Concentration Changes Over Time

SECTION 1

Historical Context & Motivation

The question of how quickly chemical substances are consumed or formed has occupied chemists since the dawn of quantitative science. Early alchemists noticed that some reactions proceeded sluggishly while others erupted almost instantaneously, yet they lacked the mathematical framework to describe these differences rigorously. The development of chemical kinetics as a formal discipline required not only precise analytical techniques for measuring concentrations but also the calculus-based tools to model their continuous evolution over time. Today, understanding how concentration varies with time is indispensable in fields ranging from pharmaceutical drug design to atmospheric chemistry and industrial catalysis.

1850

Wilhelmy's Sucrose Hydrolysis

Ludwig Wilhelmy performed one of the first quantitative kinetics experiments, showing that the rate of acid-catalyzed sucrose inversion was proportional to the remaining sucrose concentration—establishing the concept of a first-order reaction.

1864

Guldberg & Waage's Law of Mass Action

Cato Guldberg and Peter Waage formulated the law of mass action, proposing that reaction rate depends on the concentrations of reactants raised to certain powers, laying the groundwork for differential rate laws.

1889

Arrhenius Equation

Svante Arrhenius linked the temperature dependence of rate constants to an exponential activation energy term, connecting the rate constant k to thermodynamic quantities and enabling predictions of concentration profiles at different temperatures.

1913

Michaelis–Menten Kinetics

Leonor Michaelis and Maud Menten applied integrated rate law analysis to enzyme-catalyzed reactions, demonstrating that concentration-versus-time data could reveal saturation behavior and the mechanism of biological catalysts.

1950s–Present

Modern Spectroscopic & Computational Methods

Advances in stopped-flow spectrophotometry, NMR relaxation, and computational modeling now allow chemists to track concentration changes on femtosecond timescales, providing direct experimental tests of integrated rate law predictions.

The central question that this lesson addresses is deceptively simple: if you know a reaction's rate law and initial conditions, can you predict the concentration of any species at an arbitrary future time? The answer lies in the integrated rate laws—the mathematical expressions obtained by integrating the differential rate equation. These integrated forms are the bridge between instantaneous rates and the macroscopic concentration data you actually collect in the laboratory.

SECTION 2

Core Principles & Definitions

Before diving into the mathematics, it is essential to solidify the conceptual foundations that underpin concentration-versus-time analysis. A differential rate law expresses the instantaneous rate as a function of concentration (e.g., rate = k[A]ⁿ), whereas the integrated rate law relates concentration directly to elapsed time. Both contain the same information, but the integrated form is far more practical when you want to calculate how much reactant remains after a given duration or determine the reaction order from experimental [A]-versus-t data.

Reaction Order

The exponent n in rate = k[A]ⁿ dictates the functional form of the integrated rate law and determines whether concentration decays linearly, exponentially, or hyperbolically with time.

Rate Constant (k)

The proportionality constant k encodes all temperature, catalyst, and medium effects. Its units change with reaction order: s⁻¹ for first order, M⁻¹·s⁻¹ for second order, ensuring dimensional consistency.

Half-Life (t₁/₂)

The time required for the reactant concentration to fall to half its initial value. For first-order reactions t₁/₂ is constant; for second-order reactions it depends inversely on [A]₀, a key diagnostic.

Linear Plots & Order Determination

Plotting [A] vs. t, ln[A] vs. t, or 1/[A] vs. t and finding which yields a straight line is the classic graphical method for identifying zeroth-, first-, or second-order kinetics, respectively.

✦ KEY TAKEAWAY

Think of the integrated rate law as a GPS route planner for a reaction. The differential rate law tells you your speed at every instant (like a speedometer), but the integrated rate law tells you exactly where you will be at a specific future time—how much reactant is left and how much product has formed. Different reaction orders correspond to different driving styles: constant speed (zeroth order), exponential coasting (first order), or progressively slowing deceleration (second order).

SECTION 3

Concentration vs. Time Profiles

The most revealing way to compare the three common reaction orders is to plot concentration versus time on the same set of axes. The diagram below shows how a hypothetical reactant A disappears under zeroth-order, first-order, and second-order kinetics, all starting from the same initial concentration [A]₀ and using rate constants chosen so that the curves cross near the half-life region for comparison purposes.

