Integrated Rate Equations
Transform instantaneous rates into measurable time. Master the absolute mathematical forms of Zero and First-Order reactions, half-life shortcuts, and gas-phase kinetics.
Module Focus: Calculus to Reality
The differential rate equation ($-d[R]/dt = k[R]^n$) is mathematically precise but practically useless for finding out how much reactant is left after 2 hours. By integrating these equations using calculus, we obtain relations that directly link Concentration with Time. For NEET, you must memorize the final integrated formulas, their graphical slopes, and the unique half-life properties of Zero and First-order reactions.
1. Zero-Order Reactions
In a zero-order reaction, the rate is entirely independent of the concentration of the reactants. No matter how much reactant you have, the reaction proceeds at a constant, steady speed.
Differential Form:
Integrated Form:
- Graph of $[R]$ vs. time ($t$): Gives a straight line with a negative slope.
Slope = $-k$ | y-intercept = $[R]_0$ - Graph of Rate vs. $[R]$: Gives a horizontal straight line parallel to the concentration axis (because Rate = $k$, a constant).
- Classic Examples: Photochemical reactions, decomposition of gases on solid metal surfaces at high pressure ($NH_3$ on Pt).
2. First-Order Reactions
In a first-order reaction, the rate is directly proportional to the first power of the concentration of the reactant. As the reactant is consumed, the rate slows down exponentially.
Integrating $-d[R]/dt = k[R]$ yields the natural log ($\ln$) equation. For numericals, we always convert it to base-10 log ($\log_{10}$) by multiplying by 2.303.
- Graph of $\ln[R]$ vs. $t$: Straight line. Slope = $-k$.
- Graph of $\log_{10}[R]$ vs. $t$: Straight line. Slope = $-\frac{k}{2.303}$ (Most frequent NEET trap!).
- Graph of $[R]$ vs. $t$: An exponential decay curve that theoretically never touches the x-axis (infinite time to complete 100%).
- Classic Examples: ALL natural and artificial radioactive decays.
First-Order Kinetics in the Gas Phase
Instead of concentration in mol/L, we measure the Total Pressure ($P_t$) of the system over time. Consider a typical gas phase reaction: $A(g) \rightarrow B(g) + C(g)$.
| Time | $A(g)$ | $\rightarrow$ | $B(g)$ | $+$ | $C(g)$ |
| At $t=0$ | $P_i$ | $0$ | $0$ | ||
| At time $t$ | $P_i - x$ | $x$ | $x$ |
Total Pressure $P_t = (P_i - x) + x + x = P_i + x \implies \mathbf{x = P_t - P_i}$
Pressure of A at time $t$ ($P_A$) = $P_i - x = P_i - (P_t - P_i) = \mathbf{2P_i - P_t}$
3. Half-Life ($t_{1/2}$) & Ultimate Shortcuts
The half-life ($t_{1/2}$) is the time required for the concentration of the reactant to reduce to exactly half of its initial value.
| Property | Zero-Order | First-Order |
|---|---|---|
| Half-life Formula | $t_{1/2} = \frac{[R]_0}{2k}$ | $t_{1/2} = \frac{0.693}{k}$ |
| Dependence on Initial Concentration ($[R]_0$) | Directly proportional ($\mathbf{t_{1/2} \propto [R]_0}$). More reactant takes more time to halve. | Completely Independent. Halving 100M to 50M takes the exact same time as 1M to 0.5M. |
If you see these completion percentages in a numerical, DO NOT calculate the full log formula. Use these direct relationships:
To find how half-life depends on initial concentration for an nth-order reaction (where $n \neq 1$):
NEET Grand Test: Integrated Rates
15 High-Yield Questions testing fractional completion shortcuts, logarithmic slopes, and gas-phase derivations.
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