CHM2046 Kinetics: Quick Reference

Rate Fundamentals

• Reaction rate: rate of consumption of reactants or formation of products.

• For a reaction aA + bB → products, the rate can be written as
  $-\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = \frac{d[C]}{dt}! /\nu_C$

• Four factors that influence rate (controlled during the reaction):

  • Concentration: more collisions if [reactants] higher.

  • Physical state: better mixing/ contact.

  • Temperature: higher T increases kinetic energy to exceed activation barrier.

  • Catalyst: provides alternate pathway with lower activation energy.

Rate Laws and Reaction Order

• General rate law: $\text{Rate} = k [A]^m [B]^n$

  • m, n: reaction orders with respect to each reactant (can be integer or fractional).

  • Overall order = m + n.

  • k: rate constant (depends on temperature; influences speed).

• Units of k depend on the overall order N = m+n:
  $\text{Rate units} = \text{mol L}^{-1} \text{s}^{-1}$
  $k\text{ units} = \text{mol}^{1-N} \text{L}^{N-1} \text{s}^{-1}$

• Common forms:

  • Zero order: $\text{Rate} = k$ (independent of [A])

  • First order: $\text{Rate} = k[A]$

  • Second order: either $\text{Rate} = k[A]^2$ or $\text{Rate} = k[A][B]$

• Determining orders from a rate law:

  • If rate = k[NO]^2[O2], orders: 2 in NO, 1 in O2, overall order 3.

Integrated Rate Laws

• First order:
  $\ln [A]<em>t = -kt + \ln [A]</em>0$
  (plot of $\ln[A]$ vs time is linear)

• Second order:
  $\frac{1}{[A]<em>t} = kt + \frac{1}{[A]</em>0}$

• Zero order:
  $[A]<em>t = [A]</em>0 - kt$

Half-Life

• First order: $t_{1/2} = \frac{\ln 2}{k}$

• Zero order: $t<em>{1/2} = \frac{[A]</em>0}{2k}$

• Second order: $t<em>{1/2} = \frac{1}{k[A]</em>0}$

Temperature Dependence: Arrhenius Equation

• Arrhenius relation: $\ln k = \ln A - \frac{E_a}{RT}$

  • A: frequency factor (collision pre-exponential factor)

  • E_a: activation energy

  • R: gas constant; T: Kelvin

• Two-condition form (comparison):
  $\ln \frac{k<em>2}{k</em>1} = -\frac{E<em>a}{R}\left(\frac{1}{T</em>2} - \frac{1}{T_1}\right)$

Collision Theory and Activation Energy

• Reaction rate depends on collisions with sufficient energy and proper orientation.

• Fraction of collisions with enough energy increases with temperature; governed by Ea.

• Higher Ea reduces the fraction f of effective collisions at a given T.

Transition State Theory

• Reaction proceeds through an activated complex (transition state).

• Ea is the energy barrier between reactants and products; rate relates to formation and breakdown of the transition state.

• Catalyst can provide a lower-energy pathway via a different transition state.

Catalysts

• Catalysts provide alternate mechanism with lower Ea.

• They speed up both forward and reverse reactions but do not change the equilibrium composition.

• Effectively lowers Ea by creating a new, lower-energy pathway.

Elementary Steps and Reaction Mechanisms

• Elementary step molecularity:

  • Unimolecular: rate = k[A]

  • Bimolecular: rate = k[A]^2 or rate = k[A][B]

  • Termolecular: rate = k[A]^2[B]

• Overall reaction mechanism must:

  • Sum of steps yields the overall equation.

  • Have rate law that matches the observed rate (rate-determining step governs).

  • Be physically reasonable and consistent with the rate law.

Practice Concepts (Summary)

• Identify rate law from data or given rate expression and determine individual and overall orders.

• Use integrated rate laws to determine concentration changes and to extract k.

• Use Arrhenius equation to relate rate constants at different temperatures and to estimate Ea.

• Distinguish zero-, first-, second- (and higher) order behaviors via plots of [A] vs t, ln[A] vs t, or 1/[A] vs t.

• Understand how catalysts alter Ea and mechanism without changing equilibrium yields.

• Recognize that the rate-determining step controls the overall rate in multi-step mechanisms.

Quick Reference Formulas

• Rate law:
  $\text{Rate} = k [A]^m [B]^n$

• Integrated rate laws:

  • First order: $\ln [A]<em>t = -kt + \ln [A]</em>0$

  • Second order: $\frac{1}{[A]<em>t} = kt + \frac{1}{[A]</em>0}$

  • Zero order: $[A]<em>t = [A]</em>0 - kt$

• Half-lives:
  $t<em>{1/2}^{(1^{st})} = \frac{\ln 2}{k},\quad t</em>{1/2}^{(0^{th})} = \frac{[A]<em>0}{2k},\quad t</em>{1/2}^{(2^{nd})} = \frac{1}{k[A]_0}$

• Arrhenius: $\ln k = \ln A - \frac{E<em>a}{RT},\quad \ln \frac{k</em>2}{k<em>1} = -\frac{E</em>a}{R}\left(\frac{1}{T<em>2} - \frac{1}{T</em>1}\right)$

• Rate constants depend on overall order N: $k\text{ units} = \text{mol}^{1-N} \text{L}^{N-1} \text{s}^{-1}$

• Elementary steps: unimolecular, bimolecular, termolecular rate laws as above.