Integrated Rate Laws

12.3 Rate Laws

Rate Laws (Differential Rate Laws): Mathematical expressions describing the relationship between the rate of a chemical reaction and the concentration of its reactants.
General Form: $rate = k[A]^m[B]^n$
- $[A], [B]$ : Molar concentrations of reactants.
- $k$ : Rate constant, specific for a particular reaction at a particular temperature. It is independent of concentration but varies with temperature.
- $m, n$ : Reaction orders, determined experimentally. They describe the mathematical dependence of the rate on specific reactants.
Overall Reaction Order: The sum of the exponents for each reactant ( $m + n$ ).
Method of Initial Rates: An experimental approach where reaction rates are measured across multiple trials with different initial reactant concentrations to determine reaction orders and the rate constant.
Stoichiometry vs. Order: Reaction orders are not reliably predicted by stoichiometric coefficients and must be determined via experiment.
Rate Constant Units: Units vary based on the overall reaction order ( $x$ ) to ensure the rate is expressed in $mol/L/s$ . The general formula is $L^{x-1} mol^{1-x} s^{-1}$ .

12.4 Integrated Rate Laws
Integrated rate laws relate reactant concentrations to the elapsed time ( $t$ ).

Zero-Order Reactions
- Differential Rate Law: $rate = k$
- Integrated Form: [A]t = -kt + [A]sub 0
- \Linear Plot: $[A]_t$ vs. $t$ yields a straight line with slope $-k$ .
- Half-Life ( t1/2): t{1/2} = frac : [A]sub 0/2]. Half-life increases as the initial concentration increases.
First-Order Reactions
- Differential Rate Law: $rate = k[A]$
- Integrated Form: ln[A]t = -kt + ln[A]sub0
- Linear Plot: $\ln[A]_t$ vs. $t$ yields a straight line with slope $-k$ .
- Half-Life (t1/2): t1/2 = frac: ln2/k ≅ frac: 0.693/k. This half-life is independent of the reactant concentration.
Second-Order Reactions
- Differential Rate Law: $rate = k[A]^2$
- Integrated Form: frac: 1/[A]t = kt + frac:1/[A]sub 0
- Linear Plot: $\frac{1}{[A]_t}$ vs. $t$ yields a straight line with slope $k$ .
- Half-Life ( t1/2): t1/2 = frac: 1/k[A] sub 0. Half-life increases as the reaction proceeds because concentration decreases.

Summary Table of Kinetics

Order	Rate Law	Integrated Rate Law	Linear Plot	Slope	Half-life
0	$rate = k$	[A]t = -kt + [A]0	$[A]$ vs $t$	$-k$	$\frac{[A]_0}{2k}$
1	$rate = k[A]$	ln[A]t = -kt + ln[A]0	$\ln[A]$ vs $t$	$-k$	$\frac{0.693}{k}$
2	$rate = k[A]^2$	frac 1/[A]t = kt + frac 1/[A]0	$\frac{1}{[A]}$ vs $t$	$k$	$\frac{1}{k[A]_0}$