3.3 Graphical methods for rate law analysis

Graphical Methods for Rate Law Analysis

Graphical methods let you determine the order of a reaction and extract the rate constant $k$ directly from experimental data. The core idea is straightforward: if you plot concentration-vs.-time data in the right way, the correct reaction order will produce a straight line. The order that gives you linearity is the order of the reaction.

This section covers the integrated rate laws for zero-, first-, and second-order reactions, how to linearize each one, and how half-life behavior can also reveal reaction order.

Integrated Rate Laws and Linearization

Each reaction order has its own integrated rate law. When you rearrange these into the form $y = mx + b$, each one suggests a specific plot that will be linear only if the data actually follow that order.

Zero-order reactions:

• Integrated rate law: $[A]_t = -kt + [A]_0$
• Plot $[A]$ vs. $t$
• Slope = $-k$, y-intercept = $[A]_0$

This is already in $y = mx + b$ form with no transformation needed. If raw concentration vs. time is a straight line, the reaction is zero-order.

First-order reactions:

• Integrated rate law: $\ln[A]_t = -kt + \ln[A]_0$
• Plot $\ln[A]$ vs. $t$
• Slope = $-k$, y-intercept = $\ln[A]_0$

You take the natural log of each concentration value before plotting. A straight line on this plot means first-order.

Second-order reactions:

• Integrated rate law: $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$
• Plot $\frac{1}{[A]}$ vs. $t$
• Slope = $k$ (positive), y-intercept = $\frac{1}{[A]_0}$

You take the reciprocal of each concentration value. A straight line here means second-order.

How to Determine Reaction Order from Plots

When you're given concentration-vs.-time data and need to find the reaction order, follow this process:

1. Make three plots from the same data set: $[A]$ vs. $t$, $\ln[A]$ vs. $t$, and $\frac{1}{[A]}$ vs. $t$.
2. Check which plot produces a straight line (or the best linear fit if working with real, slightly noisy data).
3. The linear plot tells you the order: linear $[A]$ vs. $t$ = zero-order, linear $\ln[A]$ vs. $t$ = first-order, linear $\frac{1}{[A]}$ vs. $t$ = second-order.
4. Read $k$ from the slope of the linear plot. Remember that for zero- and first-order, the slope is $-k$ (so $k$ is the absolute value), while for second-order, the slope equals $k$ directly.

Only one of the three plots should be linear. If $\ln[A]$ vs. $t$ is straight but the other two curve, you've confirmed first-order behavior.

Half-Life and Reaction Order

Half-life ($t_{1/2}$) is the time it takes for the reactant concentration to drop to half its current value. The relationship between half-life and initial concentration is different for each order, which gives you another way to identify reaction order.

First-order:

$t_{1/2} = \frac{\ln 2}{k}$

Half-life is constant regardless of concentration. If you notice that it always takes the same amount of time for the concentration to halve (say, 20 s to go from 1.0 M to 0.5 M, and another 20 s to go from 0.5 M to 0.25 M), that's a signature of first-order kinetics.

Second-order:

$t_{1/2} = \frac{1}{k[A]_0}$

Half-life is inversely proportional to the initial concentration. As the reaction proceeds and $[A]$ decreases, each successive half-life gets longer. A reaction that starts fast and slows down dramatically may be second-order.

Zero-order:

$t_{1/2} = \frac{[A]_0}{2k}$

Half-life is directly proportional to the initial concentration. As $[A]$ decreases, each successive half-life gets shorter. The reaction runs at a steady rate until the reactant is nearly gone, then finishes quickly.