The rate at any time is the gradient of the graph: for a curve, draw a tangent and take change in y divided by change in x; a steeper gradient means a faster reaction.
This page walks through rate from a graph gradient for SPM Chemistry, step by step: the formula you need, the units to watch, and a worked example.
When you use this
A rate experiment is usually plotted as volume of gas (or mass) against time. The gradient of that graph is the rate, and reading it correctly, especially for a curve, is a standard SPM Chemistry skill.
The formula and units
rate = gradient = change in y / change in x
- For a straight line, pick any two points and divide the rise by the run.
- For a curve, the rate is changing, so you draw a tangent at the time you want and take the gradient of that tangent. This gives the instantaneous rate.
- The gradient of a chord between two points on a curve gives the average rate between them.
Units. The unit comes from the axes: a volume-time graph in cm3 and s gives cm3 s−1; a mass-time graph gives g s−1. A steeper gradient always means a faster reaction.
Worked example 1 (easy)
A straight line passes through the origin and the point (20 s, 60 cm3). Find the rate.
- gradient = change in y / change in x = 60 / 20 = 3 cm3 s−1.
Worked example 2 (medium)
On a curve of gas volume against time, the tangent drawn at the very start (t = 0) passes through (0, 0) and (10 s, 45 cm3). Find the initial rate of reaction.
- initial rate = 45 / 10 = 4.5 cm3 s−1.
Worked example 3 (SPM level)
Two experiments are drawn on the same axes. For the faster one, the tangent at t = 0 reaches 40 cm3 at 8 s; by t = 40 s its curve is horizontal. Find the initial rate and describe the rate at 40 s.
- initial rate = gradient of the tangent = 40 / 8 = 5 cm3 s−1.
- At 40 s the curve is flat, so its gradient is 0; the rate is zero and the reaction has finished.
Because the faster experiment has the steeper starting tangent, you can also state that it has the higher initial rate without any further arithmetic, the gradient alone tells you.
Reading a tangent correctly
The commonest mistake is to read two points straight off the curve and divide; that gives neither a proper average nor the instantaneous rate. Instead, lay a ruler so it just touches the curve at your chosen time and matches its slope there, extend the tangent across a wide span of the graph, and read the change in y and change in x from that line. A long tangent reduces the reading error and protects the mark.
Average and instantaneous on one graph
The same graph gives both kinds of rate. A tangent at a single time gives the instantaneous rate there; a straight line joining the start and end points gives the average rate over the whole reaction. If a question asks how the rate changes, compare tangents at the start and later on: the starting tangent is steepest, then the gradient falls to zero as the graph levels off. This matters because Paper 2 often pairs a gradient calculation with a written explanation of why the rate drops with time, and both can be read from the shape of the same curve.
Common traps
- Using points on the curve for an instantaneous rate. Only a tangent gives the rate at a single time.
- Ignoring the axis units. Read the actual scale; do not assume each square is one unit.
- Forgetting a flat line means zero rate. A horizontal graph means the reaction has stopped, not that it is steady.
- Reading a tangent over too small a span. A short tangent magnifies reading errors.
Our teachers have students draw tangents with a ruler over a wide interval and label the change in y and change in x on the graph, so both the value and its unit are secure.
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