Chemical Kinetics
Rate laws, integrated rate laws, half-lives, Arrhenius, mechanisms, and energy diagrams.
Rate of reaction
CORE RULEFor aA + bB → cC + dD: rate = −(1/a)d[A]/dt = −(1/b)d[B]/dt = +(1/c)d[C]/dt = +(1/d)d[D]/dt. The 1/coefficient prefactor makes the single rate value independent of which species you track.
Rate law
CORE RULErate = k[A]ᵐ[B]ⁿ. Orders m, n are found by EXPERIMENT, not from coefficients (except for an elementary step). k depends only on temperature. Overall order = m + n.
Integrated rate laws
REFERENCE| Order | Integrated law | Linear plot |
|---|---|---|
| 0 | [A]ₜ = [A]₀ − kt | [A] vs t (slope −k) |
| 1 | ln[A]ₜ = ln[A]₀ − kt | ln[A] vs t (slope −k) |
| 2 | 1/[A]ₜ = 1/[A]₀ + kt | 1/[A] vs t (slope +k) |
Half-life by order
REFERENCE| Order | t₁/₂ | Depends on [A]₀? |
|---|---|---|
| 0 | [A]₀ / (2k) | yes |
| 1 | ln2 / k ≈ 0.693/k | no |
| 2 | 1 / (k·[A]₀) | yes |
Fraction remaining (first order)
CORE RULEAfter N half-lives, fraction left = (1/2)ᴺ (50%, 25%, 12.5%, …), with N = t/t₁/₂. For any time: [A]ₜ/[A]₀ = e^(−kt). If X% decomposes, (100−X)% remains.
Method of initial rates
METHODPick two trials where only ONE concentration changes: rate₂/rate₁ = ([X]₂/[X]₁)^order, so order = log(rate ratio) / log(conc ratio). Repeat per reactant, then get k by plugging one trial into rate = k[A]ᵐ[B]ⁿ.
Arrhenius equation
CORE RULEk = A·e^(−Ea/RT). Two-temperature: ln(k₂/k₁) = (Ea/R)(1/T₁ − 1/T₂). Plot ln k vs 1/T: slope = −Ea/R. T in Kelvin; R = 8.314 J/(mol·K) for Ea in J/mol.
Units of k
REFERENCE| Overall order | k units |
|---|---|
| 0 | M/s |
| 1 | s⁻¹ |
| 2 | M⁻¹·s⁻¹ |
| 3 | M⁻²·s⁻¹ |
Rate-determining step
CORE RULEThe slowest (RDS) step sets the overall rate. For an ELEMENTARY step the orders DO equal its coefficients: A + 2B → products gives rate = k[A][B]². If the RDS contains an intermediate, substitute it out via the pre-equilibrium.
Intermediate vs catalyst
CORE RULEBoth have net-zero stoichiometry (not in the overall equation). Intermediate: produced first, then consumed. Catalyst: consumed first, then regenerated (it lowers Ea).
Energy diagram
CORE RULEPeak = transition state; valley between peaks = intermediate. Forward Ea = peak − reactants; ΔH = products − reactants. A catalyst gives a lower-Ea path with the SAME ΔH. RDS = the highest peak.
Pre-equilibrium
METHODIf a fast equilibrium precedes the RDS, solve the intermediate from that step: [int] = K_eq·[reactants], then substitute into the RDS rate. E.g. 2NO + Cl₂: rate = k₂K_eq[NO]²[Cl₂].