5.3 Concentration Changes Over Time

Concentration changes over time help you determine a reaction's order by testing which concentration-time graph is linear. Plot $[A]$ vs. time for zeroth order, $\ln[A]$ vs. time for first order, or $\frac{1}{[A]}$ vs. time for second order. For AP Chemistry, use the linear graph to identify the rate law and rate constant.

AP Chem 5.3: Concentration Changes Over Time

In AP Chem 5.3, concentration-time data is used to identify the reaction order and calculate the rate constant. The exam usually gives you a table or graph, then asks which plot is linear: [A] vs time for zeroth order, ln[A] vs time for first order, or 1/[A] vs time for second order.

Once you know the linear plot, use its slope to find k. For zeroth- and first-order plots, slope = -k; for a second-order plot, slope = k. First-order reactions also have a constant half-life, so t1/2 = 0.693/k only applies after you know the reaction is first order.

Why This Matters for the AP Chemistry Exam

This topic is where kinetics turns into data analysis. You take a table or graph of how a reactant's concentration changes and use it to identify the reaction order and find the rate constant. Both multiple-choice and free-response questions ask you to read concentration-versus-time data, decide which plot is linear, pull the slope, and connect that slope to k with the right units.

A common stumbling point is mixing up the units shown on the graph axes with the coefficients in the balanced equation. Keep those separate. The order comes from the experimental graph, not from the equation's coefficients. Half-life questions, especially first-order half-life and radioactive decay, also show up regularly.

Key Takeaways

• A straight-line plot tells you the order: [A] vs t is zeroth, ln[A] vs t is first, 1/[A] vs t is second.
• The slope of the linear plot gives k. Slope is negative for zeroth and first order, positive for second order.
• The three integrated rate laws are on the AP formula sheet, so practice applying them rather than memorizing derivations.
• For first-order reactions, half-life is constant and equals t1/2 = 0.693/k. This equation is also on the formula sheet.
• Radioactive decay is a clean example of first-order kinetics, with a half-life that never changes.
• Watch your units for k: they change with reaction order.

The Integrated Rate Laws

A regular rate law tells you how rate depends on concentration at one instant. An integrated rate law tells you how concentration itself changes as time passes. These are the three you need:

Zeroth order: [A]ₜ - [A]₀ = -kt

• Concentration drops at a steady, constant rate.
• Rate does not depend on concentration. Rate = k.

First order: ln[A]ₜ - ln[A]₀ = -kt

• This can also be written as ln([A]₀/[A]ₜ) = kt.
• Concentration drops quickly at first, then slows down (exponential decay).

Second order: 1/[A]ₜ - 1/[A]₀ = kt

• Concentration drops fast early on, then much more slowly.

All three appear on the AP formula sheet, so you do not need to derive them. Focus on choosing the right one and applying it correctly.

Why the Graphs Work

Each integrated rate law has the form of a straight line, y = mx + b, where time is x. That is why plotting the correct function of concentration against time gives a straight line. Once you have a straight line, the slope hands you the rate constant.

Understanding Zeroth-Order Reactions

Zeroth-order reactions have a rate that does not depend on reactant concentration. The rate stays constant as long as reactant is present, so Rate = k. These often happen when something else limits the reaction, such as a catalyst surface that is fully saturated or a fixed amount of light driving a photochemical step.

Key features:

• Constant reaction rate: the same amount of reactant is used up each second.
• A plot of [A] vs time is a straight line with slope -k.
• Half-life depends on the starting concentration: t₁/₂ = [A]₀/(2k).
• Units of k are M·s⁻¹.

An example is the decomposition of ammonia on a hot platinum surface at high pressure. The surface becomes saturated with NH₃, so adding more does not speed things up.

Determining Reaction Order from Graphs

You can find a reaction's order directly from concentration-versus-time data, even without knowing the rate law ahead of time. Plot different functions of concentration against time and see which one comes out straight.

• A straight line for [A] vs time means zeroth order. Slope = -k.
• A straight line for ln[A] vs time means first order. Slope = -k.
• A straight line for 1/[A] vs time means second order. Slope = k (positive).

