A-level Chemistry/WJEC/Module 3/Kinetics

Order and rate equations

The reaction is investigated by running several series of experiments. In each series just one of the reactants is varied so that we may observe the effect on the rate of reaction.

Effect of concentration on reaction rate showing different orders of reaction

Above: Increasing the concentration of reactant has three typical ways of affecting the reaction rate. The most common is first order where rate and concentration are directly proportional. A second order relationship is where the rate is proportional to the square of the concentration. A zero order relationship is where the rate is not affected by the concentration. Logically, the zero-order pattern does not extend to very low reactant concentrations.

For any one reactant the change in the rate of the reaction is categorised as:

• Zero order
  • Any increase in the concentration of the reactant has no effect on the rate of the reaction.

In Biology, it is well-known that for many enzyme-catalysed reactions, the rate is proportional to the reactant ("substrate") concentration at low concentrations, but reaches a maximum rate at high concentrations. A chemist would say the reaction is first-order at low concentrations but zero-order at high concentrations.

• First order
  • The rate of reaction increases in direct proportion to the increased concentration of the reactant. This can be observed by comparing runs 1 and 2. The concentration of A remains constant whilst the concentration of B is doubled (x2). Consequently the rate of the reaction is doubled from 0.08 to 0.16. We would say that this reaction is first order with respect to B.
• Second order
  • The rate of the reaction increases in proportion to the square of the concentration change. This can be observed by comparing runs 1 and 3. The concentration of B remains constant whilst the concentration of A is doubled. Consequently the rate of the reaction is quadrupled (x4) from 0.08 to 0.32. This reaction is second order with respect to A.

Run	Initial concentration of A	Initial concentration of B	Initial rate
First	0.01	0.02	0.08
Second	0.01	0.04	0.16
Third	0.02	0.02	0.32

It should be made clear that rate equations can only be determined by experiment.

If we plot concentration against time we can see different patterns for the three orders of reaction.

First order kinetics result in an exponential curve with a constant half-life. This is the same mathematics as radioactive decay.

Second order kinetics result in a geometrical curve with very fast initial reaction. This is not a curve you need to know.

Zero-order kinetics result in a straight line. If the line continued it would predict negative concentrations, so logically the zero-order line has to become a first-order curve as it approaches low reactant concentrations.

Constructing Rate Equations

Rate equations allow us to predict the rate of a reaction under different conditions. They take the form

Rate, v = k[A]^ma[B]^mb ...etc.

Where

• [A] is the concentration of reactant A (in moldm^-3) and m_a is the order of reaction with respect to A
• [B] is the concentration of reactant B (in moldm^-3) and m_b is the order of reaction with respect to B
• k is the Rate Constant. This depends on the reaction itself - how inherently fast it is - temperature, and any catalysts.

The rate equation can be rearranged in the normal manner to find any one of the required variables. At A level chemistry this is often finding k given data like above. The overall order of the reaction, m is the sum of the indexes m_a + m_b ..etc.

Arrhenius Equation

The Arrhenius equation determines a rate coefficient based on temperature and activation energy. It is surprisingly accurate and very useful. The Arrhenius equation is:

k = Ae^-Ea/RT

E_a is the activation energy for the reaction, in joules per mole. R is the Universal Gas Constant, T is the absolute temperature (in kelvin), and A is the pre-exponential factor. A is usually determined experimentally.

A very useful alternative way to write the equation is by taking natural logarithms of both sides:

ln(k) = ln(A) - E_a/RT

This form of the equation can be used to plot experimental data. If k is measured at several temperatures, we can plot a graph of ln(k) on the y-axis and 1/T on the x-axis. This gives a gradient of -E_a/R. In other words, we can calculate E_a because E_a = -R x gradient.

Plot of ln(k) on the y-axis and 1/T on the x-axis. The gradient can be used to calculate E_a and the intercept is ln(A).

Above: The gradient of the plot is -12650 K. E_a = -R x gradient = -8.31 J mol^-1 K^-1 x -12650 K = 105 122 J mol^-1 = 105 kJ mol^-1

The basic equation can be rearranged to find A:

k = Ae^-Ea/RT

A = ke^+Ea/RT

The logarithmic equation can be used to find T or E_a:

E_a = RTln(A/k)

T = E_a/[Rln(A/k)]

It is probably best to check that you can derive these versions of the formula, and derive them if you ever need them. Learning them is likely to be a waste of time - only one version, if any, will ever be needed in your exams.