Chemical Kinetics: Rate Laws and Reaction Orders

• Chemical kinetics explores reaction rates.
• Reaction rate describes changes in reactant/product concentrations over time.

Rate Expression

• For a reaction $aA + bB \rightarrow cC + dD$, the rate expression is Rate $= -\frac{1}{a} \frac{\Delta[A]}{\Delta t} = -\frac{1}{b} \frac{\Delta[B]}{\Delta t} = \frac{1}{c} \frac{\Delta[C]}{\Delta t} = \frac{1}{d} \frac{\Delta[D]}{\Delta t}$.
• $[A]$, $[B]$, $[C]$, and $[D]$ symbolize reactant/product concentrations.
• $a$, $b$, $c$, and $d$ represent stoichiometric coefficients.
• $\Delta t$ is the time change.

Rate Law

• Rate law relates reaction rate to reactant concentrations raised to powers: Rate $= k[A]^m[B]^n$.
• $k$ stands for the rate constant.
• $m$ and $n$ represent reaction orders repectively.
• $m + n$ gives overall reaction order.

Determining Reaction Order

• Reaction orders comes from experiment data.

Common Rate Laws

• Zero-Order: Rate $= k$; the rate is independent of reactant concentration.
• First-Order: Rate $= k[A]$; the rate is directly proportional to reactant concentration.
• Second-Order: Rate $= k[A]^2$ or Rate $= k[A][B]$; rate depends on square of one reactant's concentration or product of two's.

Temperature Dependence of Reaction Rates

• The Arrhenius Equation links rate constant (k) to temperature (T): $k = Ae^{-\frac{E_a}{RT}}$.
• $A$ means pre-exponential factor.
• $E_a$ is the activation energy.
• $R$ is the gas constant, $8.314 J/(mol \cdot K)$.
• $T$ is in Kelvin.
• $E_a$ is the minimum energy for reaction.

Catalysis

• A catalyst speeds up reaction without being consumed.
• Homogeneous catalysis: catalyst and reactants are in the same phase.
• Heterogeneous catalysis: catalyst and reactants are in different phases.

Reaction Mechanisms

• A reaction mechanism comprises elementary steps from reactants to products.
• Elementary step: single step in mechanism; its rate law comes from stoichiometry.
• Rate-determining step: slowest step, dictating overall reaction rate.

Reaction Rate

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

Zero-Order Reaction

• Rate $= k[A]^0 = k$
• $A \rightarrow Products$
• $[A] = [A]_0 - kt$
• $t_{1/2} = \frac{[A]_0}{2k}$

First-Order Reaction

• Rate $= k[A]^1 = k[A]$
• $A \rightarrow Products$
• $\ln[A] = \ln[A]_0 - kt$
• $t_{1/2} = \frac{0.693}{k}$

Second Order Reaction

• Rate $= k[A]^2$
• $A \rightarrow Products$
• $\frac{1}{[A]} = \frac{1}{[A]_0} + kt$
• $t_{1/2} = \frac{1}{k[A]_0}$

Arrhenius Equation

• $k = A e^{-E_a/RT}$
• $E_a$: activation energy.
• R : gas constant is 8.314 J/mol·K
• A : frequency factor
• $\ln(\frac{k_2}{k_1}) = \frac{E_a}{R} (\frac{1}{T_1} - \frac{1}{T_2})$