Integrated Rate Laws, Reaction Order, and Half-Lives in Chemical Kinetics

Chemical Kinetics: Rate Laws and Reaction Order

Introduction to Rate Laws

Chemical kinetics studies the speed at which chemical reactions occur and the factors that affect these rates. Rate laws mathematically describe how the concentration of reactants influences the rate of a reaction.

• Differential Rate Law: Expresses the rate of change of reactant concentration over time. For a reaction AB → A + B, the differential rate law for a first-order reaction in AB is: $\text{rate} = k[\text{AB}]$

• Integrated Rate Law: Relates reactant concentration to time, allowing calculation of concentrations at specific times without calculus knowledge.

Mathematical Relationship: Differential vs. Integrated Rate Laws

The differential rate law describes instantaneous rate, while the integrated rate law provides a direct relationship between concentration and time. Integration of the differential rate law yields the integrated rate law.

• Differential Rate Law: $\frac{d[\text{AB}]}{dt} = -k[\text{AB}]$

• Integrated Rate Law (First Order): $[\text{AB}]_t = [\text{AB}]_0 e^{-kt}$

• Linear Form: $\ln[\text{AB}]_t = -kt + \ln[\text{AB}]_0$

Additional info: The integrated rate law allows calculation of reactant concentration at any time t, given the initial concentration and rate constant.

Determining Reaction Order

Graphical Methods and Rate Constant Units

The order of a reaction can be inferred from the linearity of plots of concentration data and the units of the rate constant.

• Zero Order: $[A]_t = [A]_0 - kt$ (Plot of $[A]$ vs. time is linear)

• First Order: $\ln[A]_t = -kt + \ln[A]_0$ (Plot of $\ln[A]$ vs. time is linear)

• Second Order: $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$ (Plot of $1/[A]$ vs. time is linear)

Order	Integrated Rate Law	Graph	Slope
Zero	$[A]_t = [A]_0 - kt$	$[A]$ vs. $t$	$-k$
First	$\ln[A]_t = -kt + \ln[A]_0$	$\ln[A]$ vs. $t$	$-k$
Second	$\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$	$1/[A]$ vs. $t$	$k$

Units of Rate Constant (k):

• Zero order: $\text{M} \cdot \text{s}^{-1}$

• First order: $\text{s}^{-1}$

• Second order: $\text{M}^{-1} \cdot \text{s}^{-1}$

Calculating Concentrations Over Time

Using Integrated Rate Laws

Given the initial concentration and rate constant, the integrated rate law allows calculation of reactant concentration at any time.

• Example (First Order): If $[A]_0 = 1.0$ M, $k = 0.1$ s$^{-1}$, find $[A]$ after 30 s: $[A]_{30} = [A]_0 e^{-k \cdot 30} = 1.0 e^{-0.1 \times 30} = 1.0 e^{-3} \approx 0.05$ M

Reaction Half-Lives

Definition and Calculation

The half-life ($t_{1/2}$) is the time required for the concentration of a reactant to fall to half its initial value. Expressions for half-life differ by reaction order.

Order	Half-Life Expression
Zero	$t_{1/2} = \frac{[A]_0}{2k}$
First	$t_{1/2} = \frac{\ln 2}{k} \approx \frac{0.693}{k}$
Second	$t_{1/2} = \frac{1}{k[A]_0}$

• First Order Half-Life Derivation: $[A]_{t_{1/2}} = \frac{[A]_0}{2}$ $\ln\left(\frac{[A]_0}{2}\right) = -k t_{1/2} + \ln [A]_0$ $\ln\left(\frac{1}{2}\right) = -k t_{1/2}$ $-\ln 2 = -k t_{1/2}$ $t_{1/2} = \frac{\ln 2}{k}$

• Diagnostic Tool: For first-order reactions, half-life is independent of initial concentration.

Graphical Analysis and Reaction Order

Interpreting Plots

Graphical analysis of concentration vs. time data helps determine reaction order:

• Zero order: $[A]$ vs. time is linear and decreasing.

• First order: $\ln[A]$ vs. time is linear and decreasing.

• Second order: $1/[A]$ vs. time is linear and increasing.

Example: Plots of $1/[I-131]$ and $\ln[I-131]$ vs. time can reveal whether the decay of I-131 is first or second order.

Summary Table: Integrated Rate Laws and Half-Lives

Order	Integrated Rate Law	Half-Life
Zero	$[A]_t = [A]_0 - kt$	$t_{1/2} = \frac{[A]_0}{2k}$
First	$\ln[A]_t = -kt + \ln[A]_0$	$t_{1/2} = \frac{0.693}{k}$
Second	$\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$	$t_{1/2} = \frac{1}{k[A]_0}$

Key Takeaways

• Integrated rate laws allow calculation of reactant concentrations at any time.

• Graphical analysis and rate constant units help determine reaction order.

• Half-life equations are useful for quick kinetics calculations, especially for first-order reactions.

• First-order reactions have a half-life independent of initial concentration, serving as a diagnostic tool.