Chemical Kinetics revision notes

• Expressing Rate: For a hypothetical reaction $R \rightarrow P$, the rate is expressed as $-\frac{\Delta[R]}{\Delta t}$ (rate of disappearance) or $+\frac{\Delta[P]}{\Delta t}$ (rate of appearance). The negative sign ensures the rate value remains a positive quantity.
• Units of Rate: Concentration time$^{-1}$ (e.g., $\text{mol L}^{-1}\text{s}^{-1}$). For gaseous reactions using partial pressures, the unit is $\text{atm s}^{-1}$.

1. Zero Order Reactions

Rate is independent of reactant concentration.

• Equation: $[R] = -kt + [R]_0$.
• Graph: $[R]$ vs $t$ yields a straight line with slope $= -k$ and y-intercept $= [R]_0$.
• Examples: Decomposition of gaseous ammonia ($NH_3$) on a hot platinum catalyst at high pressure. (At high pressure, the metal surface saturates, making rate independent of concentration). Thermal decomposition of $HI$ on a gold surface. Certain enzyme-catalyzed reactions.

2. First Order Reactions

Rate is proportional to the first power of concentration.

• Equations: $\ln[R] = -kt + \ln[R]_0$ OR $[R] = [R]_0 e^{-kt}$ OR $k = \frac{2.303}{t} \log\frac{[R]_0}{[R]}$.
• Graphs: $\ln[R]$ vs $t$ $\rightarrow$ slope $= -k$, intercept $= \ln[R]_0$. $\log([R]_0/[R])$ vs $t$ $\rightarrow$ straight line through origin, slope $= k/2.303$.
• Examples: Hydrogenation of ethene ($C_2H_4 + H_2 \rightarrow C_2H_6$). All natural and artificial radioactive decays. Example: $^{226}_{88}Ra \rightarrow ^{222}_{86}Rn + ^{4}_{2}He$. Decomposition of $N_2O_5$ and $N_2O$.

3. First Order Gas Phase Reactions

For a typical reaction: $A(g) \rightarrow B(g) + C(g)$.

• Let $p_i$ be initial pressure of A. Total pressure at time $t$ is $p_t$.
• $p_t = (p_i - x) + x + x = p_i + x \Rightarrow x = p_t - p_i$.
• Pressure of A at time $t$: $p_A = p_i - x = 2p_i - p_t$.
• Equation: $k = \frac{2.303}{t} \log\left(\frac{p_i}{2p_i - p_t}\right)$.

The time in which reactant concentration is reduced to half its initial value.

• Zero Order: $t_{1/2} = \frac{[R]_0}{2k}$. Directly proportional to initial concentration.
• First Order: $t_{1/2} = \frac{0.693}{k}$. Independent of initial concentration.
• JEE Tip: For a first-order reaction, the time required for 99.9% completion is exactly 10 times its half-life ($t_{99.9\%} \approx 10 \times t_{1/2}$).

Reactions that are functionally first order but involve more than one reactant (higher molecularity). Occurs when one reactant is present in such large excess that its concentration change is negligible.

Examples

• Acid-catalyzed Hydrolysis of Ethyl Acetate: $CH_3COOC_2H_5 + H_2O \xrightarrow{H^+} CH_3COOH + C_2H_5OH$. (Water is in massive excess).
• Inversion of Cane Sugar (Sucrose): $C_{12}H_{22}O_{11} + H_2O \xrightarrow{H^+} C_6H_{12}O_6 (\text{Glucose}) + C_6H_{12}O_6 (\text{Fructose})$. Rate $= k[C_{12}H_{22}O_{11}]$.

• Rule of Thumb: A 10°C rise in temperature nearly doubles the rate constant.

• Reaction Coordinate & Energy: When reactants convert to products, they pass through an unstable intermediate called an "activated complex". The energy required to reach this state is the activation energy ($E_a$).

• Comparing Two Temperatures: $\log \frac{k_2}{k_1} = \frac{E_a}{2.303 R} \left[ \frac{T_2 - T_1}{T_1 T_2} \right]$.

Proposed by Max Trautz and William Lewis. Assumes molecules are hard spheres.

• Collision frequency ($Z$): Number of collisions per second per unit volume.
• Rate Equation (Basic): $\text{Rate} = Z_{AB} e^{-E_a/RT}$.
• Threshold Energy: The minimum kinetic energy molecules must possess for a collision to be effective. (Threshold Energy = Activation Energy + energy already possessed by reacting species).
• Steric Factor / Probability Factor ($P$): Not all collisions with sufficient energy yield products. Proper spatial orientation is required. For example, in the formation of methanol from bromoethane, improper orientation causes molecules to bounce back without reacting.
• Modified Rate Equation: $\text{Rate} = P \cdot Z_{AB} \cdot e^{-E_a/RT}$.
• Drawback: It ignores the structural aspect of molecules by treating them as rigid hard spheres.

• A catalyst accelerates the reaction by providing an alternate pathway or mechanism with a lower activation energy ($E_a$), thus reducing the potential energy barrier.
• Specific Catalyst Example: Addition of Manganese dioxide ($MnO_2$) considerably increases the rate of decomposition of Potassium chlorate ($KClO_3$): $2KClO_3 \xrightarrow{MnO_2} 2KCl + 3O_2$.

