2.2 Rate laws and reaction orders

Rate Laws and Reaction Orders

Concept of a Rate Law

A rate law is a mathematical expression that links the rate of a reaction to the concentrations of its reactants. You can't derive a rate law from a balanced equation alone; it has to be determined experimentally by measuring how the rate changes when you adjust concentrations.

The general form looks like this:

$\text{Rate} = k[\text{A}]^m[\text{B}]^n$

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

Why does this matter? Rate laws let you:

• Predict rate changes: if you know the order, you know exactly what happens when you change a concentration (e.g., doubling a first-order reactant doubles the rate)
• Calculate the rate constant from experimental data
• Gain clues about the reaction mechanism, since the rate law reflects which species are involved in the rate-determining step

Order of Chemical Reactions

The reaction order with respect to a given reactant is the exponent on its concentration in the rate law. Orders can be zero, first, second, or even fractional. The overall order is the sum of the individual orders.

For example, if $\text{Rate} = k[\text{A}]^2[\text{B}]$, the reaction is second order in A, first order in B, and third order overall.

Determining Reaction Order: Method of Initial Rates

This is the most common approach you'll use in problems. The idea is to compare experiments where only one reactant's concentration changes at a time.

1. Set up (or receive) data from multiple experiments with different initial concentrations.
2. Pick two experiments where only one reactant concentration changes.
3. Calculate the ratio of the rates and the ratio of the concentrations for that reactant.
4. Find the order using the relationship:

$\frac{\text{Rate}_2}{\text{Rate}_1} = \left(\frac{[\text{A}]_2}{[\text{A}]_1}\right)^m$

1. Solve for $m$. If doubling the concentration doubles the rate, $m = 1$. If doubling the concentration quadruples the rate, $m = 2$. If doubling the concentration has no effect, $m = 0$.
2. Repeat for each reactant.

Determining Reaction Order: Graphical Method

Plot $\log(\text{rate})$ vs. $\log[\text{A}]$ for each reactant. The slope of the resulting line gives the order with respect to that reactant. A slope of 1 means first order; a slope of 2 means second order; a slope of 0 means zero order.

Rate Constant Determination

The rate constant $k$ is the proportionality constant in the rate law. A few things to know about it:

• It does not depend on reactant concentrations.
• It does depend on temperature (higher temperature generally means a larger $k$).
• Its units change depending on the overall reaction order.

To find $k$, plug known values of rate and concentration into the rate law and solve. For $\text{Rate} = k[\text{A}]^m[\text{B}]^n$, the units of $k$ work out to:

$k \text{ units} = \frac{\text{concentration}^{1-(m+n)}}{\text{time}}$

Concentration Effects on Reaction Rates

Increasing a reactant's concentration generally increases the rate, but how much it increases depends entirely on the reaction order with respect to that reactant.

1. Zero-order reactions ($\text{Rate} = k$)

   • The rate doesn't change when you change the concentration. It stays constant until the reactant runs out.
   • This often shows up in enzyme-catalyzed reactions at high substrate concentration, where the enzyme is fully saturated.

2. First-order reactions ($\text{Rate} = k[\text{A}]$)

   • Rate is directly proportional to concentration. Double the concentration, double the rate.
   • Radioactive decay is the classic example: the rate of decay depends only on how much of the isotope remains.

3. Second-order reactions ($\text{Rate} = k[\text{A}]^2$)

   • Rate is proportional to the square of the concentration. Double the concentration and the rate quadruples. Triple it and the rate goes up by a factor of nine.
   • Dimerization reactions, where two identical molecules combine, often follow second-order kinetics.