Chemical Kinetics: Rate Expression and Rate Law

• Reaction rate is the change in a reactant or product's concentration over time.
• The stoichiometric coefficients of the reactants are negative in the rate expression.
• The stoichiometric coefficients of the product are positive in the rate expression.

Rate Expression

• For the 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 $[D]$ are the concentrations of reactants and products.
• $a$, $b$, $c$, and $d$ are the stoichiometric coefficients for each species.
• $\Delta t$ is the change in time.

Rate Law

• The rate equation relates reaction rate to reactant concentrations raised to a power.
• Rate $= k[A]^m[B]^n$ is the general form of a rate law.
• $k$ is the rate constant.
• $m$ and $n$ are reaction orders for reactants A and B.
• The overall reaction order is $m + n$.
• Reaction orders must be found experimentally.

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 = $k[A][B]$, the rate is proportional to the square of the concentration of one reactant, or to the product of the concentrations of two reactants.

Temperature Dependence of Reaction Rates

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

Catalysis

• A catalyst increases reaction speed without being consumed.
• Homogeneous catalysis: catalyst is in the same phase as reactants.
• Heterogeneous catalysis: catalyst is in a different phase from reactants.

Reaction Mechanisms

• A reaction mechanism consists of elementary steps that describe the pathway from reactants to products.
• An elementary step's rate equation is based directly on its stoichiometry.
• The rate-determining step is the slowest step, dictating the overall reaction rate.

Solving different orders using Calculus

• Zero Order Reaction : $[A] = [A]0 - kt$; $t{1/2} = \frac{[A]_0}{2k}$
• First Order Reaction : $\ln[A] = \ln[A]0 - kt$; $t{1/2} = \frac{0.693}{k}$
• Second Order Reaction : $\frac{1}{[A]} = \frac{1}{[A]0} + kt$; $t{1/2} = \frac{1}{k[A]_0}$

Rate Constants and Frequency Factors

• Arrhenius Equation: $k = A e^{-E_a/RT}$
• $E_a$ = activation energy.
• R = gas constant (approximated as 8.314 J/mol·K).
• A = frequency factor.
• Relationship between two rate constants : $\ln(\frac{k_2}{k_1}) = \frac{E_a}{R} (\frac{1}{T_1} - \frac{1}{T_2})$