# 1.4.6 Maths Skill: Using a Tangent to Find Initial Rate of Reaction

### Maths Skill: Using a Tangent to Find Initial Rate of Reaction

• For linear graphs (i.e. graphs with a straight-line), the gradient is the same throughout
• This makes it easy to calculate the rate of change (rate of change = change ÷ time)
• However, many enzyme rate experiments produce non-linear graphs (i.e. graphs with a curved line), meaning they have an ever-changing gradient
• They are shaped this way because the reaction rate is changing over time
• In these cases, a tangent can be used to find the reaction rate at any one point on the graph:
• A tangent is a straight line that is drawn so it just touches the curve at a single point
• The slope of this tangent matches the slope of the curve at just that point
• You then simply find the gradient of the straight line (tangent) you have drawn
• The initial rate of reaction is the rate of reaction at the start of the reaction (i.e. where time = 0)

#### Worked Example

The graph below shows the results of an enzyme rate reaction. Using this graph, calculate the initial rate of reaction.

Step 1: Estimate the extrapolated curve of the graph

Step 2: Find the tangent to the curve at 0 seconds (the start of the reaction)

The tangent drawn in the graph above shows that 72 cm3 of product was produced in the first  20 seconds.

Step 3: Calculate the gradient of the tangent (this will give you the initial rate of reaction):

Gradient = change in y-axis ÷ change in x-axis

Initial rate of reaction = 72 cm3 ÷ 20 s

Initial rate of reaction = 3.6 cm3 s-1

#### Exam Tip

When drawing tangents: always use a ruler and a pencil; make sure the line you draw is perfectly straight; choose the point where the tangent is to be taken and slowly line the ruler up to that point; try to place your ruler so that none of the line of the curve is covered by the ruler (it is much easier if the curve is entirely visible whilst the tangent is drawn).

There is a handy phrase to help you remember how to calculate the gradient of a tangent or line. Rise over run means that any increase/decrease vertically should be divided by any increase/decrease horizontally.

### Author: Amelia

While studying Biochemistry at Oxford University, Amelia started her own tutoring service, helping to connect Science tutors with students in her local area. Amelia has experience teaching the sciences and Maths at all levels to UK and international students and, as well as being our Biology Lead, designs revision resources for Chemistry.
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