After you identify the order from the linear plot, use that line's slope to find k. This graphical method is a standard experimental technique and shows up on free-response questions.

Worked Example

For the reaction A → B, three plots are made: [A] vs t, ln[A] vs t, and 1/[A] vs t. Only the 1/[A] vs t plot is linear.

a) What is the rate law? Represent the rate constant as k. Explain your reasoning.

The 1/[A] vs t plot is linear, so the reaction is second order in A. The rate law is Rate = k[A]². (Linear 1/[A] vs t is the signature of second order.)

b) Estimate k. Show your work.

Using points (0, 5) and (200, 25) from the 1/[A] vs t line:

k = slope = (25 - 5)/(200 - 0) = 20/200 = 0.1 M⁻¹·s⁻¹

For a second-order plot, slope equals k directly, and k is positive.

c) If the initial concentration of A is 0.200 M, what is the concentration after 30 seconds?

Use the second-order integrated rate law:

1/[A]ₜ - 1/[A]₀ = kt

1/[A]ₜ - 1/0.200 = 0.1(30)

1/[A]ₜ = 0.1(30) + 1/0.200 = 3 + 5 = 8

[A]ₜ = 0.125 M

Half-Life in First-Order Reactions

Half-life is the time it takes for a reactant's concentration to drop to half its current value. For first-order reactions, half-life is constant. It does not matter how much you start with, every half-life takes the same amount of time.

The first-order half-life is tied to the rate constant by:

t₁/₂ = 0.693/k

You can use this to find the half-life when you know k, or to find k when you know the half-life. This equation is on the AP formula sheet.

The constant half-life is unique to first order. Zeroth-order half-life depends on starting concentration (t₁/₂ = [A]₀/(2k)), and second-order half-life also changes as concentration changes.

Radioactive Decay as First-Order Kinetics

Radioactive decay is a clean illustration of first-order kinetics. The decay rate depends only on how many radioactive nuclei are present, so Rate = k[N]. Each isotope has its own half-life that stays the same regardless of how much material you have.

Real-world examples include carbon-14 (used in carbon dating), uranium-238, and iodine-131 (used in some medical treatments). These are applications of first-order kinetics, not separate AP content, but they show why the constant half-life of first-order processes is so useful.

How to Use This on the AP Chemistry Exam

Problem Solving

• Start by asking which plot is linear. That single fact gives you the order before you do any calculations.
• Once you know the order, grab the slope from the linear plot. Remember slope is -k for zeroth and first order, and +k for second order.
• Match the integrated rate law to the order, plug in known values, and solve for the unknown ([A]ₜ, [A]₀, k, or t).

Free Response

• When asked to justify the order, point to the specific plot that is linear and name it (for example, "ln[A] vs t is linear, so the reaction is first order").
• Show the slope calculation with two clear points and report k with correct units.
• For half-life problems, state whether the reaction is first order before using t₁/₂ = 0.693/k, since that constant-half-life shortcut only works for first order.

Common Trap

• Do not read reaction order from the coefficients in the balanced equation. Order comes from experimental data, here the linear graph.
• Keep graph-axis units separate from the units in the chemical equation. These are different things.

Common Misconceptions

• "Zeroth order means no reaction happens." It still reacts, just at a constant rate that does not depend on concentration.
• "The exponent in a rate law comes from the balanced equation." For an overall reaction, order is determined experimentally, often from which concentration plot is linear.
• "Half-life is always constant." Only for first-order reactions. Zeroth and second order have half-lives that change with concentration.
• "A positive slope means I made an error." For a second-order 1/[A] vs t plot, the slope is supposed to be positive and equals k.
• "ln[A] vs t and [A] vs t are interchangeable." They test different orders. Only the correct function gives a straight line for a given reaction.
• "The units of k are always the same." The units of k depend on the overall reaction order, so they change from one order to another.

Related AP Chemistry Guides

• Unit 5 Overview: Kinetics
• 5.1 Reaction Rates
• 5.2 Introduction to Rate Law
• 5.4 Elementary Reactions
• 5.5 Collision Model
• 5.6 Reaction Energy Profile