§2 The Idea, Step by Step
From "how fast?" to the rate law — building up gently
Start — the everyday picture
Light a match and it's over in a second; leave an iron nail in damp air and it takes years to rust. Both are the same kind of event — atoms rearranging into new substances — yet their speeds differ enormously. Reaction kinetics is the stopwatch of chemistry: it measures how fast a reaction runs, and what lets you speed it up or slow it down.
Build — naming the pieces
The thing that's running out is a reactant, and we track its concentration $[A]$ — how many moles are packed into each litre. The reaction rate is simply how fast that number falls. Chemists found a clean pattern: the rate is proportional to the concentration raised to some power, $\text{rate} = k[A]^n$. Here $k$ is the rate constant (a fixed number for a given reaction at a given temperature) and $n$ is the reaction order — how strongly the rate responds to concentration. If a reaction is first order ($n=1$) with $k = 0.01\ \text{s}^{-1}$ and you start at $[A] = 0.5$ mol/L, the opening rate is $0.01 \times 0.5 = 0.005$ mol/L·s. As $[A]$ drops, the rate drops with it — the reaction slows down as it eats its own fuel.
Deepen — the precise forms
Written exactly, the rate is a derivative: $-\frac{d[A]}{dt} = k[A]^n$. Integrating it gives the curve you actually see. First order collapses to exponential decay, $[A] = [A]_0\,e^{-kt}$, whose hallmark is a constant half-life $t_{1/2} = 0.693/k$ — the same fixed time to halve, over and over (exactly why radioactive dating works). Second order instead obeys $\frac{1}{[A]} = \frac{1}{[A]_0} + kt$, and its half-life stretches as the reaction proceeds. Temperature enters through the Arrhenius equation, $k = A\,e^{-E_a/RT}$: only molecules colliding with at least the activation energy $E_a$ can react, and that lucky fraction climbs steeply as $T$ rises. Each slider moves exactly one symbol — $[A]_0$, $k$, $n$, $T$ and $E_a$.
Try this in the sim above
(1) Set the order to 1 and open the ln[A] vs t graph — it's a straight line, the fingerprint of first order. (2) Switch to order 2 and watch 1/[A] vs t become the straight one instead. (3) In Arrhenius mode, drag the temperature up and watch $k$ climb the curve — a small rise in $T$ produces a big jump in rate.
§3 Equation Derivation
Integrated Rate Laws & Arrhenius Equation
\[\text{rate} = k[A]^n \qquad k = A\,e^{-E_a/RT}\]
| Symbol | Meaning | Unit |
|---|
| \([A]\) | Concentration of reactant A | mol L⁻¹ |
| \(k\) | Rate constant | varies with order |
| \(n\) | Reaction order | dimensionless |
| \(A\) | Pre-exponential (frequency) factor | same as k |
| \(E_a\) | Activation energy | J mol⁻¹ |
| \(R\) | Gas constant = 8.314 | J mol⁻¹ K⁻¹ |
| \(T\) | Absolute temperature | K |
| \(t_{1/2}\) | Half-life | s |
Step-by-Step Derivation
Step 1 — Differential rate law
For an \(n\)th-order reaction: \(-\dfrac{d[A]}{dt} = k[A]^n\)
Step 2 — Zero Order (n=0): separate and integrate
\(\displaystyle\int_{[A]_0}^{[A]}\!\!d[A] = -k\int_0^t dt\)
\(\Rightarrow [A] = [A]_0 - kt\)
Half-life: \(t_{1/2} = \dfrac{[A]_0}{2k}\) (depends on initial concentration)
Step 3 — First Order (n=1): separate variables
\(\dfrac{d[A]}{[A]} = -k\,dt \Rightarrow \ln[A] = \ln[A]_0 - kt\)
\(\Rightarrow [A] = [A]_0\,e^{-kt}\qquad t_{1/2} = \dfrac{\ln 2}{k} = \dfrac{0.693}{k}\) (constant!)
Step 4 — Second Order (n=2): integrate
\(\displaystyle\int_{[A]_0}^{[A]}\frac{d[A]}{[A]^2} = -k\int_0^t dt \Rightarrow \frac{1}{[A]} = \frac{1}{[A]_0} + kt\)
\(t_{1/2} = \dfrac{1}{k[A]_0}\) (depends on initial concentration)
Step 5 — Arrhenius Equation (from collision theory)
Boltzmann factor: fraction of molecules with \(E \geq E_a\) is \(e^{-E_a/RT}\).
