Chemical Kinetics: Rate Laws, Reaction Orders, and Integrated Rate Laws

Chemical Kinetics

Introduction to Chemical Kinetics

Chemical kinetics is the study of reaction rates and the factors that affect them. Understanding kinetics allows chemists to predict how fast reactions occur and how concentrations of reactants and products change over time.

• Reaction Rate: The change in concentration of a reactant or product per unit time.

• Rate Law: An equation that relates the reaction rate to the concentrations of reactants, often in the form $\text{Rate} = k [A]^m [B]^n$.

• Order of Reaction: The power to which the concentration of a reactant is raised in the rate law.

Determining Reaction Order

The order of a reaction with respect to a particular reactant is determined experimentally by observing how changes in concentration affect the rate.

• Example: If tripling the concentration of reactant A multiplies the rate by a factor of nine, the reaction is second order in A, since $3^2 = 9$.

• General Rate Law: $\text{Rate} = k [A]^m [B]^n$

• Order with respect to Na2S: For the reaction $2\text{NaCl}(aq) + 2\text{Na}_2\text{S}(aq) \rightarrow 4\text{NaCl}(aq) + \text{VS}_2(s)$, the order with respect to Na2S cannot be determined from stoichiometry alone; experimental data is required.

Experimental Determination of Rate Laws

Rate laws are determined by measuring initial rates at varying concentrations of reactants.

Experiment	[NO] (M)	[Br2] (M)	Initial Rate (M/s)
1	0.10	0.20	24
2	0.25	0.20	150
3	0.10	0.50	60
4	0.35	0.50	735

• Rate Law for 2NO(g) + Br2(g) → 2NOBr(g): $\text{Rate} = k [NO]^2 [Br_2]^1$

• Calculation of Rate Constant: Using experiment 1: $24 = k (0.10)^2 (0.20)$, so $k = 12000\ \text{M}^{-2}\text{s}^{-1}$

Stoichiometry and Rate Relationships

The rate of appearance of products and disappearance of reactants are related by stoichiometric coefficients.

• General Relationship: For $aA + bB \rightarrow cC + dD$, $\frac{1}{a} \frac{d[A]}{dt} = \frac{1}{b} \frac{d[B]}{dt} = \frac{1}{c} \frac{d[C]}{dt} = \frac{1}{d} \frac{d[D]}{dt}$

• Example: For $2NO(g) + Br_2(g) \rightarrow 2NOBr(g)$, $\frac{d[NOBr]}{dt} = -2 \frac{d[Br_2]}{dt}$

Calculating Rates from Rate Laws

Given concentrations and the rate constant, the rate of disappearance or appearance of a species can be calculated.

• Example: For $[NO] = 0.075\ \text{M}$ and $[Br_2] = 0.25\ \text{M}$, $\text{Rate} = 12000\ \text{M}^{-2}\text{s}^{-1} \times (0.075)^2 \times 0.25 = 16.8\ \text{M/s}$

• Relationship: $\text{Rate of disappearance of Br}_2 = \frac{1}{2} \text{Rate of appearance of NOBr}$

Integrated Rate Laws

Integrated rate laws describe how reactant concentrations change over time for different reaction orders.

• Zero Order: $[A] = [A]_0 - kt$

• First Order: $\ln[A] = \ln[A]_0 - kt$

• Second Order: $\frac{1}{[A]} = \frac{1}{[A]_0} + kt$

• Graphical Analysis:

  • Zero order: plot $[A]$ vs. $t$ (slope = $-k$)

  • First order: plot $\ln[A]$ vs. $t$ (slope = $-k$)

  • Second order: plot $1/[A]$ vs. $t$ (slope = $k$)

Half-Life of Reactions

The half-life ($t_{1/2}$) is the time required for the concentration of a reactant to decrease to half its initial value. The formula depends on reaction order.

Order	Half-life Formula	Dependence on [A]0
Zero	$t_{1/2} = \frac{[A]_0}{2k}$	Depends on $[A]_0$
First	$t_{1/2} = \frac{0.693}{k}$	Independent of $[A]_0$
Second	$t_{1/2} = \frac{1}{k[A]_0}$	Depends on $[A]_0$

• Example: For a first-order reaction with $k = 0.010\ \text{s}^{-1}$, $t_{1/2} = \frac{0.693}{0.010} = 69.3\ \text{s}$

Summary Table: Rate Laws and Integrated Rate Laws

Order	Rate Law	Integrated Rate Law	Graph for Straight Line	Half-life
Zero	$\text{Rate} = k$	$[A] = [A]_0 - kt$	$[A]$ vs. $t$	$t_{1/2} = \frac{[A]_0}{2k}$
First	$\text{Rate} = k[A]$	$\ln[A] = \ln[A]_0 - kt$	$\ln[A]$ vs. $t$	$t_{1/2} = \frac{0.693}{k}$
Second	$\text{Rate} = k[A]^2$	$\frac{1}{[A]} = \frac{1}{[A]_0} + kt$	$1/[A]$ vs. $t$	$t_{1/2} = \frac{1}{k[A]_0}$

Key Takeaways

• Reaction order must be determined experimentally, not from stoichiometry.

• Rate laws allow calculation of reaction rates for given concentrations.

• Integrated rate laws describe concentration changes over time and allow determination of rate constants from experimental data.

• Half-life formulas differ for zero, first, and second order reactions.

Additional info: These notes are based on lecture slides and practice problems from a General Chemistry course at Ohio State University, focusing on chemical kinetics and rate laws.