Comprehensive Study Notes: Integrated Rate Laws, Temperature Effects, and Reaction Theories

General Principles for Complex Reactions

• Rate-Determining Step: For a complex reaction, the overall order of the reaction is determined by the slowest step in the reaction mechanism.
• Molecularity and Order: The molecularity of the slowest step in a complex reaction is identical to the overall order of the reaction.

Intext Questions and Applications

• Question 3.3: For a reaction $A + B \rightarrow \text{Product}$, given the rate law $r = k [A]^{1/2} [B]^2$, find the order of the reaction.     * Calculation: The order is the sum of the powers of the concentration terms in the rate law. Total order = $\frac{1}{2} + 2 = 2.5$.
• Question 3.4: The conversion of molecule $X$ to $Y$ follows second-order kinetics. If the concentration of $X$ is increased to three times, how will it affect the rate of formation of $Y$?     * Analysis: Rate law $r = k [X]^2$. If $[X]<em>{\text{new}} = 3[X]$, then $r</em>{\text{new}} = k(3[X])^2 = 9k[X]^2$. The rate increases by 9 times.

Integrated Rate Equations: Overview

• Differential vs. Integrated Rate Equations: The concentration dependence of the rate is expressed as a differential rate equation. However, determining instantaneous rates (slopes of tangents at point $t$) is inconvenient and difficult for finding rate laws/orders.
• Purpose of Integration: Differential rate equations are integrated to establish a direct relationship between measured experimental data (concentrations at specific times) and the rate constant ($k$).
• Specificity: Integrated rate equations vary depending on the reaction order.

Zero Order Reactions: Derivation and Characteristics

• Definition: A zero-order reaction occurs when the rate of reaction is proportional to the zero power of the reactant concentration.
• Derivation:     * Reaction: $R \rightarrow P$     * Rate expression: $\text{Rate} = -\frac{d[R]}{dt} = k [R]^0 = k \times 1$     * Differential form: $d[R] = -k dt$     * Integration: Integrating both sides yields $[R] = -kt + I$, where $I$ is the integration constant.
• Determination of Integration Constant ($I$):     * At $t = 0$, $[R] = [R]_0$ (initial concentration).     * Substitution: $[R]_0 = -k(0) + I \Rightarrow I = [R]_0$.
• Final Zero Order Equation: $[R] = -kt + [R]_0$
• Graphical Representation: A plot of concentration ($[R]$) against time ($t$) yields a straight line with a slope equal to $-k$ and a y-intercept equal to $[R]_0$.
• Calculation of Rate Constant ($k$): $k = \frac{[R]_0 - [R]}{t}$

Zero Order Reactions: Examples and Mechanism

• General Nature: These reactions are uncommon and typically occur under special conditions, such as enzyme-catalyzed reactions or reactions on metal surfaces.
• Specific Examples:     * Decomposition of Gaseous Ammonia on Platinum: $2NH_3 (g) \xrightarrow[1130K]{\text{Pt catalyst}} N_2 (g) + 3H_2 (g)$. Rate $= k [NH_3]^0 = k$. At high pressure, the metal surface becomes saturated with ammonia molecules, making the reaction rate independent of further concentration increases.     * Thermal Decomposition of HI on Gold Surface: Another instance of surface-mediated zero-order kinetics.

First Order Reactions: Derivation and Analytical Representation

• Definition: The rate of reaction is proportional to the first power of the reactant concentration.
• Derivation:     * Reaction: $R \rightarrow P$     * Rate expression: $\text{Rate} = -\frac{d[R]}{dt} = k [R]$     * Differential form: $\frac{d[R]}{[R]} = -k dt$     * Integration: $\ln [R] = -kt + I$.
• Determination of Integration Constant ($I$):     * At $t = 0$, $[R] = [R]_0$.     * Substitution: \ln [R]_0 = -k(0) + I \Rightarrow I = \ln [R]_0.
• Integrated Rate Equation: $\ln [R] = -kt + \ln [R]_0$
• Alternative Forms:     * Logarithmic rearrangement: $\ln \frac{[R]}{[R]<em>0} = -kt$ or $k = \frac{1}{t} \ln \frac{[R]_0}{[R]}$     * Exponential form: $[R] = [R]_0 e^{-kt}$     * Common Logarithm ($\log</em>{10}$) form: $k = \frac{2.303}{t} \log \frac{[R]_0}{[R]}$

First Order Reactions: Comparison at Different Times

• Let concentrations be $[R]_1$ at time $t_1$ and $[R]_2$ at time $t_2$.
• Equations:     1. $\ln [R]_1 = -kt_1 + \ln [R]_0$     2. $\ln [R]_2 = -kt_2 + \ln [R]_0$
• Subtraction yields: $\ln \frac{[R]_1}{[R]_2} = k(t_2 - t_1)$.
• Rate constant solution: $k = \frac{1}{t_2 - t_1} \ln \frac{[R]_1}{[R]_2}$.

First Order Reactions: Graphical Representation and Examples

• Natural Log Plot: Plotting $\ln [R]$ against $t$ gives a straight line with slope $-k$ and intercept $\ln [R]_0$.
• Common Log Plot: Plotting $\log \frac{[R]_0}{[R]}$ vs. $t$ gives a slope equal to $\frac{k}{2.303}$.
• Examples:     * Hydrogenation of Ethene: $C_2H_4(g) + H_2(g) \rightarrow C_2H_6(g)$. Rate $= k [C_2H_4]$.     * Radioactive Decay: All natural and artificial radioactive decays of unstable nuclei follow first-order kinetics (e.g., $^{226}<em>{88}Ra \rightarrow ^4_2He + ^{222}</em>{86}Rn$).     * Decomposition: Decomposition of $N_2O_5$ and $N_2O$.

Mathematical Application: Integrated First Order Equations

• Example 3.5: Decomposition of $N_2O_5$ at $318\,K$.     * Initial concentration ($[R]_1$): $1.24 \times 10^{-2}\,mol/L$.     * Final concentration ($[R]_2$) at $t = 60\,min$: $0.20 \times 10^{-2}\,mol/L$.     * Calculation: $k = \frac{2.303}{60\,min} \log (\frac{1.24 \times 10^{-2}}{0.20 \times 10^{-2}}) = \frac{2.303}{60} \log 6.2 = 0.0304\,min^{-1}$.

Gas Phase First Order Reactions

• Reaction Type: $A(g) \rightarrow B(g) + C(g)$
• Assume initial pressure is $p_i$ and total pressure at time $t$ is $p_t$. Let $x$ be the pressure decrease in reactant $A$.
• Stoichiometry at time $t$:     * $P_A = p_i - x$     * $P_B = x$     * $P_C = x$
• Total pressure ($p_t$): $(p_i - x) + x + x = p_i + x$.
• Solving for $x$: $x = p_t - p_i$.
• Partial pressure of $A$: $P_A = p_i - (p_t - p_i) = 2p_i - p_t$.
• Gas Phase Rate Constant: $k = \frac{2.303}{t} \log \frac{p_i}{2p_i - p_t}$.

Half-Life of a Reaction ($t_{1/2}$)

• Definition: The time required for the concentration of a reactant to be reduced to exactly one-half of its initial concentration.
• Zero Order Reaction Half-Life:     * Condition: At $t_{1/2}$, $[R] = \frac{[R]<em>0}{2}$.     * Substitution into $k = \frac{[R]_0 - [R]}{t}$ yields $k = \frac{[R]_0 - [R]_0/2}{t</em>{1/2}}$.     * Formula: $t_{1/2} = \frac{[R]<em>0}{2k}$.     * Relationship: $t</em>{1/2}$ is directly proportional to initial concentration and inversely proportional to the rate constant.
• First Order Reaction Half-Life:     * Condition: At $t_{1/2}$, $[R] = \frac{[R]<em>0}{2}$.     * Substitution into $k = \frac{2.303}{t} \log \frac{[R]_0}{[R]}$ yields $k = \frac{2.303}{t</em>{1/2}} \log (2)$.     * Formula: $t_{1/2} = \frac{2.303 \times 0.301}{k} = \frac{0.693}{k}$.     * Relationship: $t_{1/2}$ is independent of the initial concentration for a first-order reaction.

Summary of Integrated Rate Laws (Table 3.4)

• Zero Order:     * Differential rate law: $\frac{d[R]}{dt} = -k$     * Integrated rate law: $kt = [R]_0 - [R]$     * Straight line plot: $[R]$ vs $t$     * Half-life: $\frac{[R]_0}{2k}$     * Units of $k$: $\text{conc time}^{-1}$ or $mol\,L^{-1}\,s^{-1}$
• First Order:     * Differential rate law: $\frac{d[R]}{dt} = -k[R]$     * Integrated rate law: $[R] = [R]_0 e^{-kt}$ or $kt = \ln \frac{[R]_0}{[R]}$     * Straight line plot: $\ln [R]$ vs $t$     * Half-life: $\frac{\ln 2}{k}$     * Units of $k$: $\text{time}^{-1}$ or $s^{-1}$

Pseudo First Order Reactions

• Definition: Reactions that are high-order but behave as first-order reactions due to specific conditions (e.g., one reactant present in large excess).
• Example: Hydrolysis of Ethyl Acetate:     * Reaction: $CH_3COOC_2H_5 + H_2O \xrightarrow{H^+} CH_3COOH + C_2H_5OH$     * Scenario: if $0.01\,mol$ of ester reacts with $10\,mol$ of water, the change in water concentration ($9.99\,mol$) is negligible.     * The rate becomes dependent only on the concentration of ethyl acetate, behaving as pseudo first-order.
• Example: Inversion of Cane Sugar:     * $C_{12}H_{22}O_{11} + H_2O \xrightarrow{H^+} C_6H_{12}O_6 + C_6H_{12}O_6$     * Rate $= k [C_{12}H_{22}O_{11}]$.

Temperature Dependence of Reaction Rates

• Observation: Reaction rates typically increase with rising temperature. For many reactions, a $10^{\circ}$ rise in temperature approximately doubles the rate constant.
• Example: Decomposition of $N_2O_5$ to half original amount:     * At $0^{\circ}C$: takes 10 days.     * At $25^{\circ}C$: takes 5 hours.     * At $50^{\circ}C$: takes 12 minutes.
• Example: Potassium Permanganate ($KMnO_4$) and Oxalic Acid ($H_2C_2O_4$): Decolourization occurs faster at higher temperatures.

The Arrhenius Equation and Activation Energy

• Formula: $k = A e^{-E_a/RT}$     * $A$: Arrhenius factor, frequency factor, or pre-exponential factor (constant specific to a reaction).     * $E_a$: Activation energy ($J\,mol^{-1}$).     * $R$: Gas constant.     * $T$: Temperature.
• Mechanism (Intermediate Complex Theory): Reactants (e.g., $H_2$ and $I_2$) collide to form an unstable intermediate known as the activated complex ($C$). The energy required to form this complex is the Activation Energy ($E_a$). Products are formed when the complex decomposes, releasing energy.
• Maxwell-Boltzmann Distribution: Describes distribution of kinetic energies among large numbers of molecules.     * Plot: Fraction of molecules ($N_E/N_t$) vs. Kinetic energy ($E$).     * Peak: Most probable kinetic energy.     * Temperature Effect: Higher temperature shifts the curve to the right and broadens it. The area representing molecules with energy $\ge E_a$ effectively doubles with a $10\,K$ rise.
• Logarithmic Arrhenius Equation: $\ln k = -\frac{E_a}{RT} + \ln A$.     * Plotting $\ln k$ vs. $1/T$ gives a straight line with slope $-\frac{E_a}{R}$ and intercept $\ln A$.
• Two-Temperature Equation: $\log \frac{k_2}{k_1} = \frac{E_a}{2.303R} [\frac{T_2 - T_1}{T_1 T_2}]$.

Effect of Catalyst on Reaction Rate

• Catalyst: A substance that increases the reaction rate without being permanently changed. Example: $MnO_2$ in the decomposition of $KClO_3$.
• Inhibitor: A substance used specifically to reduce the rate of a reaction.
• Theoretical Action: A catalyst provides an alternative reaction pathway with a lower activation energy, thereby lowering the potential energy barrier.
• Properties of Catalysts:     * Small amounts can catalyze large quantities of reactants.     * Does not alter Gibbs energy ($\Delta G$).     * Only catalyzes spontaneous reactions.     * Does not change the equilibrium constant ($K_{eq}$) but helps reach equilibrium faster by catalyzing both forward and backward reactions equally.

Collision Theory of Chemical Reactions

• Context: Developed by Max Trautz and William Lewis (1916-18), based on the kinetic theory of gases.
• Postulate: Reactants are hard spheres; reactions occur upon collision.
• Collision Frequency ($Z$): The number of collisions per second per unit volume of reaction mixture.
• Bimolecular Elementary Reaction ($A + B \rightarrow \text{Products}$):     * $\text{Rate} = Z_{AB} e^{-E_a/RT}$.
• Effective Collisions: Only collisions with sufficient kinetic energy (Threshold energy) and proper orientation result in product formation.     * Threshold Energy = Activation Energy + energy possessed by reacting species.
• Steric Factor ($P$): Introduced to account for the probability of proper orientation.     * $\text{Rate} = P Z_{AB} e^{-E_a/RT}$.     * Example: Formation of methanol from bromoethane varies based on the attack angle of the $OH^-$ ion relative to the $CH_3Br$ molecule.
• Drawbacks: Collision theory treats molecules as hard spheres, ignoring their complex structural aspects.