All three curves start at [A]₀ = 1.0 M. The zeroth-order curve (amber) is a straight line that reaches zero in finite time. The first-order curve (cyan) decays exponentially and never truly reaches zero. The second-order curve (violet) decays even more gradually at long times because the rate slows dramatically as concentration decreases.

Notice the qualitatively different behavior at long times. The zeroth-order curve reaches [A] = 0 at a definite time t = [A]₀/k, after which the model is no longer physically meaningful. The first-order curve asymptotically approaches zero but never arrives—each successive half-life removes the same fraction of remaining reactant. The second-order curve approaches zero even more slowly because the rate is proportional to [A]², so as [A] diminishes, the reaction decelerates quadratically. These differences are not merely academic: they determine how long you must wait for a reaction to reach a desired conversion, which has profound implications in reactor design and pharmacokinetics.

SECTION 4

Integrated Rate Laws — Mathematical Framework

Each integrated rate law is derived by separating variables in the differential rate expression −d[A]/dt = k[A]ⁿ and integrating from t = 0 (where [A] = [A]₀) to an arbitrary time t. The resulting expressions take distinctive algebraic forms, and recognizing these forms is the key to identifying reaction order from experimental data.

Zeroth-Order (n = 0)

ZEROTH-ORDER INTEGRATED RATE LAW

[A] = [A]₀ − kt

[A] = concentration at time t; [A]₀ = initial concentration; k = rate constant (M·s⁻¹); t = elapsed time. A plot of [A] vs. t is linear with slope −k and y-intercept [A]₀.

ZEROTH-ORDER HALF-LIFE

t₁/₂ = [A]₀ / (2k)

The half-life depends on initial concentration: higher [A]₀ means a longer half-life.

First-Order (n = 1)

FIRST-ORDER INTEGRATED RATE LAW

ln[A] = ln[A]₀ − kt ⟹ [A] = [A]₀ e⁻ᵏᵗ

k has units of s⁻¹. A plot of ln[A] vs. t is linear with slope −k. The exponential form emphasizes that concentration decays by a fixed fraction per unit time.

FIRST-ORDER HALF-LIFE

t₁/₂ = ln 2 / k ≈ 0.693 / k

The half-life is independent of [A]₀. This is a unique diagnostic feature of first-order reactions and is the basis for radioactive decay dating.

Second-Order (n = 2)

SECOND-ORDER INTEGRATED RATE LAW

1/[A] = 1/[A]₀ + kt

k has units of M⁻¹·s⁻¹. A plot of 1/[A] vs. t is linear with slope k and y-intercept 1/[A]₀.

SECOND-ORDER HALF-LIFE

t₁/₂ = 1 / (k[A]₀)

Each successive half-life is twice as long as the previous one because [A]₀ in the formula is the concentration at the start of that interval—it keeps halving.

📐 Derivation Sketch — First Order

Starting from −d[A]/dt = k[A], separate variables: d[A]/[A] = −k dt. Integrate both sides: ∫ from [A]₀ to [A] d[A]/[A] = −k ∫ from 0 to t dt, yielding ln([A]/[A]₀) = −kt. Exponentiate to obtain [A] = [A]₀ e⁻ᵏᵗ. The same separation-of-variables technique applies to zeroth and second order with the appropriate power of [A] in the denominator.

SECTION 5

Graphical Determination of Reaction Order

The power of integrated rate laws becomes most apparent when you have experimental concentration-versus-time data and need to determine the reaction order. The strategy is elegantly simple: plot the data in three different linearized forms and see which one gives a straight line. The one that produces the best linear fit reveals the order, and the slope and intercept yield the rate constant k and initial concentration [A]₀.

Each column shows the linearized plot that is diagnostic for that particular order. In practice, you prepare all three plots from the same dataset and check which yields the highest R² value (closest to a perfect straight line). The summary table beneath the plots collects the slope, intercept, and half-life formula for quick reference.

When your data produces a straight line on the [A] vs. t plot, the reaction is zeroth order; if ln[A] vs. t is linear, it is first order; and if 1/[A] vs. t is linear, it is second order. This technique is sometimes called the method of integrated rate laws or the graphical method and remains one of the most commonly used approaches in undergraduate and research laboratories alike. Note that this method works best when the reaction has a single dominant reactant or when pseudo-order conditions have been established (e.g., large excess of one reagent).

💡 Pseudo-Order Conditions

For a bimolecular reaction A + B → products with rate = k[A][B], if [B] >> [A], then [B] remains approximately constant and the rate simplifies to rate ≈ k'[A], where k' = k[B]₀. This is called a pseudo-first-order condition, and the first-order integrated rate law can be applied directly using k' in place of k.

