Molecular collisions, frequency

NMR signals are highly sensitive to the unusual behavior of pore fluids because of the characteristic effect of pore confinement on surface adsorption and molecular motion. Increased surface adsorption leads to modifications of the spin-lattice (T,) and spin-spin (T2) relaxation times, enhances NMR signal intensities and produces distinct chemical shifts for gaseous versus adsorbed phases [17-22]. Changes in molecular motions due to molecular collision frequencies and altered adsorbate residence times again modify the relaxation times [26], and also result in a time-dependence of the NMR measured molecular diffusion coefficient [26-27]. 

Elementary reactions are initiated by molecular collisions in the gas phase. Many aspects of these collisions determine the magnitude of the rate constant, including the energy distributions of the collision partners, bond strengths, and internal barriers to reaction. Section 10.1 discusses the distribution of energies in collisions, and derives the molecular collision frequency. Both factors lead to a simple collision-theory expression for the reaction rate constant k, which is derived in Section 10.2. Transition-state theory is derived in Section 10.3. The Lindemann theory of the pressure-dependence observed in unimolecular reactions was introduced in Chapter 9. Section 10.4 extends the treatment of unimolecular reactions to more modem theories that accurately characterize their pressure and temperature dependencies. Analogous pressure effects are seen in a class of bimolecular reactions called chemical activation reactions, which are discussed in Section 10.5. 

To determine the rate of change of the distribution due to molecular collisions we also need to derive an expression for the expected number of collisions. Prom this measure the molecular collision frequency can be obtained as a spin-off result. 

The simple approach discussed above is valid for steady-state Monte Carlo simulations, but dynamic simulations are also possible. In this case, the model probabilities must be updated frequently, generally after every iteration. A detailed discussion of these methods would be too lengthy to be included herein the most common algorithm for dynamic Monte Carlo simulation follows the approach proposed by Gillespie, which requires the discretization of the polymerization reactor with small control volumes and the conversion of the polymerization kinetic rates into molecular collision frequencies [97, 98],... 

In the low-pressure region the effective surface concentration of hydrogen is zero and the hydriding rate is determined by the impingement of Hj on the surface. According to kinetic theory the molecular collision frequency is proportional to The inverse... 

When the frequency of an applied electric field is the same as the collision frequency, the field and the fluctuating dipoles interact and we get relaxation or resonance phenomena. In other words, the frequency of maximum loss, of energy absorption, is the molecular collision frequency. 

There are two types of molecular theories on reaction rates collision theory and transitimr state theory. According to the coUision theory, the upper limit of a bimolecular rate constant of a gaseous reaction is the molecular collision frequency obtained by gas kinetics. The molecular coUtsimr frequency Zab between molecules, A and B, is given by... 

Dependence of Conductivity on Operational Parameters The conductivity of various substances, and in particular fuel cell materials, can depend strongly on operating parameters. For example, the electron conductivity of solid materials such as graphite and steel decreases with increasing temperature due to increased molecular collision frequency. However, the ionic conductivity of the SOFC and PEFC electrolyte is a positively related function of temperature. In the PEFC, water saturation in the electrolyte also plays a major role. These functional dependencies are highly material specific and are discussed in later chapters. 

In chemiluminescence, some of the chemical reaction products developed remain in an excited state and radiate light when the excitation is discharged. This is particularly so at low pressures, when the collision frequency is low the excitation is discharged as light radiation. The extra energy bound to the excited molecule can discharge through impact or molecular dissociation. 

Z, the collision frequency, which gives the number of molecular collisions occurring in unit time at unit concentrations of reactants. This quantity can be calculated quite accurately from the kinetic theory of gases, but we will not describe that calculation. 

The quantity / is just a further combination of constants already in Eq. (10-70). The value of Z is taken to be the collision frequency between reaction partners and is often set at the gas-phase collision frequency, 1011 L mol-1 s-1. This choice is not particularly critical, however, since / is nearly unity unless is very large. Other authors29-30 give expressions for Z in terms of the nuclear tunneling factors and the molecular dimensions. 

