10.1 Rate Laws and Reaction Order

10.1 Rate Laws and Reaction Order

Quick Answer: −r_A is the volumetric rate of disappearance of A. Power-law forms use order n so −r_A = k C_A^n (or multi-concentration products). Units of k change with n. First-order half-life is constant (ln 2 / k). Integral methods linearize C–t data to identify order.

Domain D (Chemical Reaction Engineering and Reactor Design) carries about 12% of this guide's planning allocation for the UPDA/MMUP Chemical exam—four focused sections in this chapter. Equilibrium (Chapter 6) answers “how far can the reaction go?” Kinetics answers “how fast does it go?” and “how large must the reactor be?” Confusing thermodynamic feasibility with rate is a common exam trap: a large equilibrium constant does not guarantee a fast reaction, and a catalyst changes rate, not the equilibrium constant K.

In Qatar’s refining, gas processing, fertilizer, and petrochemical plants, rate laws size reactors, set residence times, and explain why temperature, concentration, and catalysts dominate conversion and selectivity.

Definition of −r_A

For species A as a reactant, the rate of reaction of A is often written as the rate of disappearance:

[ (-r_A) = \text{moles of A reacted per unit volume per unit time} ]

Typical SI units: mol/(m³·s) or kmol/(m³·h). The minus sign in the symbol (−r_A) makes the quantity positive when A is consumed. In a design equation you always need a consistent definition of extent, stoichiometry, and volume basis (fluid volume for homogeneous liquid/gas; catalyst mass or bed volume for heterogeneous catalytic rates—exam items usually stay homogeneous unless stated).

Stoichiometric links: for a single reaction aA + bB → products,

[ \frac{-r_A}{a} = \frac{-r_B}{b} = r ]

where r is the extent-based rate. If A is the key reactant with stoichiometric coefficient 1 in A → products, then −r_A is the natural design rate.

What rate depends on: temperature (Arrhenius—Section 10.2), concentrations (or partial pressures), catalyst amount/activity, and sometimes total pressure or ionic strength. Rate does not equal conversion; conversion is an integrated result of rate, time or space time, and mixing pattern.

Concentration Dependence and Reaction Order

A common power-law rate form for irreversible reaction in A is:

[ (-r_A) = k C_A^n ]

• n = 0 — zero order: rate independent of C_A (as long as A is present and the form holds).
• n = 1 — first order: rate proportional to C_A.
• n = 2 — second order in A: rate proportional to C_A² (or first order in A and first in B: k C_A C_B).

Order is empirical (from data), not automatically equal to stoichiometric coefficients—except for elementary steps, where molecularity matches order.

Multi-reactant example: −r_A = k C_A C_B is overall second order, first order in each reactant. If B is in large excess, C_B ≈ constant and the observed rate looks pseudo-first-order in A: −r_A ≈ k' C_A with k' = k C_B.

Exam reading of stems: “rate doubles when concentration of A doubles” → first order in A (at fixed T). “Rate quadruples when C_A doubles” → second order in A. “Rate unchanged when C_A changes” → zero order in A (within the tested range).

Units of the Rate Constant k by Order

Because (−r_A) always has dimensions of concentration/time, the units of k absorb the concentration powers:

[ [k] = \frac{[\text{concentration}]^{1-n}}{[\text{time}]} ]

UPDA trap: comparing numerical k values across different orders is meaningless without converting units and forms. First-order k is a frequency (time⁻¹); second-order k is not.

Half-Life Cues—Especially First Order

Half-life t_{1/2} is the time for C_A (or amount of A in a constant-volume batch) to fall to half its initial value.

First-order cue on exams: if successive half-lives are equal as concentration falls (or if t_{1/2} does not change when you change starting concentration), the data support first order. If raising C_{A0} shortens t_{1/2}, think second order. If raising C_{A0} lengthens t_{1/2}, think zero order.

Worked half-life example (first order)

A liquid-phase irreversible reaction is first order with k = 0.12 h⁻¹ at the process temperature. Half-life:

[ t_{1/2} = \frac{\ln 2}{0.12} \approx \frac{0.693}{0.12} \approx 5.8,\mathrm{h} ]

After one half-life, C_A = 0.5 C_{A0}; after two, 0.25 C_{A0}; after three, 0.125 C_{A0}—each interval still ≈ 5.8 h in a constant-density batch. That constant-interval pattern is a strong first-order fingerprint.

Integral Method Intuition for Order Determination

The integral method assumes a rate law, integrates the constant-volume batch mole balance, and checks whether data fall on a straight line.

For constant volume (or constant density liquid), with X_A conversion of A (limiting):

[ C_A = C_{A0}(1-X_A), \quad \frac{dC_A}{dt} = r_A = -(-r_A) ]

Procedure intuition:

1. Collect C_A(t) (or conversion vs time) at fixed T in a batch reactor.
2. Plot the three classic linearizations.
3. The order whose plot is linear (random scatter about a line, not systematic curvature) is supported; slope gives k at that T.
4. Repeat at other temperatures for Arrhenius (Section 10.2).

Differential method (recognition): estimate −r_A ≈ −ΔC_A/Δt from the slope of C–t data, then plot ln(−r_A) vs ln C_A; slope ≈ order n. Integral methods are more common in textbook-style exam reasoning.

Worked conceptual: which plot?

Data at fixed T: C_A falls from 2.0 to 1.0 to 0.5 mol/L in equal 10-minute intervals. Equal half-lives at different concentrations ⇒ first order. Plot of ln C_A vs t is linear; plot of 1/C_A vs t is not the diagnostic match. Estimated k ≈ (ln 2)/t_{1/2} = 0.693/(10 min) ≈ 0.069 min⁻¹.

Connecting Rate Laws to Reactor Design

Rate laws are the constitutive equations inserted into design equations:

Higher order or lower concentration at high conversion makes (−r_A) small—so large time or volume is needed for the last stretch of conversion (especially visible in CSTRs, Section 10.3–10.4).

UPDA Exam Checklist for Section 10.1

1. −r_A = moles A reacted /(volume·time); keep stoichiometry consistent.
2. Order is from kinetics data; elementary steps are a special case.
3. Units of k: time⁻¹ only for first order; include concentration powers otherwise.
4. Constant half-life ⇒ first order; t_{1/2} ∝ 1/C_{A0} ⇒ second order; t_{1/2} ∝ C_{A0} ⇒ zero order.
5. Integral method: C_A, ln C_A, or 1/C_A vs t for orders 0, 1, 2.
6. Do not use equilibrium K as a substitute for k or −r_A.

Next: how temperature and catalysts enter through the Arrhenius form of k (Section 10.2).