Reaction Graphs Tell Us More Than What Happened

Reaction graphs do more than show how the amount of a substance changes.

They can also tell us:

how quickly the change is happening

This is the:

reaction rate

The key mathematical idea is:

gradient

A steep graph represents a rapid change.

A shallow graph represents a slower change.

A horizontal graph represents:

no net change

By calculating gradients, we can turn the shape of a graph into a numerical measurement of reaction rate.

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The Basic Rate Equation

Reaction rate compares:

change in a measured quantity

with:

change in time

The general relationship is:

average rate = change in quantity ÷ change in time

This can also be written as:

Average rate = Δy / Δt

where:

• Δy = change in the quantity shown on the y-axis
• Δt = change in time
• Δ means "change in"

The quantity might be:

• concentration
• mass
• gas volume
• amount of substance

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What Is Gradient?

The gradient of a graph tells us how quickly the y-value changes compared with the x-value.

For reaction graphs:

gradient = change in measured quantity / change in time

or:

gradient = Δy / Δx

Because time is normally on the x-axis:

gradient = Δy / Δt

This makes gradient extremely useful for measuring:

reaction rate

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Rise Over Run

Another way to remember gradient is:

gradient = rise / run

where:

rise = vertical change

and:

run = horizontal change

For a reaction graph:

vertical change = change in concentration, mass, or volume

horizontal change = change in time

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Average Rate

An average rate describes how quickly something changes over a:

time interval

Suppose a reaction produces:

60 cm³ of gas

during:

30 seconds

Average rate:

60 cm³ ÷ 30 s

= 2.0 cm³/s

Therefore:

average rate = 2.0 cm³/s

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Finding Average Rate from a Graph

Consider this illustrative product-formation data:

The graph would rise rapidly at first and gradually become horizontal.

To calculate the average rate between:

10 s and 30 s

read the corresponding values:

At 10 s:

28 cm³

At 30 s:

57 cm³

---

Step 1: Calculate the Change in Product

Δvolume = final volume − initial volume

Δvolume = 57 − 28

Δvolume = 29 cm³

---

Step 2: Calculate the Change in Time

Δtime = final time − initial time

Δtime = 30 − 10

Δtime = 20 s

---

Step 3: Calculate the Average Rate

average rate = 29 / 20

average rate = 1.45 cm³/s

Therefore:

Average rate from 10–30 s = 1.45 cm³/s

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A Common Calculation Error

Suppose we want the average rate between:

20 s and 50 s

Do not divide the product amount at 50 s by 50.

Instead, calculate the:

change between the two selected points

At 20 s:

46 cm³

At 50 s:

66 cm³

Change in product:

66 − 46 = 20 cm³

Change in time:

50 − 20 = 30 s

Therefore:

average rate = 20 / 30

= 0.67 cm³/s

approximately.

The rule is:

always calculate change in y and change in x

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Units of Reaction Rate

Rate units depend on:

what the graph measures

If the graph shows:

gas volume in cm³

and:

time in seconds

then rate units are:

cm³/s

If the graph shows:

concentration in mol/L

then:

mol L⁻¹ s⁻¹

If the graph shows:

mass in grams

then:

g/s

Always derive rate units from:

y-axis unit ÷ x-axis unit

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Positive Gradients

A product formation graph usually rises.

For example:

gas volume increases with time

Therefore:

Δy is positive

and:

gradient is positive

A positive gradient means the measured quantity is:

increasing

For a product graph, this usually represents:

product formation

---

Negative Gradients

A reactant concentration usually:

decreases

as the reaction proceeds.

Therefore, its graph may slope downward.

For example:

At 10 s:

[A] = 0.80 mol/L

At 30 s:

[A] = 0.50 mol/L

Gradient:

(0.50 − 0.80) / (30 − 10)

= −0.30 / 20

= −0.015 mol L⁻¹ s⁻¹

The negative sign tells us:

reactant concentration is decreasing

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Rate of Disappearance

Although a reactant concentration graph has a negative gradient, we often report its:

rate of disappearance

as a positive magnitude.

So if:

gradient = −0.015 mol L⁻¹ s⁻¹

we may say:

rate of disappearance = 0.015 mol L⁻¹ s⁻¹

The negative gradient describes:

direction of change

The positive magnitude describes:

how quickly the reactant disappears

---

Straight-Line Graphs

If a reaction graph is a straight line, its gradient is:

constant

This means the measured quantity changes at a:

constant rate

For example:

0 s → 0 cm³

10 s → 20 cm³

20 s → 40 cm³

30 s → 60 cm³

Every 10 seconds:

20 cm³ more product forms

Therefore:

rate = 20 / 10 = 2 cm³/s

throughout that interval.

