Rate Equations

Rate equation chemistry

The rate of a reaction is how quickly a reaction occurs. But what does that mean and what does it tell us? Well, one way of looking at it is to think about how much product is made in a period of time, which will depend on how much reactant is used up. In essence, we can say that the rate of a reaction is the speed at which reactants are converted into products.

Measuring rate of reaction

To calculate the rate of a reaction, we need to measure the change in the amount of reactant/product from the start of the reaction to the end.

We can measure this by observing things like colour change, pH change, volume of gas produced, or change in mass of a solid reactant. You should then be able to convert your data values into figures for concentration. This data is plotted on a line graph, with time on the x-axis and concentration on the y-axis. From the graph, we can find a value for the rate of reaction by working out the line's gradient. We either calculate an overall rate of reaction or an instantaneous rate of reaction. Both use the following equation:

$\mathrm{rate}\mathrm{of}\mathrm{reaction}=\frac{\mathrm{change}\mathrm{in}\mathrm{concentration}}{\mathrm{time}\mathrm{taken}}$

Overall rate of reaction

Calculating an overall rate of reaction is fairly straightforward. You divide the overall change in concentration of a reactant or product by the time taken. For a graph of concentration against time, this means dividing the change in y-values by the change in x-values. Here's an example.

To find the overall rate of reaction, we divide the change in concentration by the time taken.

Fig. 2 - A concentration-time graph used to calculate rate of reaction

Here, the concentration starts at 40 mol dm^-3 and ends at 8 mol dm^-3. This is a change of 40 - 8 = 32 mol dm^-3. The reaction takes 200 seconds. The rate of reaction is therefore $\frac{32}{200}=0.16\mathrm{mol}{\mathrm{dm}}^{-3}{\mathrm{s}}^{-1}$.

Instantaneous rate of reaction

Sometimes, finding an overall rate of reaction isn't that useful. You might instead want to know how the rate of reaction changes over time. To do this, you calculate instantaneous rates of reaction. This involves drawing a tangent to the curve at a particular point and finding its gradient. Again, this is given by the change in concentration divided by the time taken - in other words, the change in y-values divided by the change in x-values.

We first need to find the 60-second mark on the curve. We draw a tangent to the curve at this point.

Next, we calculate the gradient of this tangent by dividing the change in concentration by the time taken. You do this by turning the tangent into a right-angled triangle.

Fig. 4 - A concentration-time graph used to calculate rate of reaction

Here, we can see from our right-angled triangle that the concentration starts at 22 mol dm^-3 and ends at 6 mol dm^-3. This is an overall change of 16 mol dm^-3. This change in concentration takes place between 20 and 120 seconds, meaning it takes 100 seconds in total. The instantaneous rate of reaction is therefore $\frac{16}{100}=0.16\mathrm{mol}{\mathrm{dm}}^{-3}{\mathrm{s}}^{-1}$

Rate of reaction equation

Let's look at something different: the rate equation. The rate equation in chemistry is a formula that we can use to find the rate of a reaction using the concentration of species involved in the reaction. Here's what it looks like:

Fig. 5 - Rate equation

At first glance it certainly looks confusing, but once you understand what's going on it isn't all that bad.

• k is the rate constant.
• The letters A and B in the rate equation are used to represent species involved in the reaction. These could be reactants or catalysts.
• The square brackets around the letters represent concentration. So, [A] is used to show the concentration of species A.
• The letters m and n represent the order of the reaction with respect to a certain species. They show the power that the concentration of that species is raised to in the rate equation. Overall, [A]^m represents the concentration of A, raised to the power of m. This means that A has the order m.

The rate constant

k is the rate constant. It is used in the rate equation to link the concentrations of certain species to the rate of that reaction. The value of k changes depending on the reaction and reaction conditions. However, k is always constant for a certain reaction at a particular temperature. If you were to carry out the exact same reaction at different temperatures, k would change, but if you carried it out at the same temperature, k would stay the same: after all, it is a constant!

Orders of reaction

In chemical reactions, reactants and catalysts (if there are any) have an order of reaction. The sum of the individual orders of species in a reaction equals the overall order of the equation.

