S12E3 - How to Determine the Form of the Rate Law: The Method of Initial Rates

TL;DR

Write the differential rate law with unknown exponents (orders) for each reactant, e.g., rate = k[NH4+]^n[NO2−]^m.

Briefing Cornell Notes

Briefing

Determining the form of a reaction’s rate law becomes a matter of experiment rather than guesswork: the method of initial rates extracts the reaction orders (the exponents) by comparing how the measured rate changes when reactant concentrations change, then uses one experiment to solve for the rate constant k. For the reaction NH4+ + NO2− → N2 + 2H2O, the rate law is written in the differential form as rate = k[NH4+]^n[NO2−]^m, where n and m are found from concentration–rate data collected at the start of the reaction (time zero), before concentrations shift appreciably.

The core technique is to take ratios of rates from experiments that vary only one reactant at a time. Using experiments 1 and 2, the ratio rate2/rate1 is set up so that the unknown k cancels and one concentration term cancels as well, leaving a single power relationship. After substituting the tabulated concentrations and simplifying, the comparison yields m = 1. A second ratio using experiments 2 and 3 similarly isolates the other exponent, giving n = 1. With both orders in hand, the rate law reduces to rate = k[NH4+][NO2−]. The final step plugs values from any one experiment into the simplified law and solves for k; doing so produces k = 0.54 L/mol·s for this specific case. The units of k are emphasized as the hardest part because they depend on the overall reaction orders.

A second, more comprehensive example generalizes the same workflow to a reaction involving three reactants: BrO3− + Br− + H+ (with coefficients in the balanced equation that do not determine the orders). The differential rate law is written as rate = k[BrO3−]^n[Br−]^m[H+]^p. With four experiments available, the method again uses rate ratios chosen to cancel out unwanted terms. Comparing experiments 1 and 2 isolates n and gives n = 1 (first order in BrO3−). Repeating the ratio logic for Br− and H+ yields m = 1 and p = 2. The overall reaction order is then obtained by adding the individual orders: 1 + 1 + 2 = 4, so the reaction is fourth order overall.

Once n, m, and p are known, k is found exactly like before: substitute concentrations and the measured rate from any single experiment into rate = k[BrO3−]^1[Br−]^1[H+]^2. The calculation gives k = 8.0 L^3/(mol^3·s) (with units again reflecting the total order). The takeaway is procedural and practical: identify orders via initial-rate ratios that eliminate variables, then compute k using one experiment, while treating k’s units as a dynamic consequence of the exponents. Integrated rate laws are saved for later, since they introduce time dependence and different solution strategies.

Cornell Notes

Initial-rate experiments determine the exponents in a differential rate law by comparing how the measured rate changes when reactant concentrations change. For NH4+ + NO2− → N2 + 2H2O, the method uses ratios like rate2/rate1 so that k cancels and chosen concentration terms cancel, isolating m and n. The data yield m = 1 and n = 1, giving rate = k[NH4+][NO2−], and substituting one experiment gives k = 0.54 L/mol·s. A three-reactant example (BrO3−, Br−, H+) uses the same ratio strategy across four experiments to find n = 1, m = 1, p = 2, so the overall order is 4; then k is computed from one experiment (k = 8.0 L^3/(mol^3·s)). Units of k depend on the orders and must be handled carefully.

How does the method of initial rates determine the order with respect to a reactant?

It compares measured rates from experiments that start at the same initial time (time zero) and differ in reactant concentrations. By taking a ratio such as rate2/rate1, the rate constant k cancels. Then the experiments are selected so that one concentration term cancels too, leaving an equation involving only one unknown exponent. Solving that simplified power relationship gives the order for that reactant.

Why do the balanced-equation coefficients (like 5 for Br− or 6 for H+) not determine the reaction orders?

Reaction orders come from experimental rate data, not from stoichiometric coefficients. In the BrO3−/Br−/H+ example, the rate law is written with unknown exponents n, m, and p, and those exponents are determined by rate ratios. The coefficients in the balanced equation are ignored when setting up the rate law because they do not control how rate scales with concentration.

What is the practical workflow once n and m (or n, m, p) are found?

After determining the exponents, the rate law is fully specified except for k. Then k is found by substituting concentrations and the measured rate from any one experiment into the simplified rate law. The calculation is straightforward algebra, but the units of k must be derived carefully from the exponents and concentration units.

How is the overall reaction order determined in the three-reactant example?

The overall order is the sum of the individual orders in the differential rate law. For BrO3− + Br− + H+ (with rate = k[BrO3−]^n[Br−]^m[H+]^p), the example finds n = 1, m = 1, and p = 2, so the overall order is 1 + 1 + 2 = 4.

Why is k’s unit often described as the hardest part?

Because k’s units change with the total order. Once the rate law is known, k must carry whatever units are needed so that the product of k and the concentration terms matches the units of rate (concentration per time). In the NH4+/NO2− example, k ends up in L/mol·s, while in the three-reactant example with overall order 4, k becomes L^3/(mol^3·s).

Review Questions

In a differential rate law rate = k[A]^n[B]^m, what specific ratio of experiments would you choose to isolate n, and what must be true about the experiments for the cancellation to work?
For a reaction with rate = k[BrO3−]^1[Br−]^1[H+]^2, what is the overall reaction order and how would you compute k from one experiment’s data?
Why can two reactions with the same balanced-equation coefficients have different rate laws and different k units?

Key Points

1
Write the differential rate law with unknown exponents (orders) for each reactant, e.g., rate = k[NH4+]^n[NO2−]^m.
2
Use the method of initial rates: rely on measured rates taken at the start of the reaction so concentrations can be treated as effectively constant for the comparison.
3
Take ratios of rates from experiments (e.g., rate2/rate1) so k cancels, then choose experiments so one concentration term cancels and only one exponent remains.
4
Solve for each exponent by repeating the ratio strategy with different experiment pairs until all needed orders are determined.
5
After substituting the found exponents back into the rate law, compute k using any single experiment’s rate and concentrations.
6
Derive k’s units from the final rate law; k’s units depend on the orders and cannot be memorized universally.
7
Compute overall reaction order by summing the individual orders in the differential rate law.

Highlights

Orders come from experimental rate ratios at time zero, not from stoichiometric coefficients.

For NH4+ + NO2− → N2 + 2H2O, the initial-rate data yield n = 1 and m = 1, giving rate = k[NH4+][NO2−].

In the BrO3−/Br−/H+ example, the method finds n = 1, m = 1, p = 2, so the overall order is 4.

k is computed after orders are known, and its units shift with the total order (e.g., L/mol·s vs. L^3/(mol^3·s)).

Topics

Initial Rates
Differential Rate Laws
Reaction Orders
Rate Constant Units
Rate Ratio Method