Concentration-Time Graphs

Learning Objectives (OCR Spec 5.1.1)

By the end of this lesson you should be able to:

• Sketch concentration-time graphs for zero, first and second order reactions
• Determine the rate of reaction from the gradient of a tangent on a [A]-t graph
• Recognise the distinctive shapes of each order
• Understand how concentration-time curves are generated in the lab

What Is a Concentration-Time Graph?

A concentration-time (or [A]-t) graph plots the concentration of a reactant (y-axis) against time (x-axis). The gradient at any point is the rate of reaction at that instant:

rate = -d[A]/dt

The negative sign ensures a positive rate value (concentration is falling). Different orders give characteristic shapes.

Zero Order — Straight Line

For a zero-order reactant, rate = k is constant. Since -d[A]/dt = k, integration gives:

[A] = [A]0 - kt

This is a straight line with negative gradient -k, starting at [A]0 on the y-axis and reaching zero at t = [A]0/k.

Sketch description: Straight line, constant downward slope.

graph LR
  A["[A] at time 0"] -- constant gradient --> B["[A] = 0 at t = [A]0/k"]

• The rate is the magnitude of the gradient, which equals k directly.
• Rate does not change as [A] decreases — that is the defining feature.

First Order — Exponential Decay

For a first-order reactant, rate = k[A]. Integration gives:

[A] = [A]0 e^(-kt)

This is an exponential decay: starts at [A]0, decreases quickly at first and then more slowly, approaching zero asymptotically but never quite reaching it.

Sketch description: Smooth curve, steep at first, flattening out, never touching the x-axis.

• The gradient (rate) is steepest at t = 0 where [A] is largest.
• The gradient becomes shallower as [A] falls — in direct proportion.
• Constant half-life (see Lesson 5) — [A] halves in equal time intervals.

Second Order — Steeper Curve

For a second-order reactant, rate = k[A]^2. Integration gives:

1/[A] = 1/[A]0 + kt

or equivalently [A] = [A]0 / (1 + [A]0 k t). This is also a decaying curve but with a different shape from first order: steeper decrease at the start, but a longer "tail" as [A] approaches zero (half-life increases as [A] falls).

Sketch description: Very steep at t = 0, quickly becomes shallow, with a long tail approaching the x-axis.

Comparison of Shapes

Order	[A]-t shape	Distinguishing feature
0	Straight line	Constant gradient
1	Exponential decay	Constant half-life
2	Steeper curve with long tail	Half-life doubles each time

graph TD
  subgraph "Concentration-Time Graphs"
    A[Zero order: straight line falling linearly]
    B[First order: exponential decay, constant half-life]
    C[Second order: initial steep decay, increasing half-life]
  end

Determining Rate from a Tangent

For a curved graph (first or second order), you cannot just use a gradient of the whole curve. Instead:

1. Pick the time t at which you want the rate.
2. Draw a tangent to the curve at that point.
3. Measure the gradient of the tangent: (change in [A]) / (change in t).
4. The rate at time t is the magnitude of this gradient.