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Curved Reaction Graphs

Most reaction graphs are not perfectly straight.

They are:

curved

A curved graph means:

the gradient is changing

Therefore:

the reaction rate is changing

This is why we need to distinguish between:

average rate

and:

instantaneous rate

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Average Rate vs Instantaneous Rate

Average Rate

Measures the rate over:

a time interval

For example:

average rate between 10 s and 30 s

Instantaneous Rate

Measures the rate at:

one particular moment

For example:

rate at exactly 20 s

These are different quantities.

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Why Do We Need Instantaneous Rate?

Imagine driving a car.

Your average speed during a one-hour journey might be:

60 km/h

But at one particular moment, your speedometer might show:

85 km/h

Reaction rates work similarly.

An average rate tells us about:

an interval

An instantaneous rate tells us about:

one moment

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Finding Instantaneous Rate

For a curved graph, we cannot simply use the gradient of the entire curve.

Instead, we draw:

a tangent

at the point we want to investigate.

A tangent is a straight line that touches the curve at that point and follows its direction there.

The gradient of the tangent estimates:

the instantaneous rate

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How to Draw a Tangent

Suppose you want to find the reaction rate at:

30 seconds

Step 1

Find:

30 s

on the x-axis.

Step 2

Find the corresponding point on the reaction curve.

Step 3

Draw a straight line that:

just touches the curve at that point

and follows the direction of the curve.

Step 4

Extend the tangent so that you can select:

two widely separated points

on the tangent.

Step 5

Calculate:

gradient = Δy / Δt

The result estimates the:

instantaneous rate at 30 s

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Use Points on the Tangent

This is extremely important.

When calculating instantaneous rate:

use points on the tangent

not necessarily points on the original curve.

The tangent is a:

straight line

so its gradient can be calculated accurately.

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Use a Large Triangle

When finding the gradient of a tangent, choose points that are:

far apart

on the tangent.

Why?

A larger gradient triangle generally reduces the effect of:

small reading errors

So instead of choosing two points very close together:

use as much of the tangent as practical

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Worked Example: Instantaneous Rate

Suppose a tangent drawn at 20 s passes through:

(10 s, 25 cm³)

and:

(30 s, 55 cm³)

Change in product:

55 − 25 = 30 cm³

Change in time:

30 − 10 = 20 s

Gradient:

30 / 20

= 1.5 cm³/s

Therefore:

instantaneous rate at 20 s ≈ 1.5 cm³/s

The approximation symbol is useful because the tangent itself is:

an estimate

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Comparing Rates at Different Times

A reaction often begins quickly and then slows.

Suppose the instantaneous rates are:

At 10 s:

2.8 cm³/s

At 30 s:

1.4 cm³/s

At 50 s:

0.3 cm³/s

We can conclude:

reaction rate decreases with time

The graph becomes progressively:

less steep

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Why Does Rate Decrease?

As the reaction proceeds:

reactants are consumed

Therefore:

reactant concentrations decrease

There are fewer reactant particles available for:

successful collisions

As a result:

successful collision frequency decreases

and:

reaction rate decreases

This produces the typical curved shape.

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Connecting Gradient to Collision Theory

The relationship can be summarized:

high reactant concentration

↓

frequent successful collisions

↓

fast reaction

↓

steep graph

As the reaction proceeds:

lower reactant concentration

↓

fewer successful collisions

↓

slower reaction

↓

shallower graph

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Comparing Early and Late Gradients

Consider a product graph.

Near the beginning:

large gradient

Later:

smaller gradient

Finally:

gradient ≈ 0

This corresponds to:

fast → slower → no further net product formation

The gradient therefore provides a mathematical description of:

reaction progress

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The Gradient at Completion

When a product graph becomes horizontal:

Δy = 0

Therefore:

gradient = 0 / Δt

and:

gradient = 0

For a simple irreversible reaction, this means:

no additional product is being formed

The reaction has effectively:

reached completion

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Reaction Graph Regions

A typical reaction graph can be interpreted in three regions.

Region 1 — Steep

large gradient

The reaction is:

fast

Region 2 — Curving

decreasing gradient

The reaction is:

slowing

Region 3 — Plateau

gradient ≈ 0

There is:

no further net change

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Comparing Two Reactions

Suppose Reaction A and Reaction B are plotted on the same graph.

