11.1 Gas-phase reaction kinetics

Gas-phase reactions are all about molecules colliding. Collision theory explains how these collisions lead to reactions, considering factors like energy, orientation, and frequency. The rate of a reaction depends on these collision factors and the concentrations of reactants.

Temperature and pressure play crucial roles in gas-phase reactions. Higher temperatures increase molecular energy and collision frequency, while higher pressures increase concentrations. Both effects typically speed up reactions. For complex reactions, the slowest step often determines the overall rate.

Collision Theory and Elementary Gas-Phase Reactions

Rate expressions in collision theory

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• Collision theory explains reactions occur when reactant molecules collide with enough energy (activation energy) and proper orientation
• Elementary bimolecular gas-phase reaction (H2 + I2 -> 2HI):
  • Rate expression: $\text{rate} = k[\ce{H2}][\ce{I2}]$, $k$ is rate constant, $[\ce{H2}]$ and $[\ce{I2}]$ are concentrations
  • $k$ depends on collision frequency ($Z_{AB}$) and steric factor ($P$): $k = PZ_{AB}e^{-E_a/RT}$
    • $Z_{AB}$ involves collision cross-section ($\sigma_{AB}$), temperature ($T$), and reduced mass ($\mu$): $Z_{AB} = \sigma_{AB}\sqrt{\frac{8\pi k_BT}{\mu}}$
    • $P$ considers proper orientation of colliding molecules
    • $E_a$ is activation energy, $R$ is gas constant, $T$ is temperature
• Elementary unimolecular gas-phase reaction (N2O5 -> 2NO2 + 1/2O2):
  • Rate expression: $\text{rate} = k[\ce{N2O5}]$, $k$ is rate constant, $[\ce{N2O5}]$ is concentration

Factors Affecting Gas-Phase Reaction Rates

Temperature and pressure effects on reactions

• Temperature effects:
  • Higher temperature increases average kinetic energy of molecules, causing more collisions with enough energy to overcome activation energy barrier
  • Rate constant $k$ increases exponentially with temperature according to Arrhenius equation: $k = Ae^{-E_a/RT}$
    • $A$ is pre-exponential factor, $E_a$ is activation energy, $R$ is gas constant, $T$ is temperature
• Pressure effects:
  • Higher pressure increases reactant concentrations, leading to more frequent collisions and faster reaction rate
  • For gas-phase reaction $\ce{aA + bB -> products}$, rate is proportional to $[\ce{A}]^a[\ce{B}]^b$
    • Doubling pressure doubles concentrations, increasing rate by factor of $2^{a+b}$

Multi-Step Gas-Phase Reaction Mechanisms

Rate-determining steps in reaction mechanisms

• In multi-step reaction mechanism, slowest step is rate-determining step (RDS)
• Overall reaction rate controlled by rate of RDS
• Determining RDS:
  1. Write rate expression for each elementary step
  2. Compare rates of each step; slowest step is RDS
• Overall rate law consistent with rate law of RDS
  • Reactants not in RDS do not appear in overall rate law
• Steady-state approximation derives overall rate law from elementary steps
  • Assumes concentrations of reaction intermediates remain constant (formation rate = consumption rate)

Experimental Determination of Reaction Order

Experimental determination of reaction order

• Reaction order with respect to reactant is power its concentration is raised to in rate law
• For gas-phase reaction $\ce{aA + bB -> products}$ with rate law $\text{rate} = k[\ce{A}]^m[\ce{B}]^n$, $m$ and $n$ are orders for $\ce{A}$ and $\ce{B}$
• Methods for determining reaction order:
  • Initial rates method: Compare initial rates of experiments varying concentration of one reactant while holding others constant
    • For reactant $\ce{A}$: $\frac{\text{rate}_2}{\text{rate}_1} = \left(\frac{[\ce{A}]_2}{[\ce{A}]_1}\right)^m$, solve for $m$
  • Graphical methods: Plot log of initial rate vs log of initial concentration of reactant; slope is order for that reactant
  • Integrated rate laws: Plot concentration vs time data using integrated rate law equations for different orders; best linear fit indicates correct order