5.3 Concentration Changes Over Time

Integrated rate laws

Each rate law has an integrated form that shows concentration vs. time. Zero-order: [A] = -kt + [A]₀ (linear plot of [A] vs. t). First-order: ln[A] = -kt + ln[A]₀ (linear plot of ln[A] vs. t). Second-order: 1/[A] = kt + 1/[A]₀ (linear plot of 1/[A] vs. t). Graphing data in the right form reveals the order as a straight line.

Using graphs to find order

To determine reaction order from data, plot [A], ln[A], and 1/[A] each against time. Whichever plot is linear reveals the true order — its slope equals ±k (rate constant). This graphical method is a key way experimental data connects to rate law identification.

Half-life

Half-life (t½) is the time for concentration to drop to half its value. For first-order reactions, t½ = 0.693/k and is constant, independent of concentration — a hallmark of first-order kinetics. Zero- and second-order half-lives depend on initial concentration, so they change over the course of the reaction.

Key terms

Zero-order integrated rate law

[A] = -kt + [A]₀; linear plot of [A] vs. time, slope = -k.

First-order integrated rate law

ln[A] = -kt + ln[A]₀; linear plot of ln[A] vs. time, slope = -k.

Second-order integrated rate law

1/[A] = kt + 1/[A]₀; linear plot of 1/[A] vs. time, slope = k.

First-order half-life

t½ = 0.693/k; constant, independent of concentration.

How to determine reaction order graphically

Plot [A], ln[A], and 1/[A] each vs. time — whichever graph is linear identifies the order (zero, first, or second, respectively).

Try an AP-style question

For a first-order reaction, which plot is linear?

1. ln[A] vs. time
2. [A]² vs. time
3. 1/[A] vs. time
4. [A] vs. time