SECTION 6

Worked Example — Determining Order and Predicting Concentration

The decomposition of nitrogen dioxide, 2 NO₂(g) → 2 NO(g) + O₂(g), is studied at 300 °C. The following concentration data are collected:

Experimental data for NO₂ decomposition at 300 °C

Time (s)	[NO₂] (M)	ln[NO₂]	1/[NO₂] (M⁻¹)
0	0.01000	−4.605	100.0
50	0.00787	−4.845	127.1
100	0.00649	−5.038	154.1
200	0.00481	−5.337	207.9
300	0.00380	−5.573	263.2

Determine the Reaction Order, k, and [NO₂] at t = 500 s

Step 1 — Construct Diagnostic PlotsPlot [NO₂] vs. t, ln[NO₂] vs. t, and 1/[NO₂] vs. t. Inspecting the third and fourth columns: the ln[NO₂] values do not change by a constant amount per equal time interval (the differences are −0.240, −0.193, −0.299, −0.236), but the 1/[NO₂] values change by approximately 27.0 ± 0.5 per 50 s increment, indicating a linear relationship.

1/[NO₂] vs. t is linear → second-order reaction

Step 2 — Extract the Rate Constant kFor a second-order reaction, the slope of 1/[A] vs. t equals k. Using the first and last data points: k = (263.2 − 100.0) M⁻¹ / (300 − 0) s = 163.2 / 300 = 0.544 M⁻¹·s⁻¹. A least-squares fit to all five points would give a more precise value, but this estimate is sufficient for our purposes.

k ≈ 0.544 M⁻¹·s⁻¹

Step 3 — Predict [NO₂] at t = 500 sApply the second-order integrated rate law: 1/[A] = 1/[A]₀ + kt. Substituting: 1/[NO₂] = 100.0 + (0.544)(500) = 100.0 + 272.0 = 372.0 M⁻¹. Therefore [NO₂] = 1/372.0 = 2.69 × 10⁻³ M.

[NO₂] at 500 s ≈ 2.69 × 10⁻³ M

Step 4 — Calculate the Half-LifeFor second order: t₁/₂ = 1/(k[A]₀) = 1/(0.544 × 0.01000) = 1/0.00544 ≈ 184 s. Notice this half-life applies only to the first half-life interval; the next half-life will be twice as long (≈ 368 s) because [A] has halved.

t₁/₂ ≈ 184 s (for first interval)

SECTION 7

Comparing Rate Law Forms — Strengths & Limitations

Each integrated rate law has domains where it excels and situations where its assumptions break down. The table below summarizes the practical strengths and limitations of zeroth-, first-, and second-order integrated rate laws, helping you choose the appropriate model for a given experimental scenario.

Comparison of integrated rate law properties for common reaction orders

Feature	Zeroth Order	First Order	Second Order
Physical examples	Enzyme-catalyzed reactions at saturation; surface-catalyzed decomposition	Radioactive decay; unimolecular gas-phase decomposition	Bimolecular gas-phase reactions; dimerization
Half-life behavior	Decreases as [A] falls; each successive t₁/₂ is shorter	Constant regardless of concentration	Increases as [A] falls; each successive t₁/₂ doubles
Strengths	Simple arithmetic; physically intuitive linear decay	Constant half-life simplifies predictions; widely applicable	Captures bimolecular kinetics directly
Limitations	Predicts negative concentration; model breaks at [A] = 0	Assumes first-order dependence; fails for multi-reactant systems without pseudo-order	Limited to equal-concentration or pseudo-order cases for simple form
Units of k	M·s⁻¹	s⁻¹	M⁻¹·s⁻¹

✦ KEY TAKEAWAY

Choosing between rate law models is analogous to selecting the right regression model in data science. You do not assume which model is correct a priori; instead, you let the data speak by testing each candidate against the experimental curve. The model that produces the best linear fit—highest R², most randomly distributed residuals—wins. This empirical, model-selection approach is a cornerstone of scientific reasoning that extends far beyond kinetics into spectroscopy, thermodynamics, and even machine learning.