Multiplied by frequency factor A: \(k = Ae^{-E_a/RT}\)
Linear form: \(\ln k = \ln A - \dfrac{E_a}{RT}\). Plot \(\ln k\) vs \(1/T\) → slope \(= -E_a/R\)
Step 6 — Two-temperature form (useful in lab)
\(\ln\dfrac{k_2}{k_1} = -\dfrac{E_a}{R}\!\left(\dfrac{1}{T_2}-\dfrac{1}{T_1}\right)\)
Simulation Variable Mapping
- A₀ slider → \([A]_0\) — initial concentration in mol/L
- k slider → rate constant \(k\); units change with order
- n slider → reaction order \(n\); changes graph linearity
- T slider → temperature \(T\) in K; updates k via Arrhenius
- Eₐ slider → activation energy; steepens Arrhenius plot
- Collision mode → shows Maxwell-Boltzmann particle velocities
Worked Example — N₂O₅ Decomposition
Given: 1st-order, \(k = 6.93 \times 10^{-3}\) s⁻¹ at 65°C, \([A]_0 = 0.500\) mol/L
\(t_{1/2} = \dfrac{0.693}{6.93\times10^{-3}} = 100\) s
After 300 s (3 half-lives): \([A] = 0.500 \times \left(\dfrac{1}{2}\right)^3 = 0.0625\) mol/L
Rate at t = 0: \(\text{rate} = k[A]_0 = 6.93\times10^{-3} \times 0.500 = 3.47\times10^{-3}\) mol/L·s ✓
Reference: Atkins & de Paula — Physical Chemistry, 11th Ed. (Oxford, 2018), Chapter 17: "Chemical Kinetics", §17A–17D
§4 Frequently Asked Questions
🔬 SimulationWhat exactly is each simulation mode showing?
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The 1st/2nd/Zero Order modes show animated reactant molecules disappearing over time at the correct rate, with a live graph plotting the relevant linear function ([A] vs t for 0th, ln[A] vs t for 1st, 1/[A] vs t for 2nd). The Collision Model shows gas-phase particles with Maxwell-Boltzmann speed distribution — only particles colored red (high energy, E ≥ Eₐ) can react. The Arrhenius mode shows how ln k vs 1/T is a straight line with slope −Eₐ/R. All calculations use real integrated rate law formulas.
Key: The graph tabs reveal which plot gives a straight line — the order that gives a straight line is the correct order for that reaction.
🌍 Real LifeWhere does reaction kinetics appear in real life?
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Kinetics controls drug metabolism in the body — most drugs follow 1st-order elimination, so doctors use the half-life concept to set dosing intervals. In industry, the Haber process for ammonia synthesis requires careful temperature control: too high speeds up kinetics but worsens thermodynamics. In food science, the Arrhenius equation predicts shelf-life at different storage temperatures. Radioactive decay (1st order) uses the same mathematics for carbon dating and nuclear medicine.
Key: Any time something changes with time (decay, growth, dosing), reaction kinetics is almost certainly the underlying math.
🧪 ConceptualWhy does 1st-order half-life not depend on initial concentration?
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For a 1st-order reaction, \(t_{1/2} = 0.693/k\). The initial concentration \([A]_0\) cancels out during the derivation because the rate is proportional to [A]: when [A] is large, the rate is large, so it falls by half just as fast as when [A] is small. This is a unique mathematical property of exponential decay — the fractional rate of change is constant. In contrast, 2nd order half-life is \(1/(k[A]_0)\): it grows as [A]₀ decreases, so successive half-lives get longer.
Key: Constant half-life is the signature of 1st-order kinetics — it's why radioactive half-lives are so useful for dating.
🧮 MathematicalHow do you determine reaction order experimentally?
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Plot [A] vs t, ln[A] vs t, and 1/[A] vs t. Whichever gives a straight line reveals the order: linear [A] vs t → 0th order; linear ln[A] vs t → 1st order; linear 1/[A] vs t → 2nd order. The slope gives k directly. For example, if ln[A] vs t has slope −0.00693 s⁻¹, then k = 0.00693 s⁻¹ and the order is 1st. This "graphical method" is the standard laboratory approach. The simulation's graph tabs let you toggle between these three plots in real time.
Key: The order is identified by which linearisation gives a straight line — try all three in the lab (and in this simulation).
💡 Non-ObviousDoes increasing temperature always speed up a reaction?
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Almost always yes for kinetics — the Arrhenius equation guarantees k increases with T. However, for reversible reactions, raising T can shift the equilibrium backwards (for exothermic reactions), reducing the yield even though the forward rate increases. Enzymes are a crucial exception: above ~37–40°C, enzymes denature and reaction rate collapses. Additionally, some reactions have negative activation energies (association reactions) where rate decreases with temperature. So "more T = faster" is a kinetic rule with important thermodynamic and biological exceptions.
Key: Temperature always increases the rate constant k via Arrhenius, but equilibrium position and enzyme stability may counteract this.
🌌 DeepWhat is the molecular basis of the Arrhenius equation?
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The Boltzmann distribution tells us the fraction of molecules with energy ≥ Eₐ is \(e^{-E_a/RT}\). The pre-exponential factor A accounts for collision frequency and the fraction of collisions with the correct orientation (steric factor p in collision theory). Transition state theory (Eyring equation) provides a more rigorous derivation: \(k = \frac{k_BT}{h}e^{-\Delta G^\ddagger/RT}\), where ΔG‡ is the Gibbs energy of activation. This links kinetics directly to thermodynamics.
