Calculating Gradients

Section 1 of 7Why tangents are needed (HT only)

Curves vary

• Curves vary Most rate graphs start straight but then curve as reactant is used up.
• The rate is different at each point on a curve.

Tangent

• Tangent A tangent is a straight line touching the curve at one point;
  • its gradient equals the rate at that point.

Initial rate

• Initial rate A tangent at t = 0 gives the initial rate, the fastest rate of the reaction.

Section 2 of 7Drawing a good tangent (HT only)

1

Use a sharp pencil and a clear ruler.

2

Place the ruler on the chosen point so it just touches the curve there.

3

Rotate the ruler until the gap between ruler and curve is equal on both sides of the point.

4

Draw a straight line, extending it well beyond the point, across as much of the graph as possible.

5

Make sure the curve is still visible and not hidden under the ruler.

Section 3 of 7Calculating the gradient (HT only)

1Choose two points

• Choose two points Pick two points on the tangent line, far apart, where it crosses grid lines.

2Rise

• Rise Find the change in the quantity on the y-axis.

3Run

• Run Find the change in time on the x-axis.

4Divide

• Divide Gradient = change in product ÷ change in time, with units such as cm³/s.

Section 4 of 7Worked example (HT only)

Stage	Working
Graph	Volume of gas against time, curve levelling off
Tangent	Drawn at t = 0 touching the curve at the origin
Two points	(0 s, 0 cm3) and (20 s, 60 cm3) on the tangent
Gradient	60 ÷ 20 = 3.0
Answer	Initial rate = 3.0 cm3/s

Section 5 of 7Using tangents to compare

Rate at different times

Early on

• Early on Steep tangent, high rate.

Mid-reaction

• Mid-reaction Gentler tangent, lower rate as reactants are used.

Near the end

• Near the end Almost horizontal tangent, rate close to zero.

Different conditions

• Different conditions Compare tangents at the same time for two curves to find which run is faster.

Section 6 of 7Key terms

Term	Meaning
Tangent	A straight line that touches a curve at one point without crossing it.
Gradient	The steepness of a line, equal to change in y divided by change in x.
Initial rate	The rate of reaction at time zero.
Rise over run	A memory aid for gradient: vertical change divided by horizontal change.
Extrapolate	To extend a line beyond the plotted data to read values.

Section 7 of 7Exam technique

1

Using two points on the curve rather than on the tangent to calculate the gradient.

2

Drawing the tangent so it cuts through the curve instead of just touching it.

3

Choosing two points too close together, which increases uncertainty.

4

Leaving the unit off the answer or giving the rate as a volume rather than volume per second.