Chapter 3 / 3.3 Integrated Rate Equations

🚀 Integrated Rate Equations – Quick-Start Notes

These notes walk you through zero-order and first-order rate laws, half-life ideas, and the handy shortcuts exam setters love. Let’s dive in! 🎉

Rate law: \( \text{Rate} = -\dfrac{d[R]}{dt} = k \) :contentReference[oaicite:0]{index=0}
Integrated form: \( [R] = [R]_0 – k t \) (straight-line 📈 when you plot [R] vs t; slope = –k) :contentReference[oaicite:1]{index=1}
Rate constant from a graph: \( k = \dfrac{[R]_0 – [R]}{t} \) :contentReference[oaicite:2]{index=2}
Half-life: \( t_{1/2} = \dfrac{[R]_0}{2k} \) (depends on starting concentration) ⏱️ :contentReference[oaicite:3]{index=3}
Classic examples: Decomposition of NH₃ on hot Pt, thermal decomposition of HI on Au – rate stays constant once the metal surface is “full”. 💡 :contentReference[oaicite:4]{index=4}

Rate law: \( \text{Rate} = -\dfrac{d[R]}{dt} = k[R] \) :contentReference[oaicite:5]{index=5}
Integrated form (natural log): \( \ln[R] = \ln[R]_0 – k t \) (plot ln[R] vs t; slope = –k) :contentReference[oaicite:6]{index=6}
Integrated form (base-10 log): \( k = \dfrac{2.303}{t}\,\log\!\Big(\dfrac{[R]_0}{[R]}\Big) \) :contentReference[oaicite:7]{index=7}
Alternative “pressure method” (for A → B + C gases): \( k = \dfrac{2.303}{t}\,\log\!\Bigg(\dfrac{p_i}{2p_i – p_t}\Bigg) \) where \(p_i\) is the initial pressure and \(p_t\) the total pressure at time t. :contentReference[oaicite:8]{index=8}
Half-life: \( t_{1/2} = \dfrac{0.693}{k} \) (constant, independent of concentration) ⏱️ :contentReference[oaicite:9]{index=9}
Handy tip: Time to reach 99.9 % completion = \(10\,t_{1/2}\). 🔟 ⏱️ :contentReference[oaicite:10]{index=10}
Famous first-order cases: hydrogenation of ethene, radioactive decay (e.g., ²²⁶Ra → ²²²Rn + α), and decomposition of N₂O₅. 🌟 :contentReference[oaicite:11]{index=11}

Some higher-order reactions act like first-order because one reactant is in huge excess (its concentration barely changes). Examples:

In both, the rate law simplifies to \( \text{Rate} = k[\text{limiting reactant}] \). :contentReference[oaicite:12]{index=12}

Order	Integrated Eqn	Straight-Line Plot	Half-Life
Zero	\( [R] = [R]_0 – k t \)	[R] vs t	\( t_{1/2} = \dfrac{[R]_0}{2k} \)
First	\( \ln\!\big(\tfrac{[R]_0}{[R]}\big) = k t \)	\(\ln[R]\) vs t	\( t_{1/2} = \dfrac{0.693}{k} \)

Data summary adapted from the integrated-law table. :contentReference[oaicite:13]{index=13}

Memorise the half-life formulas: \( t_{1/2} = \dfrac{[R]_0}{2k} \) (zero) and \( t_{1/2} = 0.693/k \) (first). These pop up in MCQs all the time. ⏱️:contentReference[oaicite:14]{index=14}
Slope tricks: Graph choice tells reaction order—[R] vs t (zero) vs ln[R] vs t (first). 📈:contentReference[oaicite:15]{index=15}
First-order half-life is concentration-independent—great for “same t_1/2 at any [R]” questions. 🌟:contentReference[oaicite:16]{index=16}
Pressure method saves time when only total pressure data are given (common in gas-phase decomposition problems). 💨:contentReference[oaicite:17]{index=17}
Pseudo first-order concept—identify the reactant in excess to simplify kinetics. 🤓:contentReference[oaicite:18]{index=18}

✨ Keep practicing with sample graphs and half-life calculations—soon these equations will feel like second nature. Happy studying! ✨