Calculating rates of reactions
Reaction rate graphs
A reaction rate graph shows how the quantity of product formed or reactant used up changes as time passes. Its shape shows how quickly the reaction progresses at different stages.
Drawing a reaction rate graph
Put time on the horizontal axis and the measured quantity on the vertical axis. Label both axes with their units. Plot the measured quantity at each recorded time, then draw a suitable line or smooth curve through the points.
As product forms, the quantity of product increases. Similarly, the total quantity of reactant used up increases as the reaction proceeds.
Interpreting the graph
The steepness, or gradient, shows how quickly the measured quantity is changing:
- A steep section shows a faster reaction: the quantity changes by more in a given time.
- A less steep section shows a slower reaction: the quantity changes by less in the same time.
- A flat section shows that the quantity is no longer changing because the reaction has stopped.
The steepness can change along the graph, so the reaction may be faster at some times than at others. This lets the graph show changes in reaction progress that a single overall average rate would not reveal.
Rate at a specific time from a tangent
On a quantity–time graph, the gradient shows how quickly the quantity changes. If the graph is curved, its gradient changes with time, so a gradient measured across an interval gives a mean rate rather than the rate at one specific moment.
To find the rate at a chosen time, draw a tangent: a straight line that touches the curve at that point and follows the curve’s direction there. The gradient of the tangent gives the instantaneous rate at that time.
Calculating the tangent gradient
- Find the chosen time on the horizontal axis and locate the corresponding point on the curve.
- Draw a tangent at that point. Extend the line so that its gradient can be measured accurately.
- Choose two well-separated points on the tangent line. They do not have to lie on the curve.
- Divide the change in quantity between those points by the change in time:
For example, suppose a tangent drawn at $$30\ \mathrm{s}$$ passes through $$(20\ \mathrm{s}, 30\ \mathrm{cm^3})$$ and $$(40\ \mathrm{s}, 50\ \mathrm{cm^3})$$. Its gradient is:
The instantaneous rate at $$30\ \mathrm{s}$$ is therefore $$1.0\ \mathrm{cm^3,s^{-1}}$$. The two points were used to calculate the gradient of the tangent, not the mean rate along the curve between those times.
If the measured quantity decreases, the tangent has a negative gradient. The rate of disappearance is usually stated as a positive value: the magnitude of that gradient.
Mean rate of reaction
A reaction uses up reactants and forms products. Its mean rate over a chosen time interval describes how much of a reactant is used up, or how much of a product is formed, per unit of time.
Use the change in quantity of the chosen substance during the interval:
- For a reactant, subtract its final quantity from its initial quantity to find the amount used.
- For a product, subtract its initial quantity from its final quantity to find the amount formed.
The numerical rates calculated for different substances in the same reaction need not be equal. State which substance the rate refers to.
Mean rate units
Quantity can be measured as mass in grams or volume in $$\text{cm}^3$$. If time is measured in seconds, the corresponding mean rate units are $$\text{g/s}$$ for mass and $$\text{cm}^3/\text{s}$$ for volume.
Calculating a mean rate
Suppose the mass of a reactant falls from $$5.0\ \text{g}$$ to $$3.5\ \text{g}$$ in $$30\ \text{s}$$. The mass used is $$5.0 - 3.5 = 1.5\ \text{g}$$, so:
This is the average rate over the $$30\ \text{s}$$ interval. The reaction may have been faster during some parts of the interval and slower during others.