it is the concentration of a reactant or product that is measured - not the time rate of change as given by the differential rate laws. For instance, a spectrophotometer may measure the absorbance at particular times thus providing us with the the concentration of a reactant versus time.

To theoretically predict how the concentration varies with time we must integrate the differential rates laws to obtain the integrated rate equations. For each of these reactions an analytical expression for the integrated rate equation can be derived.

For example, by integrating the differential rate equation for the first order reaction we can obtain the equation which describes the variation of concentration with time:

$$ln [A] = -kt + ln [A]_0$$

where $[A]_0$ is the concentration of A at time=0.

Integration of the second order rate equation is more complex and there are different equations depending on whether the initial concentration of A is the same as B or not.

If $[A]_0 \ne [B]_0$ then:

$$\frac{1}{B_o - A_o} \ln \frac{A_o(B_o-x)}{B_o(A_o-x)} = kt$$

For the case where $[A]_0 = [B]_0$ the reaction behaves as if it were second order in a single reactant and we obtain:

$$\frac{1}{A} - \frac{1}{A_0} = kt$$

again $[A]_0$ is the concentration of A or B at time=0.

For the consecutive reaction the closed form integrated rate equation can only be derived with extreme difficulty [3]. Herein lies the elegance of the Monte Carlo method. The computer can simulate the reaction taking place and save the concentration data as a function of time. The data can then be plotted in the same way as you would for a real experiment. For more complex reactions than the above, no closed form rate equations can be obtained, yet the Monte Carlo method still enables one to obtain the variation of concentration with time.

Bibliography

1

J.M. Hammersley and D.C. Handscomb, Monte Carlo Methods, Methuen, London (1964).

2

B. Rabinovitch, Journal of Chemical Education, 46, 262 (1969).

3

A.A. Frost and R.G. Pearson, Kinetics and Mechansim, Wiley, New York (1961).

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