Rate-Concentration Graphs

Learning Objectives (OCR Spec 5.1.1)

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

• Sketch rate-concentration graphs for zero, first and second order reactions
• Use a rate-[A] plot to deduce the order of reaction
• Determine the rate constant k from the gradient of a first-order rate-[A] plot
• Recognise the advantage of rate-concentration graphs over concentration-time graphs

What Is a Rate-Concentration Graph?

Whereas a concentration-time graph shows how [A] falls during the reaction, a rate-concentration graph (rate-[A]) plots instantaneous rate on the y-axis against [A] on the x-axis. The shape of this plot depends directly on the order:

• Zero order: horizontal straight line
• First order: straight line through the origin
• Second order: upward curve (parabola)

Rate-[A] graphs are usually produced by:

1. Measuring rate at several times using tangents on a concentration-time graph.
2. Plotting those rates against the corresponding [A] values.

Zero Order — Horizontal Line

For zero order, rate = k — rate does not depend on [A]. The plot is a horizontal straight line at height k.

Sketch description: Flat horizontal line starting at (0, k) extending rightwards.

• The y-intercept gives k directly.
• The gradient is zero, confirming that [A] does not appear in the rate equation.

First Order — Straight Line Through Origin

For first order, rate = k[A]. The plot is a straight line passing through the origin with gradient k.

Sketch description: Straight line from (0,0) with positive gradient.

• The line passes through the origin — when [A] = 0, rate = 0.
• The gradient equals k, giving a direct route to the rate constant.
• This linear plot is the clearest evidence of first-order kinetics (alongside constant half-life on a [A]-t graph).

Worked Example — Finding k from a First-Order Plot

A rate-[A] plot is straight, passing through the origin. At [A] = 0.50 mol dm^-3 the rate is 0.020 mol dm^-3 s^-1. What is k?

Gradient k = rate / [A] = 0.020 / 0.50 = 0.040 s^-1

Units check: (mol dm^-3 s^-1) / (mol dm^-3) = s^-1 — correct for first order.

Second Order — Upward Curve

For second order, rate = k[A]^2. The plot is a parabolic curve passing through the origin, starting shallow and becoming steeper as [A] increases.

Sketch description: Curved line starting at origin, increasingly steep.

• Does not give a straight line.
• To extract k, you can plot rate against [A]^2 instead — this gives a straight line through the origin with gradient k.

Getting k for Second Order

Plot rate (y) against [A]^2 (x). Gradient = k.

Example: A rate-[A]^2 plot has gradient 5.0 mol^-1 dm^3 s^-1. Then k = 5.0 mol^-1 dm^3 s^-1.

Summary Table

Order	rate-[A] plot	Gradient	y-intercept
0	Horizontal line	0	k
1	Straight line through origin	k	0
2	Upward curve (parabola)	varies	0

Why Bother with Two Types of Graph?

You might wonder: if concentration-time graphs show order from their shape, why also use rate-concentration graphs? Two reasons: