Reaction Rates and Rate Laws

Reaction Rate

• Indicates the change in reactant or product concentration over time.

Rate Expression

• Describes how reaction rate relates to reactant and product concentrations.

General Rate Expression

• For the reaction $aA + bB \rightarrow cC + dD$, the rate can be expressed as: $-\frac{1}{a} \frac{d[A]}{dt} = -\frac{1}{b} \frac{d[B]}{dt} = \frac{1}{c} \frac{d[C]}{dt} = \frac{1}{d} \frac{d[D]}{dt}$.

Instantaneous Rate

• Defined as rate = $k[A]^x [B]^y$, where k is the rate constant, and x and y are the reaction orders with respect to A and B.

Overall Order of Reaction

• Calculated as the sum of the individual orders: overall order = $x + y$.

Rate Laws

• Mathematical relationships that describe how reaction rates depend on reactant concentrations.

Zero Order Reactions

• Rate is constant and unaffected by reactant concentration.
• Rate Law: rate = $k[A]^0 = k$
• Integrated Rate Law: $[A]_t = -kt + [A]_0$
• Half-life: $t_{1/2} = \frac{[A]_0}{2k}$
• A plot of $[A]_t$ vs. t yields a straight line with a slope of -k.

First Order Reactions

• Rate is directly proportional to the concentration of one reactant.
• Rate Law: rate = $k[A]^1 = k[A]$
• Integrated Rate Law: $ln[A]_t = -kt + ln[A]_0$
• Half-life: $t_{1/2} = \frac{0.693}{k}$
• A plot of $ln[A]_t$ vs. t is linear with a slope of -k.
• Half-life is constant, regardless of the initial concentration.

Second Order Reactions

• Rate is proportional to either the square of one reactant's concentration or the product of two reactants' concentrations.
• Rate Law: rate = $k[A]^2$ or rate = $k[A][B]$
• Integrated Rate Law: $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$
• Half-life: $t_{1/2} = \frac{1}{k[A]_0}$
• A plot of $\frac{1}{[A]_t}$ vs. t is linear with a slope of k.
• Half-life depends on the initial concentration of the reactant.

Activation Energy

• The minimum energy required for a reaction to occur.

Arrhenius Equation

• Describes the relationship between the rate constant, temperature, and activation energy: $k = Ae^{-E_a/RT}$.
• k is the rate constant, A is the frequency factor, $E_a$ is the activation energy, R is the gas constant ($8.314 J/(mol \cdot K)$), and T is the temperature in Kelvin.

Determining Activation Energy

• Can be found using rate constants at different temperatures: $ln(\frac{k_2}{k_1}) = -\frac{E_a}{R} (\frac{1}{T_2} - \frac{1}{T_1})$.

Reaction Mechanisms

• Step-by-step sequence of elementary reactions by which overall chemical change occurs

Elementary Steps

• Each individual step in a reaction mechanism.

Rate-Determining Step

• The slowest step in a reaction mechanism, which dictates the overall reaction rate.

Catalyst

• A substance that speeds up a reaction without being consumed.

Homogeneous catalyst

• Exists in the same phase as the reactants.

Heterogeneous catalyst

• Exists in a different phase than the reactants.

Summary Table

• Useful cheat sheet of the important information

Order 0

• Rate Law: $rate = k$
• Integrated Rate Law: $[A]_t = -kt + [A]_0$
• Half-life: $t_{1/2} = [A]_0 / 2k$
• Plot of $[A]_t$ vs. t for Linearity

Order 1

• Rate Law: $rate = k[A]$
• Integrated Rate Law: $ln[A]_t = -kt + ln[A]_0$
• Half-life: $t_{1/2} = 0.693 / k$
• Plot of $ln[A]_t$ vs. t for Linearity

Order 2

• Rate Law: $rate = k[A]^2$
• Integrated Rate Law: $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$
• Half-life: $t_{1/2} = 1 / k[A]_0$
• Plot of $\frac{1}{[A]_t}$ vs. t for Linearity