Reaction A has a steeper initial slope.

Reaction B has a shallower initial slope.

We can conclude:

Reaction A has a greater initial rate

provided the graphs show comparable measured quantities and scales.

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Same Final Amount, Different Rates

Suppose both reactions eventually produce:

80 cm³ of gas

but:

Reaction A reaches 80 cm³ in 40 s

while:

Reaction B reaches 80 cm³ in 90 s

Reaction A is:

faster

but both produce:

the same final amount

This demonstrates:

gradient → rate

while:

plateau height → final amount

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Different Rates at the Same Time

Suppose at 20 s:

Reaction A has a tangent gradient of:

2.0 cm³/s

Reaction B has a tangent gradient of:

1.1 cm³/s

Therefore, at 20 s:

Reaction A is producing gas more rapidly

This is a comparison of:

instantaneous rates

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Different Rates Within the Same Reaction

We can also compare the reaction with:

itself

at different times.

For example:

At 10 s:

rate = 3.0 cm³/s

At 30 s:

rate = 1.5 cm³/s

At 50 s:

rate = 0.4 cm³/s

This tells us:

the reaction is slowing down

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Percentage Decrease in Rate

Sometimes we can quantify how much the rate changes.

Suppose:

Initial rate:

2.0 cm³/s

Later rate:

0.5 cm³/s

Decrease:

2.0 − 0.5 = 1.5 cm³/s

Percentage decrease:

1.5 / 2.0 × 100

= 75%

The rate has decreased by:

75%

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Product Graphs

For a product formation graph:

positive gradient → product forming

large positive gradient → rapid product formation

small positive gradient → slow product formation

zero gradient → no further net product formation

The gradient generally decreases as the reaction progresses.

---

Reactant Graphs

For a reactant concentration graph:

negative gradient → reactant being consumed

steep negative gradient → rapid reactant consumption

shallow negative gradient → slower consumption

zero gradient → concentration no longer changing

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Mass-Loss Graphs

Some reactions release gas.

If the gas escapes from the reaction vessel:

measured mass decreases

For example:

acid + carbonate → salt + water + carbon dioxide

As CO₂ escapes:

mass decreases

The graph therefore slopes:

downward

The magnitude of the gradient tells us:

how quickly mass is being lost

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Gas Volume Graphs

If the gas is collected instead:

gas volume increases

The graph slopes:

upward

Even though one graph decreases and the other increases, both can measure the rate of:

the same reaction

This is why you must always read:

the axes

before interpreting the graph.

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Units from Different Graphs

The units come directly from:

y-axis units ÷ x-axis units

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Graph Scale Matters

Two graphs can look very different simply because they use:

different axis scales

A visually steeper curve does not automatically mean a faster reaction if the graphs use different scales.

For reliable comparisons:

• check the x-axis scale
• check the y-axis scale
• check the units
• calculate gradients when necessary

Never compare graph steepness without first checking:

the axes

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Worked Example 1: Average Rate

A gas-producing reaction gives:

Calculate the average rate from:

20 s to 60 s

Change in volume:

76 − 40 = 36 cm³

Change in time:

60 − 20 = 40 s

Rate:

36 / 40

= 0.90 cm³/s

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Worked Example 2: Overall Average Rate

Using the same data, calculate the average rate from:

0 to 80 s

Change in volume:

80 − 0 = 80 cm³

Change in time:

80 − 0 = 80 s

Average rate:

80 / 80

= 1.0 cm³/s

Notice that this does not mean the reaction rate was:

1.0 cm³/s at every moment

It is only the:

average over the entire interval

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Worked Example 3: Reactant Concentration

A reactant concentration decreases from:

1.20 mol/L

at 10 s to:

0.60 mol/L

at 40 s.

Gradient:

(0.60 − 1.20) / (40 − 10)

= −0.60 / 30

= −0.020 mol L⁻¹ s⁻¹

Therefore:

gradient = −0.020 mol L⁻¹ s⁻¹

and the average rate of disappearance is:

0.020 mol L⁻¹ s⁻¹

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Worked Example 4: Tangent

A tangent drawn at 25 s passes through:

(10 s, 18 cm³)

and:

(40 s, 60 cm³)

Change in volume:

60 − 18 = 42 cm³

Change in time:

40 − 10 = 30 s

Gradient:

42 / 30

= 1.4 cm³/s

Therefore:

instantaneous rate at 25 s ≈ 1.4 cm³/s

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Worked Example 5: Comparing Rates

Reaction A has an initial gradient of:

2.4 cm³/s

Reaction B has an initial gradient of:

1.6 cm³/s

Difference:

2.4 − 1.6 = 0.8 cm³/s

Therefore, Reaction A initially produces product:

0.8 cm³/s faster

than Reaction B.

