Chemical Kinetics

Rate of Reaction: Change in concentration of reactant or product per unit time.

For: $aA + bB \to cC + dD$: $$\text{Rate} = -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = +\frac{1}{c}\frac{d[C]}{dt} = +\frac{1}{d}\frac{d[D]}{dt}$$

• Reactants: $-$ (decreasing)
• Products: $+$ (increasing)
• Units: mol/L/s (M/s) or M/min

Average vs Instantaneous Rate:

• Average: Over a finite time interval
• Instantaneous: $dt \to 0$; tangent slope at a point

Rate Law (Differential): Relates rate to concentrations of reactants: $$\text{Rate} = k[A]^x[B]^y$$

• $k$: rate constant; depends on T (not concentration)
• $x, y$: orders w.r.t. A, B (determined experimentally)
• Overall order = $x + y$

Order of Reaction:

• Can be 0, 1, 2, 3, or fractional
• Determined experimentally, NOT from stoichiometry
• Cannot exceed 3 for elementary reactions

Examples:

• $2N_2O_5 \to 4NO_2 + O_2$: rate = $k[N_2O_5]$ → first order
• $2NO + Cl_2 \to 2NOCl$: rate = $k[NO]^2[Cl_2]$ → third order overall

Molecularity:

• Number of reactant molecules in the elementary step
• Always integer (1, 2, rarely 3)
• Defined only for elementary reactions
• e.g., $NH_4NO_2 \to N_2 + 2H_2O$ is unimolecular but kinetically can be first or zero order

Differences:

Units of Rate Constant:

• Zero order: mol/L/s
• First order: 1/s (or s⁻¹)
• Second order: L/mol/s
• nth order: $\text{(mol/L)}^{1-n} \cdot \text{s}^{-1}$

Zero Order Reactions: Rate = $k$ (independent of concentration).

Integrated: $[A] = [A]_0 - kt$

Plot of $[A]$ vs $t$: straight line, slope $= -k$, intercept $= [A]_0$.

Half-life ($t_{1/2}$): Time for $[A]$ to decrease to half: $$t_{1/2} = [A]_0/(2k)$$ Note: $t_{1/2}$ depends on initial concentration.

First Order Reactions: Rate = $k[A]$.

Integrated: $$\ln\frac{[A]_0}{[A]} = kt$$ $$[A] = [A]_0 e^{-kt}$$ $$\log[A] = \log[A]_0 - kt/2.303$$

Plot of $\log[A]$ vs $t$: straight line, slope $= -k/2.303$.

Half-life: $$t_{1/2} = \frac{0.693}{k}$$ Independent of initial concentration (characteristic of first order).

Second Order Reactions: Rate = $k[A]^2$.

Integrated: $$\frac{1}{[A]} - \frac{1}{[A]_0} = kt$$

Plot of $1/[A]$ vs $t$: straight line, slope $= k$.

Half-life: $t_{1/2} = 1/(k[A]_0)$ (depends on initial concentration).

Summary Table:

Pseudo-Order Reactions: Reactions with multiple reactants behave as lower order when one reactant is in large excess. e.g., hydrolysis of ester in water: $CH_3COOC_2H_5 + H_2O \to CH_3COOH + C_2H_5OH$

• True rate = $k[ester][H_2O]$ (2nd order)
• With excess water, $[H_2O]$ ≈ constant: rate = $k'[ester]$ (pseudo first order)

Temperature Effect on Rate:

• Rate increases with $T$
• Empirical rule: rate ~ doubles per $10°$C rise (rough)

Arrhenius Equation: $$k = A e^{-E_a/RT}$$

• $A$: pre-exponential (frequency) factor; related to collision frequency
• $E_a$: activation energy
• $R$: gas constant ($8.314$ J/mol·K)
• $T$: absolute temperature

Logarithmic Form: $$\ln k = \ln A - E_a/(RT)$$ $$\log k = \log A - E_a/(2.303 RT)$$

Plot of $\log k$ vs $1/T$: straight line, slope $= -E_a/(2.303R)$, intercept $= \log A$.

At Two Temperatures: $$\log\frac{k_2}{k_1} = \frac{E_a}{2.303 R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right) = \frac{E_a}{2.303 R} \cdot \frac{T_2 - T_1}{T_1 T_2}$$

Activation Energy ($E_a$): Minimum energy reactants must have for successful reaction. Energy difference between reactants and transition state (activated complex).

Threshold energy: Total energy of activated complex. Activation energy = Threshold energy − Energy of reactants.

Energy Profile Diagram:

• Endothermic reaction: products at higher energy
• Exothermic reaction: products at lower energy
• $E_a$ (forward) and $E_a$ (reverse) related to $\Delta H$:

$\Delta H = E_a^{forward} - E_a^{reverse}$

Collision Theory: For reaction to occur:

1. Reactants must collide
2. Collision must have sufficient energy ($\geq E_a$) — energy condition
3. Collision must be in correct orientation — orientation condition

Rate $\propto$ (collision frequency) × (fraction with $E \geq E_a$) × (steric/orientation factor): $$\text{Rate} = Z \cdot e^{-E_a/RT} \cdot P$$

Arrhenius factor $A = ZP$ where $P$ is orientation/steric factor.

Reaction Mechanism:

• Sequence of elementary steps that combine to give overall reaction
• Rate-determining step (RDS): Slowest step; determines overall rate

For multi-step reactions, rate law is derived from RDS.

Example: $NO_2 + CO \to NO + CO_2$. Mechanism: Step 1 (slow): $NO_2 + NO_2 \to NO_3 + NO$ Step 2 (fast): $NO_3 + CO \to NO_2 + CO_2$

Rate law = $k[NO_2]^2$ (from RDS).

Pre-equilibrium Approximation: For a fast equilibrium followed by slow step, use $K_{eq}$ to eliminate intermediates.

Catalysis: Substance that increases rate without being consumed.

Types:

Properties of Catalysts:

1. Don't appear in net equation
2. Provide alternative pathway with lower $E_a$
3. Same effect on forward and reverse rates (doesn't shift equilibrium)
4. Small amount enough
5. Catalyst-specific to certain reactions

Adsorption Theory (Heterogeneous Catalysis):

1. Reactants adsorb on catalyst surface
2. Adsorbed species more reactive (weak bonds form/break)
3. Reaction occurs on surface
4. Products desorb

Examples of Catalysts in Industry:

Promoters: Increase activity of catalyst (e.g., $K_2O$, $Al_2O_3$ promote $Fe$ in Haber).

Poisons: Decrease activity (e.g., $As$, $S$ poison Pt).