• Decomposition of Hydrogen Peroxide: $2H_2O_2 \xrightarrow{I^-, \text{ Alkaline medium}} 2H_2O + O_2$. Rate $= k[H_2O_2][I^-]$. (Overall 1st order in reactant, 1st order in catalyst $\Rightarrow$ 2nd order overall). Mechanism: Step 1 (Slow, RDS): $H_2O_2 + I^- \rightarrow H_2O + IO^-$. Step 2 (Fast): $H_2O_2 + IO^- \rightarrow H_2O + I^- + O_2$. Note: $IO^-$ (hypoiodite) is the reaction intermediate.

1. Average Rate: $r_{av} = \frac{-\Delta[Reactant]}{\Delta t} = \frac{+\Delta[Product]}{\Delta t}$.
2. General Rate Law: For $aA + bB \rightarrow cC$, $Rate = k[A]^x[B]^y$.
3. Units of $k$: $mol^{1-n} L^{n-1} s^{-1}$ (where $n$ is order).
4. Zero Order Kinetics:
   • $k = \frac{[R]_0 - [R]}{t}$
   • $t_{1/2} = \frac{[R]_0}{2k}$
5. First Order Kinetics:
   • $k = \frac{2.303}{t} \log\frac{[R]_0}{[R]}$
   • $[R] = [R]_0 e^{-kt}$
   • $t_{1/2} = \frac{0.693}{k}$
6. Arrhenius Equation: $k = A e^{-E_a/RT}$
   • $\ln k = -\frac{E_a}{RT} + \ln A$
   • $\log \frac{k_2}{k_1} = \frac{E_a}{2.303 R} \left[ \frac{T_2 - T_1}{T_1 T_2} \right]$
7. Collision Theory Equation: $\text{Rate} = P \cdot Z_{AB} \cdot e^{-E_a/RT}$.

• Zero Order Reactions & Concentration: While normally reaction rates decrease as reactants are consumed, zero-order reactions maintain a completely constant rate regardless of how much reactant is left, up until the reactant is completely exhausted. Why? They occur under special conditions (like metal surface catalysis at high pressure) where the surface is saturated. The reaction rate is limited by the available surface area, not the bulk gas concentration.
• Pseudo First-Order Concentration Anomaly: In reactions like the hydrolysis of ethyl acetate ($CH_3COOC_2H_5 + H_2O \xrightarrow{H^+} CH_3COOH + C_2H_5OH$), the reaction is bimolecular but behaves as first-order. Why? Water is present in such massive stoichiometric excess (e.g., 10 moles vs 0.01 moles) that its concentration is virtually unchanged during the reaction, rendering the rate independent of it.
• Fractional Order: $CHCl_3 + Cl_2 \rightarrow CCl_4 + HCl$ has an order of 1.5 ($[Cl_2]^{1/2}$). Why? It proceeds via a complex mechanism resulting in a fractional power dependence.
• The "Termolecular" Limit Anomaly: While we can write balanced equations with 4, 5, or even 10 reactants, reactions with a molecularity greater than 3 are exceedingly rare and do not proceed in a single step. Why? The statistical probability of more than three molecules colliding simultaneously with proper orientation and sufficient kinetic energy is practically zero.
• The Complex Reaction Molecularity Exception: Molecularity is rigorously defined for elementary steps, but the overall molecularity of a complex (multi-step) reaction has absolutely no meaning.
• "Negative Catalyst" Terminology Error: It is a common misnomer to call a substance that reduces the rate of a reaction a "negative catalyst". The word catalyst must not be used in this context; the correct and only term is inhibitor.
• Thermodynamic Feasibility vs. Kinetic Deadlock: A reaction can be highly thermodynamically spontaneous ($\Delta G < 0$) but functionally never occur. Why? The activation energy is so high that the reaction rate is imperceptible. Example: The conversion of diamond to graphite is thermodynamically favored, but kinetically "frozen" at room temperature.
• Catalysts and Equilibrium: A catalyst lowers the activation energy and speeds up the reaction, but it does not alter the Gibbs free energy ($\Delta G$) or the equilibrium constant ($K_{eq}$). Why? It lowers the activation energy of both the forward and backward reactions by the exact same amount, merely helping the system reach the exact same equilibrium state faster.
• Catalysts and Non-Spontaneous Reactions: A catalyst can only accelerate spontaneous reactions; it cannot catalyze or force a non-spontaneous reaction to occur.

• Integrated Rate Equation Graphs: Determining order and $k$ from plots of $[R]$ vs $t$, $\ln[R]$ vs $t$, and the slope/intercept values.
• Arrhenius Parameter Calculations: Using two-point temperatures ($T_1, T_2$) and $k_1, k_2$ values to solve for $E_a$ or plotting $\ln k$ vs $1/T$ to match linear formats.
• Catalyst Thermodynamics: True/False questions verifying that catalysts do NOT shift equilibrium ($K_{eq}$) or alter $\Delta H$, $\Delta G$.
• Gas Phase First Order Kinetics: Calculating total pressure $p_t$ or $k$ after time $t$ using variable substitution (the ICE table method).
• Decomposition Mechanisms: Identifying intermediates ($IO^-$) and the rate determining step in reactions like the $I^-$ catalyzed $H_2O_2$ decomposition.