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Worked Example 6: Rate at Completion

A product graph becomes horizontal after:

70 seconds

What is the gradient after 70 seconds?

Because the product amount remains constant:

Δy = 0

Therefore:

gradient = 0

For a simple irreversible reaction:

no further product is being formed

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Experimental Data and Scatter

Real experimental results rarely produce perfectly smooth curves.

Data may contain:

scatter

Small variations may occur because of:

• measurement uncertainty
• timing uncertainty
• temperature fluctuations
• reading errors
• limitations of equipment

Scientists therefore often draw:

a smooth curve of best fit

rather than connecting every data point with straight lines.

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Tangents and Experimental Data

When using experimental data:

1. Draw an appropriate curve of best fit.
2. Identify the required time.
3. Draw a tangent to the curve.
4. Choose two widely separated points on the tangent.
5. Calculate the gradient.

This provides an estimate of:

instantaneous reaction rate

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Why Tangent Rates Are Estimates

A tangent drawn by hand will not be:

perfectly exact

Two students may draw slightly different tangents and calculate slightly different rates.

This is acceptable if:

• the tangent is reasonable
• a large gradient triangle is used
• calculations are correct
• units are included

Therefore, instantaneous rates obtained graphically are often reported as:

approximate values

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Interpreting Graphical Evidence

Calculations are only part of graph analysis.

You should also be able to explain:

what the graph tells us about the reaction

For example:

The curve is steepest during the first 10 seconds, indicating the greatest reaction rate. The gradient decreases with time as reactants are consumed. After approximately 60 seconds, the graph becomes horizontal, showing that there is no further net product formation.

This combines:

observation + calculation + chemical explanation

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A Strong Graph Analysis Structure

Use:

Observe → Quantify → Explain

Observe

Describe what the graph does.

"The curve becomes less steep."

Quantify

Use numbers or gradients.

"The rate decreases from approximately 2.5 cm³/s to 0.6 cm³/s."

Explain

Use chemistry.

"Reactants are being consumed, so successful collisions become less frequent."

This creates a much stronger scientific response.

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Comparing Reaction Conditions

Suppose two curves represent the same reaction:

Curve A: higher temperature

Curve B: lower temperature

If Curve A has a greater initial gradient:

higher temperature produced a faster initial reaction

This can be explained because particles have:

greater kinetic energy

and a larger fraction of collisions can overcome:

activation energy

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Catalyst Graphs

Suppose two experiments differ only by the presence of a catalyst.

The catalyzed reaction generally has:

a greater gradient

and reaches the plateau:

sooner

If both experiments contain the same reactant amounts and reach the same completion:

final product amount remains the same

The catalyst changes:

rate

not the stoichiometric amount of product available.

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Concentration and Graph Gradient

Higher reactant concentration usually increases:

collision frequency

Therefore, the initial gradient may be:

greater

A graph can therefore provide evidence about how concentration affects:

reaction rate

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Surface Area and Graph Gradient

For reactions involving solids:

greater surface area

means more particles are exposed.

This increases collision opportunities.

Therefore:

greater surface area → greater initial gradient

For example:

powdered calcium carbonate

will generally react faster than the same mass of:

large calcium carbonate pieces

under otherwise identical conditions.

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Common Misconception: Rate Is the Height of the Graph

Rate is not determined by:

how high the graph is

Rate is determined by:

gradient

A graph can be very high but completely horizontal.

In that case:

rate of change = 0

Remember:

height → amount

gradient → rate

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Common Misconception: Use y ÷ x for Every Gradient

This only works if your interval begins at:

(0,0)

For any two general points, you must calculate:

(y₂ − y₁) / (x₂ − x₁)

Always use:

change in y ÷ change in x

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Common Misconception: Use Curve Points to Calculate a Tangent Gradient

For instantaneous rate:

draw the tangent first

Then select two points:

on the tangent

They do not have to be points on the original reaction curve.

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Common Misconception: A Steep Downward Graph Means a Slow Reaction

A steep downward slope has a:

large negative gradient

Its magnitude is large.