SECTION 8

Connection to Advanced Kinetic Theory

The simple integrated rate laws for isolated reactions with a single reactant are the entry point to a much richer landscape. In more advanced treatments, concentration-versus-time analysis extends to complex mechanisms involving consecutive reactions (A → B → C), parallel reactions (A → B and A → C simultaneously), and reversible reactions where forward and reverse steps compete. Each of these scenarios requires solving coupled differential equations, often necessitating numerical methods or matrix exponentials rather than simple closed-form solutions.

Simple vs. advanced concentration-time analysis

Topic	This Lesson (Simple Rate Laws)	Advanced Treatment
Number of steps	Single elementary step or pseudo-order simplification	Multi-step mechanisms; steady-state and pre-equilibrium approximations
Mathematical tools	Separation of variables; direct integration	Systems of ODEs; Laplace transforms; numerical integration (Runge–Kutta)
Equilibrium	Not addressed; irreversible reaction assumed	Reversible kinetics; K_eq = k_f/k_r emerges naturally from forward and reverse integrated laws
Temperature dependence	k is treated as constant at fixed T	Arrhenius and Eyring equations predict k(T); transition-state theory connects to molecular properties
Typical application	Simple decompositions; radioactive decay; introductory lab experiments	Enzyme catalysis; atmospheric ozone chemistry; polymerization kinetics; pharmacokinetic compartment models

As you advance in physical chemistry or biochemistry, you will encounter the steady-state approximation, in which the concentration of a reactive intermediate is assumed to change negligibly over time, yielding simplified rate expressions for complex mechanisms. The Lindemann–Hinshelwood mechanism for unimolecular reactions and the Michaelis–Menten model for enzyme kinetics are both elegant applications of concentration-versus-time reasoning applied to multi-step mechanisms. Mastering the simple integrated rate laws in this lesson provides the essential algebraic and conceptual toolkit for tackling these more complex systems.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

A reaction's half-life is observed to decrease as the initial concentration of reactant increases. What does this tell you about the reaction order? Explain your reasoning by referencing the half-life expressions for zeroth-, first-, and second-order reactions.

PROBLEM 2 — BASIC CALCULATION

A first-order reaction has a rate constant k = 3.50 × 10⁻² s⁻¹. If the initial concentration of reactant A is 0.250 M, what is [A] after 45.0 s? Also calculate the half-life.

PROBLEM 3 — INTERMEDIATE

The following data were collected for the decomposition of a reactant X at constant temperature: t = 0 s, [X] = 0.500 M; t = 100 s, [X] = 0.250 M; t = 200 s, [X] = 0.167 M; t = 300 s, [X] = 0.125 M. Determine the reaction order with respect to X and calculate the rate constant k.

PROBLEM 4 — APPLIED

A pharmaceutical company needs a drug to maintain at least 75% of its initial concentration for 24 hours at body temperature (37 °C). Lab tests show that the drug degrades via first-order kinetics with k = 1.20 × 10⁻⁶ s⁻¹ at 37 °C. Does the drug meet the stability requirement? What fraction remains after 48 hours?

PROBLEM 5 — CRITICAL THINKING

Consider a reaction A → Products that appears to follow second-order kinetics when studied under normal conditions. A researcher suspects that the true mechanism involves two elementary steps: A + A → A₂ (slow, rate-determining) followed by A₂ → Products (fast). However, another colleague proposes that the reaction might actually be first order but appears second order because the experiment inadvertently uses conditions where [A] changes the ionic strength of the solution, thereby changing k during the run. Design an experiment that could distinguish between these two hypotheses, and explain what concentration-versus-time data would look like under each scenario.

SUMMARY

Lesson Summary

This lesson developed the mathematical and conceptual tools for predicting how reactant concentrations evolve over time by integrating differential rate laws. For a zeroth-order reaction, concentration decreases linearly: [A] = [A]₀ − kt. For a first-order reaction, concentration decays exponentially: [A] = [A]₀ e⁻ᵏᵗ, with a characteristic constant half-life of t₁/₂ = 0.693/k. For a second-order reaction, the reciprocal of concentration increases linearly: 1/[A] = 1/[A]₀ + kt, and successive half-lives double.

The graphical method for determining reaction order involves plotting [A] vs. t, ln[A] vs. t, and 1/[A] vs. t, then identifying which yields a straight line. The slope of the linear plot gives the rate constant k (with a sign that depends on the order), and the intercept confirms [A]₀. These integrated rate laws provide the essential foundation for advanced topics including consecutive reactions, reversible kinetics, and enzyme catalysis, all of which build upon the same principle of integrating rate equations to connect concentration with time.