Rate Equations

This lesson covers rate equations, orders of reaction, determining orders from experimental data, the rate constant k, half-life, and the rate-determining step.

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The Rate Equation

For a reaction A + B → products, the rate equation is:

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

where:

• k is the rate constant
• [A] and [B] are concentrations of reactants (mol dm⁻³)
• m and n are the orders with respect to A and B
• The overall order = m + n

Key Definition: The order of reaction with respect to a reactant is the power to which its concentration is raised in the rate equation. Orders can only be determined experimentally — they cannot be deduced from the balanced equation.

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Orders of Reaction

Order	Effect of doubling [A] on rate	Concentration–time graph	Rate–concentration graph
Zero (m = 0)	No effect	Straight line (linear decrease)	Horizontal line
First (m = 1)	Rate doubles	Exponential decay curve	Straight line through origin
Second (m = 2)	Rate quadruples	Steeper curve than first order	Parabola through origin

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Determining Orders: The Initial Rates Method

Use experimental data where one concentration is changed at a time while others are held constant.

Example 1: Given the following data for 2A + B → C:

Experiment	[A] / mol dm⁻³	[B] / mol dm⁻³	Initial rate / mol dm⁻³ s⁻¹
1	0.10	0.10	2.0 × 10⁻⁴
2	0.20	0.10	8.0 × 10⁻⁴
3	0.10	0.20	4.0 × 10⁻⁴

Finding order with respect to A: Compare experiments 1 and 2. [A] doubles (×2), [B] constant. Rate quadruples (×4). Since 2² = 4, the order with respect to A is 2.

Finding order with respect to B: Compare experiments 1 and 3. [B] doubles (×2), [A] constant. Rate doubles (×2). Since 2¹ = 2, the order with respect to B is 1.

Rate equation: rate = k[A]²[B]

Overall order: 2 + 1 = 3

Finding k: Using experiment 1: k = rate / ([A]²[B]) = 2.0 × 10⁻⁴ / (0.10² × 0.10) = 2.0 × 10⁻⁴ / 1.0 × 10⁻³ = 0.20 dm⁶ mol⁻² s⁻¹

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