Chemical Kinetics – Lessons 19 & 20

Collision Theory

• Fundamental assertion: every chemical reaction requires a successful collision between reactant particles.

  • “Successful” means:

  • Sufficient kinetic energy $\ge Ea$ (activation energy).

  • Correct molecular orientation so that the appropriate bonds can form / break.

• Sequence of events (illustrated with $\text{CO} + \text{NO}<em>2 \;\rightarrow\; \text{CO}</em>2 + \text{NO}$):

  1. Gas-phase molecules move randomly with a spectrum of speeds/kinetic energies.

  2. Low-energy or mis-oriented collisions → elastic bounce; molecules remain intact.

  3. High-energy, properly oriented collision → formation of an activated (transition-state) complex.

  • Simultaneous bond-breaking and bond-forming.

  • Extremely unstable; exists for only an instant.

  1. Activated complex relaxes → stable product molecules.

• Energy Profile Diagram:

  • Vertical axis = Potential Energy; horizontal = Reaction Progress / Coordinate.

  • Peak corresponds to activated complex.

  • $Ea$ (activation energy) = energy gap between reactants and peak.

  • Only those reactant molecules with kinetic energy >Ea can surmount the barrier and form products.

Maxwell–Boltzmann Distribution & Factors Affecting Rate

Maxwell–Boltzmann Curve Basics

• At any fixed temperature, molecules possess a distribution of kinetic energies.

• Curve shape: rises sharply from zero energy, peaks (most probable energy), then decays with a long tail.

• The area under the curve = total number of molecules.

• Plotting a vertical line at $Ea$ divides:

  • Left region: molecules with E<Ea (cannot react upon collision).

  • Right region: molecules with E>Ea (can react if orientation is correct).

Parameters that Alter Reaction Rate

1. Surface Area (for heterogeneous solids or large particles):

   • Smaller particle size ⇒ larger surface area per unit mass ⇒ more exposed reactant sites ⇒ higher collision frequency ⇒ faster rate.

2. Concentration (solutions) / Pressure (gases):

   • Higher concentration or pressure ⇒ more particles per unit volume ⇒ higher collision frequency.

   • Maxwell–Boltzmann picture: curve does not move, but the number of collisions with E>Ea per unit time increases.

3. Temperature:

   • Raising $T$ shifts the entire Boltzmann curve rightwards & flattens it.

   • Consequences:

     • Average kinetic energy increases.

     • Significantly larger fraction of molecules possess E>Ea.

     • Both collision frequency and fraction of successful collisions rise ⇒ rate increases markedly.

4. Catalyst:

   • Provides an alternative reaction pathway with lower activation energy $Ea<em>12$.

   • Boltzmann curve unchanged, but vertical $Ea$ line moves left ⇒ many more molecules have E> Ea_1 ⇒ higher successful-collision frequency.

Rate of Reaction

• Generic stoichiometric equation: $aA + bB \;\rightarrow\; cC + dD$.

• Rate is expressed as time derivative of concentration: $rate = -\frac{1}{a}\frac{\Delta[A]}{\Delta t} = -\frac{1}{b}\frac{\Delta[B]}{\Delta t} = \frac{1}{c}\frac{\Delta[C]}{\Delta t} = \frac{1}{d}\frac{\Delta[D]}{\Delta t}$

  • Negative signs on reactants ensure positive rate values (reactant concentrations decrease).

Empirical Rate Law (rate equation)

• Form: $rate = k[A]^m[B]^n$

  • $[A], [B]$ = instantaneous molar concentrations.

  • $k$ = rate constant (temperature dependent only).

  • $m, n$ = orders with respect to each reactant – obtained experimentally, NOT from stoichiometry (except for elementary steps).

• Typical integer orders: 0, 1, 2 (can occasionally be non-integer or fractional).

Determining Reaction Order – Example (tert-butyl bromide hydrolysis)

• Reaction: $(CH3)3CBr + OH^- \rightarrow (CH3)3COH + Br^-$

• Experimental data:

  • Doubling $[(CH3)3CBr]$ doubles rate → first order in $[(CH3)3CBr]$.

  • Doubling $[OH^-]$ leaves rate unchanged → zero order in $[OH^-]$.

• Rate law: $rate = k[(CH3)3CBr]^1[OH^-]^0 = k[(CH3)3CBr]$.

• Overall order: $1+0 = 1$ (first order overall).

Practice Dataset (A + B₂ → AB₂)

• Use initial-rate method to determine:

  1. Order in $[A]$.

  2. Order in $[B_2]$.

  3. Overall order (sum).

  4. Rate law.

  5. Calculate $k$ with units; predict rate change when both reactant concentrations are doubled.

  • (Dataset supplied in transcript: three experiments with specified [A], [B₂], rates.)