In the rate equation, the order of a reaction with respect to a species is shown using a power. For example, in the rate equation we looked at above, the order of A is represented by the letter m. The order of a reaction with respect to a species tells us how the concentration of that particular species affects the reaction rate. Some species don't affect the rate whatsoever, while other species affect it dramatically.

Zero-order reactants

The concentration of a zero-order reactant doesn't affect the rate of reaction. If you double its concentration, the rate stays the same. This is because 2⁰ = 1. Because they have no effect on the rate of reaction, zero-order reactants don't appear in the rate equation.

First-order reactants

The concentration of first-order reactants is directly proportional to the rate of reaction. If you double the concentration of a first-order reactant, the rate of reaction also doubles. This is because 2¹ = 2.

If a reactant is first-order, it appears in the rate equation raised to the power of 1. However, we don't tend to write the number 1 because raising something to the power of 1 has no effect on its value. You'll see first-order reactants in the rate equation as [A], where A represents the species.

Second-order reactants

The concentration of second-order reactants has an exponential effect on the rate of reaction. Doubling the concentration of a second-order reactant causes the rate of reaction to quadruple. This is because 2² = 4.

If a reactant is second-order then we put it in the rate equation raised to the power of 2; in other words, squared. You'll see it in the rate equation as [A]².

Order of a reaction

The overall order of a reaction is the sum of all the individual reactant orders. Remember that the order of a reactant is the power that it is raised to in the rate equation. If you're ever asked to find the overall order of reaction, simply add together all of the powers present in the equation and you'll reach your final answer.

Understanding orders of reaction is a bit tricky, so let's look at an example to help you understand.

Determining the rate equation

There are a few different methods we can use to determine the rate equation for a reaction. The basic principles come down to determining the species involved in the rate equation and then finding each of their orders. The main methods for doing this are:

• The initial rates method.
• Using rate-concentration graphs.
• Finding first-order reactants from their half-life.
• Inspecting the reaction mechanism.

Initial rates

The initial rates method involves measuring the rate of the same reaction over several experiments, each with different starting concentrations of a particular reactant. This method allows us to see numerically how the concentration of the reactant affects the rate of the reaction. We do this for each reactant, and can use the information to determine the reactant's order.

Rate-concentration graphs

Earlier in the article, we looked at how you use graphs showing concentration of a species against time to calculate rate of reaction at a specific instant. You can then take the values for instantaneous rate of reaction and plot them against concentration to make a rate-concentration graph. These take specific shapes, depending on the order of the species involved.

• A horizontal straight line shows that the rate of reaction is unaffected by the concentration of the species. The species is therefore zero-order.
• A sloping straight line through the origin shows that the rate is directly proportional to the concentration of the species. The species is therefore first-order.
• A curved line through the origin shows that the rate is exponentially proportional to the concentration of the species. The species is second-order or higher.

Fig. 6 - Rate-concentration graphs for reactants with different orders

Half-life equations

The half-life, ${\mathrm{t}}_{1/2}$, of a reactant is the time it takes for the concentration of that reactant to become half of what it was where you started measuring from. There's an interesting feature of first-order reactants: they have a constant half-life. This means that it takes the same amount of time to get from, say, a concentration of 1.0 to a concentration of 0.5 mol dm^-3, as it does to get from a concentration of 0.8 to 0.4 mol dm^-3. In both cases, the concentration has halved.

You can measure half-life using concentration-time graphs. Pick any point on the graph and look at the concentration for that time value. Then, see how long it takes to halve the concentration. Repeat this again to find multiple half-lives for a species. If all of the half-lives are the same, the species is first-order.

Fig. 7 - A concentration-time graph used to show half-life

Half-life and rate constant

The half-life of a first-order reactant relates to the rate constant, k, using the following equation:

$\mathrm{k}=\frac{\ln \left(2\right)}{{\mathrm{t}}_{1/2}}$

This means that once you know the half-life of a first-order reactant, you can easily find k.

Reaction mechanism

Reactions can have mechanisms with one step or multiple steps. Each step happens at a different speed, and the rate of reaction is determined by the slowest step. We call the slowest step in a reaction the rate-determining step, and it gives us an idea of what the rate equation is likely to look like. This is because the rate equation is only made up of reacting species found in the steps up to and including the rate-determining step. The number of moles of each species relates to its order.

If you know the reaction mechanism and the rate-determining step of a reaction, you can predict the rate equation!