Key: Arrhenius is an empirical equation; its molecular foundation comes from Boltzmann statistics and transition state theory.
🧮 MathematicalHow do you calculate Eₐ from two rate constants?
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Use the two-temperature form: \(\ln(k_2/k_1) = -(E_a/R)(1/T_2 - 1/T_1)\). Example: k₁ = 2.0×10⁻⁵ s⁻¹ at T₁ = 300 K, k₂ = 1.6×10⁻⁴ s⁻¹ at T₂ = 350 K. Then ln(8) = 2.08, and 1/300 − 1/350 = 4.76×10⁻⁴ K⁻¹. So Eₐ = −R × 2.08/(−4.76×10⁻⁴) = 8.314 × 4370 ≈ 36.3 kJ/mol. This is the standard exam calculation in HSC and university courses.
Key: Just two rate constants at two temperatures are enough to calculate the activation energy — you don't need the full Arrhenius plot.
Reference: LibreTexts Chemistry — "Reaction Kinetics" https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules/Kinetics | Khan Academy — Chemical kinetics https://www.khanacademy.org/science/ap-chemistry-beta/x2eef969c74e0d802:kinetics
§5 Common Misconceptions
❌ Misconception: "Reaction order can be determined from the balanced equation's stoichiometric coefficients."
✅ Correction: Reaction order must be determined experimentally — it is NOT derived from the balanced equation. For example, H₂ + I₂ → 2HI is second order (first order in each reactant), but many reactions with similar equations have completely different orders due to their mechanism. Stoichiometric coefficients are only equal to rate law exponents for elementary (single-step) reactions, which are rare.
📖 Reference: Atkins & de Paula — Physical Chemistry, 11th Ed., §17A.1: "The rates of reactions" — explicitly states orders must be experimentally determined
❌ Misconception: "A catalyst increases the rate by providing more energy to reactants."
✅ Correction: A catalyst lowers the activation energy Eₐ by providing an alternative reaction pathway — it does not supply energy. The Arrhenius equation k = Ae^(−Eₐ/RT) shows that reducing Eₐ increases k exponentially. The catalyst is not consumed and does not change ΔH or the equilibrium position — it only speeds up both the forward and reverse reactions equally.
📖 Reference: Zumdahl — Chemical Principles, 8th Ed., Chapter 12.7: "Catalysis" — clear distinction between energy supply and pathway alteration
❌ Misconception: "Rate constant k and reaction rate are the same thing."
✅ Correction: k is a temperature-dependent constant specific to a reaction; the rate is k multiplied by the concentration terms (rate = k[A]^n). At a given temperature, k is fixed, but the rate decreases as [A] falls during the reaction. Doubling [A] doubles the rate (1st order) but k stays the same. Only changing temperature or adding a catalyst changes k.
📖 Reference: Silberberg — Chemistry: The Molecular Nature of Matter and Change, 9th Ed., Chapter 16.3: "Expressing the Reaction Rate"
❌ Misconception: "A reaction with a large k must have a large rate."
✅ Correction: The rate depends on both k and the concentration(s). A reaction can have a large k but operate at extremely low concentration (e.g. atmospheric trace-gas chemistry), resulting in a tiny rate. Conversely, a small k with a very high concentration can give a substantial rate. This is why industrial processes use high pressures and concentrations even for reactions with modest k values.
📖 Reference: Levine — Physical Chemistry, 6th Ed., Chapter 17.1: "Reaction Kinetics" — distinction between rate constant and rate
❌ Misconception: "Increasing temperature always doubles the reaction rate."
✅ Correction: The common rule-of-thumb that rate doubles per 10°C is only a rough approximation for reactions with Eₐ ≈ 50 kJ/mol near 300 K. For high activation energies (Eₐ > 100 kJ/mol), a 10°C rise can triple or quadruple the rate. For low Eₐ reactions, the effect is much smaller. The exact relationship is given by the Arrhenius equation, not a simple doubling rule.
📖 Reference: Nakhleh — J. Chem. Educ. 69, 191 (1992): "Why some students don't learn chemistry" — temperature-rate misconceptions explicitly discussed
❌ Misconception: "Half-life always refers to the time for [A] to fall to exactly half — so second half-life is the same as first."
✅ Correction: Only 1st-order reactions have a constant half-life. For 2nd-order reactions, t₁/₂ = 1/(k[A]) — as [A] decreases after the first half-life, the next half-life is twice as long. For zero-order, t₁/₂ = [A]/(2k), so successive half-lives get shorter. The constancy of half-life is unique to 1st-order and is a diagnostic tool for determining reaction order.
📖 Reference: Atkins & de Paula — Physical Chemistry, 11th Ed., §17A.2 Table 17A.2: "Half-lives for different orders" — explicit comparison table
Section 4 reference: Taber, K.S. — Chemical Misconceptions: Prevention, Diagnosis and Cure (RSC, 2002) Vol. 2, Ch. 5 | Cakmakci, G. et al. — Students' understanding of solution concentration. J. Chem. Educ. 2005, 82, 1074