Integrated Rate Laws

• Provide concentration–time relationships that integrate the differential rate laws.

• Linearisation makes it easy to identify reaction order experimentally – plot each of the three suggested graphs and see which is linear.

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

• Definition: time required for [reactant] to drop to half of $C_0$.

• Utility: examining how half-life varies with $C_0$ reveals reaction order swiftly.

Worked Practice Highlights (mentioned in slides)

1. Cyclopropane → propene (first order, $k = 6.7\times10^{-4}\;s^{-1},\;C_0 = 0.05\;M$):

   • Use $\ln C = -kt + \ln C_0$ to compute [cyclopropane] after 30 min.

   • Rearrange for $t$ when $C = 0.01\;M$.

   • $t_{1/2}=\ln 2/k$ (convert seconds to minutes).

2. $2\,NOBr\rightarrow 2\,NO + Br2$ (second order; $k = 0.810\;M^{-1}s^{-1}$): compute [NOBr] after 10 min and half-life at given $C0$ using $1/C = kt + 1/C_0$.

Reaction Mechanisms

• Elementary step: single molecular-level event with its own rate law whose order equals stoichiometric coefficients.

• Reaction mechanism: sequence of elementary steps summing to overall reaction.

• Intermediate: produced in one step, consumed in a later step; not present initially or in final equation.

• Catalyst: present at start and regenerated at end; participates but not consumed.

• Rate-determining step (RDS): slowest elementary step; its rate law becomes overall rate law (assuming any preceding steps are at equilibrium or fast).

Example: Iodide-catalysed $\text{H}2\text{O}2$ Decomposition

Overall: $\text{H}2\text{O}2 + I^- \rightarrow \text{H}2\text{O} + IO^-$ (Step 1, slow) $\text{H}2\text{O}2 + IO^- \rightarrow \text{H}2\text{O} + O_2 + I^-$ (Step 2, fast)

• Intermediate: $IO^-$.

• Catalyst: $I^-$ (consumed Step 1, regenerated Step 2).

• Observed rate law $rate = k[H2O2][I^-]$ matches slow Step 1, confirming it as RDS.

Criteria for a Valid Mechanism

1. Sum of elementary steps reproduces overall balanced equation.

2. Predicted overall rate law (from RDS and any pre-equilibria) matches experimentally determined rate law.

Practice Scenarios from Slides

1. Hypochlorite self-oxidation (two-step proposition): write individual rate equations, infer overall order from half-life constancy (constant $t_{1/2}$ ⇒ first order), identify RDS accordingly.

2. $2H2 + 2NO \rightarrow N2 + 2H_2O$ with three alternative mechanisms:

   • For each, write rate law from slow step; compare to empirical $rate = k[H_2][NO]^2$ to decide which mechanism agrees (students should deduce Mechanism III etc.).

Arrhenius Equation

• Quantifies temperature dependence of rate constant $k$: $k = A e^{-Ea/RT}$ where:

  • $A$ = frequency factor (collision frequency × orientation factor).

  • $Ea$ = activation energy (kJ mol$^{-1}$).

  • $R = 8.314 \times 10^{-3}\;\text{kJ mol}^{-1}\,\text{K}^{-1}$.

  • $T$ = absolute temperature (K).

Implications

• Higher $T$ → exponent less negative → larger $k$ → faster reaction.

• Higher $Ea$ (for given $T$) → smaller $k$ → slower reaction.

• Presence of catalyst lowers $Ea$ (and may slightly affect $A$) → increases $k$.

Linear Form (for data analysis)

• Taking natural log: $\ln k = \ln A - \frac{Ea}{R}\,\frac{1}{T}$.

  • Plot of $\ln k$ vs $1/T$ is straight line: slope $= -Ea/R$, intercept $=\ln A$.

Practice Example (Reactions P & Q)

• Energy profiles provided: $EaP = 25\;\text{kJ mol}^{-1},\;EaQ = 50\;\text{kJ mol}^{-1}$ at $T = 298\,K$.

• Plug into Arrhenius form to compute $kP, kQ$ at 298 K (illustrated in slides), then repeat at 348 K (298 K + 50 K increase) to show more pronounced increase for reaction with higher $Ea$.

• In presence of catalyst: $Ea$ decreases, $k$ increases (value of $A$ commonly similar but may vary slightly).

• Apply collision theory to rationalise how surface area, concentration/pressure, temperature, catalysts influence reaction rate.

• Formulate the empirical rate law, determine individual and overall reaction orders experimentally.

• Utilise integrated rate laws and half-life relationships to classify