Rate laws and reaction order
A rate law shows exactly how rate depends on concentrations. It must be found by experiment, and it reveals clues about how a reaction actually happens.
The rate law
For \(a\text{A} + b\text{B} \to \text{products}\), the rate law has the form rate \(= k[\text{A}]^m[\text{B}]^n\). The exponents \(m\) and \(n\) are the orders with respect to each reactant, and \(k\) is the rate constant. The orders must be determined experimentally; they are not simply the coefficients of the balanced equation.
\text{rate} = k[\text{A}]^m[\text{B}]^nFinding orders from data
Compare experiments where only one concentration changes. If doubling [A] doubles the rate, the order in A is 1; if it quadruples the rate, the order is 2; if the rate is unchanged, the order is 0. This method of initial rates is a standard laboratory technique.
Worked example
Doubling [A] with [B] fixed multiplies the rate by 4; doubling [B] with [A] fixed doubles the rate. So the rate law is rate \(= k[\text{A}]^2[\text{B}]\), and the overall order is 3.
Integrated rate laws
For a first-order reaction, \(\ln[\text{A}] = \ln[\text{A}]_0 - kt\): concentration decays exponentially. For zero order, \([\text{A}] = [\text{A}]_0 - kt\), a straight line. For second order, \(\frac{1}{[\text{A}]} = \frac{1}{[\text{A}]_0} + kt\). Plotting data each way and finding which gives a straight line identifies the order.
Half-life
A first-order reaction has a constant half-life \(t_{1/2} = \frac{\ln 2}{k}\), independent of concentration. Radioactive decay and many drug eliminations are first order: a drug with a 6-hour half-life drops to 25% after 12 hours whatever the starting dose.
t_{1/2} = \frac{\ln 2}{k}Reaction mechanisms
Most reactions occur in a series of elementary steps, the mechanism. The slowest step, the rate-determining step, controls the overall rate, like the narrowest point of a funnel. A proposed mechanism must add up to the overall equation and be consistent with the observed rate law.
Intermediates and catalysts in mechanisms
An intermediate is formed in one step and used up in a later one; it does not appear in the overall equation. A catalyst is used in an early step and regenerated later. Detecting intermediates spectroscopically is strong evidence for a proposed mechanism.
Common mistakes
Using balanced-equation coefficients as orders without data. Forgetting that the half-life is constant only for first-order reactions. Confusing intermediates with catalysts. Mixing units of \(k\), which depend on the overall order.
Key takeaways
Rate laws \(k[\text{A}]^m[\text{B}]^n\) are found experimentally, often by initial rates. Integrated rate laws give straight-line plots that identify the order. First-order reactions have constant half-life \(\ln 2/k\). Mechanisms consist of elementary steps, and the slowest step determines the rate.
Flashcard
\text{rate} = k[A]^m[B]^n
Check your understanding
Doubling [A] quadruples the rate. The reaction is what order in A?