Distribution of Molecular Energy Levels Where: E = E i – E j & e - E/kT = Boltzman Factor If Boltz. FactorComment E << kT Close to 1Ratio of population is equal E ~ kT 1/e = 0.368Upper level drops suddenly E >> kT About 0Zero upper level population
The Barometric Formulation Calculate the pressure at mile high city (Denver, CO). [1 mile = 1610 m] P o = 101.325 kPa, T = 300. K. Assume 20.0 and 80.0 mole % of oxygen gas and nitrogen gas, respectively.Calculate
Molecular Temperature DistributionMeasurement of Vibrational Temp. in Hot Gases, Plasmas, Explosions Rotational Low Temp. in Interstellar Gases Electronic High Stellar Temp. of Atoms and Ions
The Kinetic Molecular Model for Gases ( Postulates ) Gas consists of large number of small individual particles with negligible size Particles in constant random motion and collisions No forces exerted among each other Kinetic energy directly proportional to temperature in Kelvin
Collision Properties ( Ref: Barrow ) Z I = collision frequency = number of collisions per molecule = mean free path = distance traveled between collisions Z II = collision rate = total number of collisions Main Concept => Treat molecules as hard-spheres
Collision Frequency ( Z I ) Interaction Volume ( V I ): ( d = interaction diameter ) Define: N* = N/V = molecules per unit volume
Kinetic-Molecular-Theory Gas Properties - Collision Parameters @ 25 o C and 1 atm Species Collision diameter Mean free path Collision Frequency Collision Rate d / 10 -10 md / Å / 10 -8 m Z I / 10 9 s -1 Z II / 10 34 m -3 s -1 H2H2 2.73 12.414.317.6 He2.18 188.8.131.52 N2N2 3.74 6.567.28.9 O2O2 3.57 184.108.40.206 Ar3.62 6.995.77.0 CO 2 4.56 4.418.610.6 HI5.56 2.967.510.6
The Arrhenius Equation Arrhenius discovered most reaction-rate data obeyed the Arrhenius equation: Including natural phenomena such as: Chirp rates of crickets Creeping rates of ants Arrhenius Concept
Extended Arrhenius Equation Experimentally, m cannot be determined easily! Implication: both A & Ea vary quite slowly with temperature. On the other hand, rate constants vary quite dramatically with temperature,
Collision Theory Main Concept: Rate Determining Step requires Bimolecular Encounter (i.e. collision) Rxn Rate = (Collision Rate Factor) x (Activation Energy) Z II (from simple hard sphere collision properties) Fraction of molecules with E > Ea : e -Ea/RT (Maxwell-Boltzmann Distribution)
Transition State Theory Concept: Activated Complex or Transition State ( ‡ ) 3D Potential Energy Surface Saddle point HH DD HH DD HH DD H 2 + D 2 2 HD H 2 + D 2 2 HD Activated Complex or Transition State ( ‡ )
Potential Energy Surfaces Consider:D + H 2 DH + H D HAHA HBHB r2r2 r1r1 r 1 = d H-D r 2 = d H-H Most favorable at: = 0 o, 180 o Calculate energy of interaction at different r 1, r 2 and . Get 3D Energy Map.3D Energy Map Reaction coordinate = path of minimum energy leading from reactants to products.
Reactions in Solutions Compared to gaseous reactions, reactions in solutions require diffusion through the solvent molecules. The initial encounter frequencies should be substantially higher for gas collisions. However, in solutions, though initial encounters are lower, but once the reactants meet, they get trapped in “solvent cages”, and could have a great number of collisions before escaping the solvent cage.
Diffusion Controlled Solutions Smoluchowski (1917): D = diffusion coefficient a = radius; = viscosity