Chemical Kinetics

1. Reaction Rates & Stoichiometric Relations

• Rate Definition: The change in molar concentration of either a reactant or a product per unit time.
  • Rate of consumption of reactant: $-\frac{\Delta [R]}{\Delta t}$
  • Rate of formation of product: $+\frac{\Delta [P]}{\Delta t}$
  • Instantaneous rate as $\Delta t \rightarrow 0$: $r_{\text{inst}} = -\frac{d[R]}{dt} = +\frac{d[P]}{dt}$

• Stoichiometric Equivalence: For a general chemical reaction:

  $$a\text{A} + b\text{B} \longrightarrow c\text{C} + d\text{D}$$

  The unique rate of reaction is expressed by dividing each component's rate of change by its stoichiometric coefficient:

  $$\text{Rate} = -\frac{1}{a}\frac{d[\text{A}]}{dt} = -\frac{1}{b}\frac{d[\text{B}]}{dt} = +\frac{1}{c}\frac{d[\text{C}]}{dt} = +\frac{1}{d}\frac{d[\text{D}]}{dt}$$

• Example: For $5\text{Br}^-(aq) + \text{BrO}_3^-(aq) + 6\text{H}^+(aq) \rightarrow 3\text{Br}_2(aq) + 3\text{H}_2\text{O}(l)$:

  $$\text{Rate} = -\frac{1}{5}\frac{d[\text{Br}^-]}{dt} = -\frac{d[\text{BrO}_3^-]}{dt} = -\frac{1}{6}\frac{d[\text{H}^}{dt} = +\frac{1}{3}\frac{d[\text{Br}_2]}{dt}$$

2. Rate Law, Order of Reaction & Molecularity

• Differential Rate Law:

  $$\text{Rate} = k[\text{A}]^x [\text{B}]^y$$

  Where exponents $x$ and $y$ represent the order with respect to $\text{A}$ and $\text{B}$. The overall order is $n = x + y$.
• Units of Rate Constant ($k$):

  $$\text{Unit of } k = (\text{mol L}^{-1})^{1-n} \text{ s}^{-1}$$

  • Zero Order ($n = 0$): $\text{mol L}^{-1}\text{ s}^{-1}$
  • First Order ($n = 1$): $\text{s}^{-1}$
  • Second Order ($n = 2$): $\text{L mol}^{-1}\text{ s}^{-1}$
  • Third Order ($n = 3$): $\text{L}^2\text{ mol}^{-2}\text{ s}^{-1}$

Order vs Molecularity Comparison

3. Integrated Rate Laws for Zero & First Order Reactions

Zero-Order Kinetics ($R \rightarrow P$)

• Differential form: $-\frac{d[R]}{dt} = k[R]^0 = k$
• Integrated form:

  $$[R] = -kt + [R]_0 \quad \iff \quad k = \frac{[R]_0 - [R]}{t}$$

• Linear plot: $[R]$ vs $t$ yields a straight line with $\text{Slope} = -k$ and $\text{Intercept} = [R]_0$.

First-Order Kinetics ($R \rightarrow P$)

• Differential form: $-\frac{d[R]}{dt} = k[R]$
• Integrated logarithmic form:

  $$k = \frac{2.303}{t}\log\left(\frac{[R]_0}{[R]}\right) \quad \iff \quad \ln[R] = -kt + \ln[R]_0$$

4. Half-Life Period & Gas-Phase First-Order Kinetics

• Half-Life ($t_{1/2}$): Time required for reactant concentration to reduce to one-half of its initial value.
  • For Zero-Order Reactions:

    $$t_{1/2} = \frac{[R]_0}{2k} \quad \implies \quad t_{1/2} \propto [R]_0$$

  • For First-Order Reactions:

    $$t_{1/2} = \frac{2.303 \log 2}{k} = \frac{0.693}{k} \quad \implies \quad t_{1/2} \text{ is independent of } [R]_0$$

• Gas-Phase Reaction Kinetics ($\text{A}(g) \rightarrow \text{B}(g) + \text{C}(g)$):
  Let $p_i$ be initial pressure of $\text{A}$, and $p_t$ be total pressure at time $t$:

  $$p_t = p_{\text{A}} + p_{\text{B}} + p_{\text{C}} = (p_i - x) + x + x = p_i + x \implies x = p_t - p_i$$

  $$p_{\text{A}} = p_i - x = p_i - (p_t - p_i) = 2p_i - p_t$$

  Substituting into the first-order integrated rate law:

  $$k = \frac{2.303}{t}\log\left(\frac{p_i}{2p_i - p_t}\right)$$

5. Pseudo First-Order Reactions

Reactions that are second order or higher, but behave kinetically as first order because one reactant is present in large excess.

1. Acid-Catalysed Hydrolysis of Ethyl Acetate:

   $$\text{CH}_3\text{COOC}_2\text{H}_5 + \text{H}_2\text{O} \xrightarrow{\text{H}^+} \text{CH}_3\text{COOH} + \text{C}_2\text{H}_5\text{OH}$$

   $$\text{Rate} = k'[\text{CH}_3\text{COOC}_2\text{H}_5][\text{H}_2\text{O}]$$

   Because $[\text{H}_2\text{O}]$ remains virtually constant:

   $$\text{Rate} = k[\text{CH}_3\text{COOC}_2\text{H}_5] \quad \text{where } k = k'[\text{H}_2\text{O}]$$

2. Inversion of Cane Sugar:

   $$\text{C}_{12}\text{H}_{22}\text{O}_{11} + \text{H}_2\text{O} \xrightarrow{\text{H}^+} \text{C}_6\text{H}_{12}\text{O}_6\text{ (glucose)} + \text{C}_6\text{H}_{12}\text{O}_6\text{ (fructose)}$$

   $$\text{Rate} = k[\text{C}_{12}\text{H}_{22}\text{O}_{11}]$$

6. Temperature Dependence & The Arrhenius Equation

For every $10^\circ\text{C}$ rise in temperature, the reaction rate constant approximately doubles.

• Arrhenius Equation:

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

  Where $A$ is the pre-exponential / frequency factor, $E_a$ is activation energy ($\text{J mol}^{-1}$), and $e^{-E_a/RT}$ represents the fraction of molecules with kinetic energy $\ge E_a$.
• Logarithmic Form:

  $$\ln k = \ln A - \frac{E_a}{RT} \iff \log k = \log A - \frac{E_a}{2.303 RT}$$

• Two-Temperature Form:

  $$\log\left(\frac{k_2}{k_1}\right) = \frac{E_a}{2.303 R}\left[\frac{T_2 - T_1}{T_1 T_2}\right]$$

7. Catalysis & Collision Theory

• Catalytic Action: A catalyst accelerates a reaction by providing an alternative mechanism with a lower activation energy barrier ($E_a' < E_a$).
  • It accelerates forward and backward reactions equally.
  • It does not alter $\Delta H$, $\Delta G$, or the equilibrium constant $K_{\text{eq}}$.

• Collision Theory for Bimolecular Reactions:

  $$\text{Rate} = P \cdot Z_{\text{AB}} \cdot e^{-E_a / RT}$$

  • $Z_{\text{AB}}$ is collision frequency (number of collisions per second per unit volume).
  • $P$ is the probability or steric factor, accounting for correct geometric orientation of colliding species.
• Criteria for an Effective Collision:
  1. Energy Barrier: Colliding species must possess energy $\ge$ threshold energy ($E_{\text{threshold}} = E_a + E_{\text{reactants}}$).
  2. Orientation Barrier: Atoms that will form new bonds must point directly toward each other during the collision event.