Chemistry

5.3 - Calculating Rate from Graphs

9 min

Mean rate

Mean rate from a graph

Drawing a tangent

The gradient of a tangent

note 1 of 4

Calculating mean rate from a graph

A rate-of-reaction graph plots the quantity of a reactant used up or product formed (y-axis) against time (x-axis).

Method

Choose the two times between which you want the mean rate.
Read the quantity plotted at each of the two times.
Subtract the earlier reading from the later one to find the change in quantity.
Subtract the earlier time from the later time to find the change in time.
Divide the change in quantity by the change in time.

Formula for mean rate of reaction

$Mean rate of reaction = \frac{change in quantity}{change in time}$

To calculate the mean rate over the whole reaction, measure from the start (0 s) to where the line becomes flat (when the reaction has finished).

A curve of quantity of product formed against time levels off at a point labelled reaction finished; a dashed right angle between two points on it shows change in time and change in quantity.

note 2 of 4

Worked example - Mean rate from a graph

The graph shows the volume of hydrogen gas produced when magnesium reacts with excess dilute hydrochloric acid. Calculate the mean rate of reaction between 20 s and 60 s.

A curve of volume of hydrogen produced in cm³ against time in seconds rises and levels off; dashed lines from 20 s and 60 s meet it at 40 cm³ and 60 cm³.

Step 1 - Read the quantity at each time

Volume at 20 s = 40 cm³
Volume at 60 s = 60 cm³

Step 2 - Find the change in quantity

60 cm³ - 40 cm³ = 20 cm³

Step 3 - Find the change in time

60 s - 20 s = 40 s

Step 4 - Divide change in quantity by change in time

$\begin{matrix} Mean rate of reaction & = \frac{20 {cm}^{3}}{40 s} \\ = 0.5 {cm}^{3} /s \end{matrix}$

Step 5 - Interpretation

Over the first interval (0 s to 20 s), the volume rises from 0 cm³ to 40 cm³, giving a mean rate of $\frac{40 {cm}^{3}}{20 s} = 2.0 {cm}^{3} /s$ .
The reaction was four times as fast over the first interval as over the later one. Comparing mean rates over successive intervals shows how the rate falls as reactants are used up.

note 3 of 4

Finding the rate at one moment using a tangent

The rate at a specific moment is the instantaneous rate.
The instantaneous rate equals the gradient of a tangent drawn to the curve at that point.

Method

Find the point on the curve at the chosen time.
Draw a straight line that touches the curve at that point without crossing it (the tangent), extending it as far as space allows.
Choose two points on the tangent as far apart as possible.
Calculate the gradient by dividing the change in quantity by the change in time between those two points.

A tangent touches a curve of quantity of product formed against time at a point labelled tangent touches here; a dashed right angle off the tangent shows change in time and change in quantity.

Formula for the rate at a point

$Rate at that point = gradient of the tangent = \frac{change in quantity}{change in time}$

Tangents are drawn by eye, so mark schemes accept a small range of answers.

note 4 of 4

Worked example - The tangent's gradient

The graph shows the volume of hydrogen gas produced when magnesium reacts with excess dilute hydrochloric acid. A tangent has been drawn to the curve at 20 s. Use the tangent to calculate the rate of reaction at 20 s.

A tangent touches a curve of volume of hydrogen produced in cm³ against time at 20 s; dashed lines read change in time = 50 s and change in quantity = 58 cm³.

Step 1 - Read the values from the tangent

Change in quantity between the two marked points = 58 cm³
Change in time between the same two points = 50 s

Step 2 - Divide change in quantity by change in time

$Rate at 20 s = \frac{58 {cm}^{3}}{50 s} = 1.16 {cm}^{3} /s$

Step 3 - Interpretation

This is higher than the mean rate over 20 s to 60 s on the same graph (0.5 cm³/s, worked out in the mean rate example), which fits the graph's shape.
The reaction is fastest early on and slows as reactants are used up, so the instantaneous rate at the start of an interval is always higher than the mean rate across it.

key terms

TermDefinition

Instantaneous rateThe speed of a reaction at one specific moment in time.

TangentA straight line that touches a curve at a single point, used to find the gradient of the curve at that point.

end of notes

Key points to remember

Calculating mean rate from a graph

A rate-of-reaction graph plots the quantity of a reactant used up or product formed (y-axis) against time (x-axis).

Method

Choose the two times between which you want the mean rate.

Read the quantity plotted at each of the two times.

Subtract the earlier reading from the later one to find the change in quantity.

Subtract the earlier time from the later time to find the change in time.

Divide the change in quantity by the change in time.

Formula for mean rate of reaction

$Mean rate of reaction = \frac{change in quantity}{change in time}$

To calculate the mean rate over the whole reaction, measure from the start (0 s) to where the line becomes flat (when the reaction has finished).

Finding the rate at one moment using a tangent

The rate at a specific moment is the instantaneous rate.

The instantaneous rate equals the gradient of a tangent drawn to the curve at that point.

Method

Find the point on the curve at the chosen time.

Draw a straight line that touches the curve at that point without crossing it (the tangent), extending it as far as space allows.

Choose two points on the tangent as far apart as possible.

Calculate the gradient by dividing the change in quantity by the change in time between those two points.