12.4 Integrated Rate Laws

Integrated Rate Laws

Integrated rate laws connect reactant concentration to time. While a differential rate law tells you the rate right now, an integrated rate law lets you zoom out and answer bigger questions: what will the concentration be after 10 minutes? How long until half the reactant is gone? Each reaction order (zero, first, second) has its own integrated rate law with a distinct equation, graph shape, and half-life behavior.

Purpose of Integrated Rate Laws

A differential rate law tells you how rate depends on concentration at a single moment. An integrated rate law is what you get when you integrate that expression over time, giving you a direct equation linking concentration to time.

This is useful because you can:

• Calculate the concentration of a reactant at any future time
• Figure out how long it takes to reach a specific concentration
• Determine the reaction order from experimental data (more on this below)

Without integrated rate laws, you'd need to continuously monitor the reaction (using spectroscopy, titration, etc.) instead of just plugging into an equation.

Calculations with Integrated Rate Laws

Each reaction order produces a different integrated rate law. The key is recognizing which quantity plots as a straight line against time.

Zero-order reactions:

• Integrated rate law: $[A]_t = -kt + [A]_0$
• Concentration decreases linearly with time. A plot of [A] vs. t gives a straight line with slope $-k$.
• Rate constant units: M/s (concentration per time)
• Think of it this way: the reaction chews through the same amount of reactant every second, regardless of how much is left.

First-order reactions:

• Integrated rate law: $\ln[A]_t = -kt + \ln[A]_0$
• The natural log of concentration decreases linearly with time. A plot of ln[A] vs. t gives a straight line with slope $-k$.
• Rate constant units: $s^{-1}$ (per time)
• Here, a constant fraction of the reactant is consumed each second. More reactant means a faster rate, but the percentage lost per unit time stays the same.

Second-order reactions:

• Integrated rate law: $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$
• The reciprocal of concentration increases linearly with time. A plot of 1/[A] vs. t gives a straight line with slope $+k$.
• Rate constant units: $M^{-1}s^{-1}$
• Notice the slope is positive here, unlike the other two. That's because $1/[A]$ gets larger as $[A]$ shrinks.

All three equations follow the form $y = mx + b$, which is why graphing is such a powerful tool for identifying reaction order.

Half-Life in Chemical Reactions

Half-life ($t_{1/2}$) is the time it takes for the reactant concentration to drop to half its initial value. Each reaction order has a different half-life expression:

1. Zero-order: $t_{1/2} = \frac{[A]_0}{2k}$
2. First-order: $t_{1/2} = \frac{\ln 2}{k}$
3. Second-order: $t_{1/2} = \frac{1}{k[A]_0}$

The most important thing to notice is what each half-life depends on:

• Zero-order: Half-life is proportional to initial concentration. Start with more reactant, and it takes longer to get halfway. As the reaction proceeds and $[A]$ drops, successive half-lives get shorter.
• First-order: Half-life is constant. It doesn't depend on concentration at all. This is the signature feature of first-order kinetics, and it's why radioactive decay (a first-order process) has a single, fixed half-life.
• Second-order: Half-life is inversely proportional to initial concentration. Start with more reactant, and the half-life is actually shorter. As $[A]$ drops, successive half-lives get longer.

Determination of Reaction Order

Given experimental concentration-vs.-time data, here's how to figure out the reaction order:

Graphical method (most common):

1. Make three plots from your data:

   • [A] vs. t (tests for zero-order)
   • ln[A] vs. t (tests for first-order)
   • 1/[A] vs. t (tests for second-order)

2. Whichever plot gives a straight line tells you the reaction order.

3. The slope of that straight line gives you the rate constant $k$ (negative slope for zero- and first-order, positive slope for second-order).

Half-life method (alternative):

If you have half-life data at different initial concentrations, compare them:

• Half-life stays the same regardless of $[A]_0$ → first-order
• Half-life increases when $[A]_0$ increases → zero-order
• Half-life decreases when $[A]_0$ increases → second-order

Reaction Kinetics and Rate Laws

A quick summary of how the pieces fit together:

• The differential rate law expresses rate as a function of concentration (e.g., $\text{Rate} = k[A]^n$). It describes the instantaneous rate.
• The integrated rate law is derived from the differential rate law by integrating with respect to time. It describes concentration as a function of time.
• The rate constant ($k$) appears in both forms. Its value depends on temperature and the specific reaction, but its units depend on the reaction order.
• Reaction kinetics is the broader study of how fast reactions occur and what factors (concentration, temperature, catalysts) affect that speed. Integrated rate laws are one of the central tools in that study.