Rates of reaction - kinetics (A2)
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Logarithms and the rate equation
If you are not sure about logarithms it would be a good idea to read the tutorial on logarithms before reading the rest of this tutorial. It can be found at Logarithms for chemists
As you know the general form of the rate equation for a reaction
xA + yB -> products is
rA = k[A]a[B]b[C]c
rA is the rate of the reaction with respect to reactant A. k is the rate constant at a particular temperature. [A], [B] and [C] are the concentrations of the reactants A and B and another species C (this could be a catalyst) - which is in some way involved in the reaction. The indices a, b and c are known as the orders of the reaction with respect to each of the species involved in the rate equation.
Equations for reactions
The equation for the reaction (known as the stoichiometric equation) simply tells you what the reactants are, what the products are, and the proportions in which they react and are formed. The equation for the reaction tells you nothing about the route the reaction takes in getting from reactants to products.
The rate equation, on the other hand, depends on the route the reaction takes (known as the mechanism). Particularly with reactions involving covalent molecules, this often involves several steps. As a result, the only way to find out what the rate equation for a particular reaction is to do a series of experiments.
Experimental methods
Various experimental methods are possible. Often this involves measuring the change in concentration with time of each of the species involved in the reaction in turn (including any catalyst); meanwhile keeping the concentration of the other species involved constant. This is commonly done by making sure that the concentration of all the species but the one under investigation are in large excess.
Processing results
Graphs of concentration against time can then be plotted. According to their shape, the order with respect to the reactant can sometimes be deduced - when the order is a simple whole number. In practice it can be difficult to decide by simple inspection what order the shape of the curve represents. It can be impossible if the order is not a simple positive whole number.
Another way of processing the results, which often makes interpreting them easier, can be used if the rate equation is converted into its logarithmic form. In order to achieve this, remember the following. If numbers are being multiplied together, then their logarithms are added. The logarithm of a number raised to a power is the logarithm of the number multiplied by the power.
Applying these principles to the rate equation
rA = k[A]a[B]b[C]c
gives the following result:
lg rA = lg (k) +a lg ([A]) + b lg([B]) + c lg ([C])
This could equally well be written using ln rather than lg.
At a particular temperature, k is a constant so its logarithm is also a constant. In addition, as stated above, it is normally the case that in a practical investigation the concentration of all but one the species involved are in excess, so that they are effectively constant. As a result their logarithms will also be constants.
An example
Suppose that, in a particular experiment, this applies to [B] and [C]. Then we can rewrite the logarithmic form of the rate equation as follows:
lg (rA) = a lg ([A]) + constantlg (k) + constantb lg([B]) + constantc lg ([C])
The sum of three constants must also be a constant so we can simplify this to lg (rA) = a lg ([A]) + constant
You should recognise this as the equation of a straight line as it is of the form y = mx + c, where
y = lg (rA), x = lg ([A]), m = a and c = constant.
(If you are not sure about the equation of a straight line graph then the tutorial Equation of a straight line graph should help you.)
From this you should be able to see that, if you plot a graph of lg (rA) against lg ([A]), you should get a straight line. The slope of that line will be a, the order of the reaction with respect to species A.
You can now repeat your experiment with first [B] varying with [A] and [C] constant, and then [C] varying with [A] and [B] constant, to get the values of b and c by plotting the corresponding logarithms of rates against the logarithms of the concentrations.
But how do you get the values of the rates at different times during the reaction? That is the subject of the tutorial Rates from concentration
You may also find it helpful to look at the tutorial How does 'y = mx + c' relate to the Arrhenius equation? which goes on from describing the equation to a straight line to show how, using the Arrhenius equation, activation energies can be calculated from the slope of a graph of ln (rate) against 1/T (the absolute temperature).
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updated: 12 January 2007
