Rate Constant k

Learning Objectives (OCR Spec 5.1.1)

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

• Understand that the rate constant k is a proportionality constant in a rate equation
• Determine the units of k from the rate equation
• Calculate k from experimental data
• Explain qualitatively how k varies with temperature and with a catalyst

What Is k?

The rate constant k appears in any rate equation:

rate = k[A]^m[B]^n

It is the proportionality constant linking rate to the concentration terms. At a given temperature, for a given reaction with a given catalyst (or none), k has a single, fixed value. k does not depend on the concentrations of any reactants.

Key properties of k:

• Constant at constant temperature.
• Different for different reactions.
• Increases with temperature (see Arrhenius, Lesson 9).
• Increased by a catalyst (because a catalyst lowers Ea).
• Has units that depend on the overall order of reaction.

Units of k

This is where students most often lose marks. The units are worked out from the rate equation by inserting the units of each quantity:

rate [mol dm^-3 s^-1] = k [units?] x concentration [mol dm^-3] raised to the overall order

Rearrange:

k = rate / ([concentrations]^overall order)

Order 0

rate = k

Units of k = units of rate = mol dm^-3 s^-1

Order 1

rate = k[A]

k = rate / [A] = (mol dm^-3 s^-1) / (mol dm^-3) = s^-1

Order 2

rate = k[A]^2 (or k[A][B])

k = rate / [A]^2 = (mol dm^-3 s^-1) / (mol dm^-3)^2 = mol^-1 dm^3 s^-1

Order 3

rate = k[A]^2[B] (or k[A][B]^2, or k[A][B][C])

k = rate / [concentration]^3 = (mol dm^-3 s^-1) / (mol dm^-3)^3 = mol^-2 dm^6 s^-1

Table Summary

Overall order	Units of k
0	mol dm^-3 s^-1
1	s^-1
2	mol^-1 dm^3 s^-1
3	mol^-2 dm^6 s^-1

You can remember the pattern: each extra order divides by another mol dm^-3, so each extra order subtracts one power of mol dm^-3 from the units.

Worked Example 1 — Calculating k with Units

rate = k[A][B]^2. Experimental data: [A] = 0.25 mol dm^-3, [B] = 0.10 mol dm^-3, rate = 3.0 x 10^-4 mol dm^-3 s^-1.

Overall order = 1 + 2 = 3.

k = rate / ([A][B]^2)
  = (3.0 x 10^-4) / (0.25 x 0.010)
  = (3.0 x 10^-4) / (2.5 x 10^-3)
  = 0.12 mol^-2 dm^6 s^-1

Worked Example 2 — First Order

rate = k[A]. When [A] = 0.40, rate = 8.0 x 10^-3 mol dm^-3 s^-1.

k = 8.0 x 10^-3 / 0.40 = 2.0 x 10^-2 s^-1.

Worked Example 3 — Zero Order

rate = k. Rate measured = 5.0 x 10^-4 mol dm^-3 s^-1.