3.2 Initial rates method

Initial Rates Method

Initial rates method fundamentals

The initial rates method is an experimental technique for determining the rate law of a reaction. You run the same reaction multiple times, changing one reactant's starting concentration at a time, and measure how fast the reaction proceeds right at the beginning (before products build up or reverse reactions kick in).

The general form of a rate law is:

$Rate = k[A]^m[B]^n$

where $k$ is the rate constant, $[A]$ and $[B]$ are reactant concentrations, and $m$ and $n$ are the reaction orders with respect to each reactant.

Reaction orders tell you how sensitive the rate is to changes in concentration:

• Zero-order in a reactant: changing that reactant's concentration has no effect on the rate
• First-order: doubling the concentration doubles the rate
• Second-order: doubling the concentration quadruples the rate ($2^2 = 4$)

The overall order of the reaction is the sum of the individual orders ($m + n$).

Analysis of initial rates data

The core idea is to isolate the effect of one reactant at a time. You do this by finding two experiments where only one reactant's concentration changes while everything else (the other reactant's concentration, temperature, pressure) stays constant.

Here's the step-by-step process:

1. Organize your data into a table with columns for experiment number, initial concentrations of each reactant, and measured initial rate.
2. Pick two experiments where only one reactant's concentration differs. Call the ratio of concentrations $\frac{[A]_2}{[A]_1}$ and the ratio of rates $\frac{Rate_2}{Rate_1}$.
3. Set up the relationship: $\frac{Rate_2}{Rate_1} = \left(\frac{[A]_2}{[A]_1}\right)^m$
4. Solve for $m$. If the concentration ratio is a clean number, you can often determine $m$ by inspection. For example, if $[A]$ triples and the rate increases by a factor of 9, then $3^m = 9$, so $m = 2$.
5. Repeat for each reactant using a different pair of experiments where only that reactant changes.
6. Write the complete rate law using the determined orders.

Rate constant and order calculations

Once you know the reaction orders, you can calculate the rate constant $k$:

1. Rearrange the rate law to isolate $k$: $k = \frac{Rate}{[A]^m[B]^n}$
2. Plug in the initial rate and concentrations from any single experiment in your data table.
3. Calculate $k$. You can check your work by substituting values from a different experiment; you should get the same value of $k$ (within experimental error).

The units of $k$ depend on the overall reaction order. For a reaction with overall order $p$:

$k \text{ units} = M^{1-p} \cdot s^{-1}$

Some common cases:

• Overall first-order ($p = 1$): $k$ has units of $s^{-1}$
• Overall second-order ($p = 2$): $k$ has units of $M^{-1}s^{-1}$
• Overall third-order ($p = 3$): $k$ has units of $M^{-2}s^{-1}$

Getting the units right is a quick way to check whether your calculated orders make sense.

Mechanism determination using initial rates

The initial rates method does more than give you numbers. It connects experimental observations to the molecular-level picture of how a reaction actually proceeds.

Every proposed reaction mechanism has a rate-determining step, the slowest step that acts as a bottleneck. The rate law for the overall reaction is governed by this step. Different mechanisms predict different rate laws, so comparing the experimentally determined rate law to each prediction lets you rule mechanisms in or out.

The process works like this:

1. Determine the experimental rate law using the initial rates method.
2. For each proposed mechanism, derive the rate law implied by its rate-determining step.
3. Compare. If the experimental rate law matches the prediction from a particular mechanism, that mechanism is supported by the data.

A match doesn't prove a mechanism is correct (other mechanisms might predict the same rate law), but a mismatch does rule a mechanism out. If no proposed mechanism fits the experimental rate law, the actual pathway is likely more complex than what's been considered.