Therefore, it represents:

rapid decrease

For a reactant:

steep downward slope = rapid reactant consumption

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Common Misconception: Negative Rate Means the Reaction Is Going Backward

A negative gradient usually means:

the measured quantity is decreasing

For example:

reactant concentration decreases

or:

mass decreases because gas escapes

It does not automatically mean the chemical reaction is:

running backward

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Common Misconception: Average Rate Equals Rate at Every Moment

If the graph is curved:

rate changes continuously

The average rate describes the entire selected interval.

It does not necessarily equal the rate at:

any particular moment

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Common Misconception: The Steepest-Looking Graph Is Always Fastest

Before comparing curves on different graphs, check:

• axis scales
• units
• quantities measured
• time intervals

Visual appearance alone can be:

misleading

Calculating gradients provides stronger evidence.

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Graph Interpretation Checklist

When given a reaction graph, ask:

1. What is on the x-axis?

Usually:

time

2. What is on the y-axis?

Concentration? Mass? Gas volume?

3. Is the graph increasing or decreasing?

This tells you the:

direction of change

4. How steep is the graph?

This tells you the:

rate

5. Does the gradient change?

This tells you whether the reaction:

speeds up or slows down

6. Does the graph become horizontal?

This indicates:

no further net change in the measured quantity

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Check Your Understanding

1. What does the gradient of a reaction graph represent?

2. A reaction produces 48 cm³ of gas in 24 s. Calculate the average rate.

3. Product volume increases from 30 cm³ at 10 s to 70 cm³ at 30 s. Calculate the average rate between these times.

4. A reactant concentration decreases from 0.90 mol/L to 0.50 mol/L over 20 s. Calculate the gradient.

5. Why is the gradient of a reactant concentration graph usually negative?

6. Explain the difference between average rate and instantaneous rate.

7. Describe how a tangent can be used to estimate instantaneous reaction rate.

8. Why should two widely separated points be selected when calculating a tangent gradient?

9. A reaction graph is steep at 10 s but almost horizontal at 50 s. What does this tell you about how the reaction rate has changed?

10. Explain why reaction graphs often become less steep as the reaction proceeds.

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Key Terms

• Reaction rate: Measure of how quickly reactants are consumed or products are formed.
• Gradient: Change in the y-value divided by the corresponding change in the x-value.
• Average rate: Rate calculated over a specified time interval.
• Instantaneous rate: Rate at one particular moment.
• Initial rate: Reaction rate at the beginning of a reaction.
• Tangent: Straight line touching a curve at a particular point and following its direction there.
• Rate of formation: Speed at which a product is produced.
• Rate of disappearance: Speed at which a reactant is consumed.
• Positive gradient: Gradient produced when the measured quantity increases.
• Negative gradient: Gradient produced when the measured quantity decreases.
• Plateau: Horizontal region where the measured quantity remains approximately constant.
• Curve of best fit: Smooth curve representing the overall pattern of experimental data.
• Collision theory: Model explaining reaction rates through collisions between reacting particles.
• Successful collision: Collision that results in a chemical reaction.
• Activation energy: Minimum energy required for a successful reaction.

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Key Takeaways

• Reaction rates can be determined from the gradients of reaction graphs.
• Average rate = change in measured quantity ÷ change in time.
• Always calculate Δy ÷ Δt, not simply y ÷ t unless the interval begins at the origin.
• Rate units depend on the quantities shown on the graph.
• Gas-volume graphs may give rates in cm³/s.
• Concentration graphs may give rates in mol L⁻¹ s⁻¹.
• Mass graphs may give rates in g/s.
• Product formation graphs usually have positive gradients.
• Reactant concentration graphs usually have negative gradients.
• The magnitude of a gradient indicates how quickly the measured quantity is changing.
• Steep gradient = rapid change.
• Shallow gradient = slow change.
• Horizontal graph = zero net change in the measured quantity.
• A curved graph indicates that reaction rate is changing.
• Average rate describes a time interval.
• Instantaneous rate describes one particular moment.
• Instantaneous rate can be estimated by drawing a tangent to the curve.
• Calculate the gradient using two widely separated points on the tangent.
• Tangent-based rates are estimates and should usually be treated as approximate.
• Reaction rates often decrease with time because reactants are consumed.
• Lower reactant concentrations generally lead to fewer successful collisions.
• Different reaction rates can be compared using gradients at the same time or over equivalent intervals.
• Graph scales and units must be checked before comparing curves.
• Graph height tells us amount; graph gradient tells us rate.
• Strong graph analysis uses Observe → Quantify → Explain.