Luminescence lifetime spectroscopy. In addition to the nanosecond lifetime measurements that are now rather routine, lifetime measurements on a femtosecond time scale are being attained with the intensity correlation method (124), which is an indirect technique for investigating the dynamics of excited states in the time frame of the laser pulse itself. The sample is excited with two laser pulse trains of equal amplitude and frequencies nl and n2 and the time-integrated luminescence at the difference frequency (nl - n2 ) is measured as a function of the relative pulse delay. Hochstrasser (125) has measured inertial motions of rotating molecules in condensed phases on time scales shorter than the collision time, allowing insight into relaxation processes following molecular collisions. 

When bounding walls exist, the particles confined within them not only collide with each other, but also collide with the walls. With the decrease of wall spacing, the frequency of particle-particle collisions will decrease, while the particle-wall collision frequency will increase. This can be demonstrated by calculation of collisions of particles in two parallel plates with the DSMC method. In Fig. 5 the result of such a simulation is shown. In the simulation [18], 2,000 representative nitrogen gas molecules with 50 cells were employed. Other parameters used here were viscosity /r= 1.656 X 10 Pa-s, molecular mass m =4.65 X 10 kg, and the ambient temperature 7 ref=273 K. Instead of the hard-sphere (HS) model, the variable hard-sphere (VHS) model was adopted in the simulation, which gives a better prediction of the viscosity-temperature dependence than the HS model. For the VHS model, the mean free path becomes ... 

The first possibility is that the attractive potential associated with the solid surface leads to an increased gaseous molecular number density and molecular velocity. The resulting increase in both gas-gas and gas-wall collision frequencies increases the T1. The second possibility is that although the measurements were obtained at a temperature significantly above the critical temperature of the bulk CF4 gas, it is possible that gas molecules are adsorbed onto the surface of the silica. The surface relaxation is expected to be very slow compared with spin-rotation interactions in the gas phase. We can therefore account for the effect of adsorption by assuming that relaxation effectively stops while the gas molecules adhere to the wall, which will then act to increase the relaxation time by the fraction of molecules on the surface. Both models are in accord with a measurable increase in density above that of the bulk gas. 

Particle collision frequency due to Brownian motion was estimated to be less than 1% of the collision frequency due to shear. The effects of Brownian motion could therefore be neglected in the flocculation rate calculations. However, for the smallest molecular size, radius of gyration 14 nm (see Table I), the effect of Brownian motion on the particle-polymer collision efficiency was of the same order of magnitude as the effect of shear. These two contributions were assumed to be additive in the adsorption rate calculations. Additivity is not fundamentally justified (23) but can be used as an interpolating... 

The transmission coefficient k is approximately 1 for reactions in which there is substantial (>4kJ) electronic coupling between the reactants (adiabatic reactions). Ar is calculable if necessary but is usually approximated by Z, the effective collision frequency in solution, and assumed to be 10" M s. Thus it is possible in principle to calculate the rate constant of an outer-sphere redox reaction from a set of nonkinetic parameters, including molecular size, bond length, vibration frequency and solvent parameters (see inset). This represents a remarkable step. Not surprisingly, exchange reactions of the type... 

A further development is possible by noting that the high frequency shear modulus Goo is related to the mean square particle displacement (m ) of caged fluid particles (monomers) that are transiently localized on time scales ranging between an average molecular collision time and the structural relaxation time r. Specifically, if the viscoelasticity of a supercooled liquid is approximated below Ti by a simple Maxwell model in conjunction with a Langevin model for Brownian motion, then (m ) is given by [188]... 

Transport of the gas to the surface and the initial interaction. The first step in heterogeneous reactions involving the uptake and reaction of gases into the liquid phase is diffusion of the gas to the interface. At the interface, the gas molecule either bounces off or is taken up at the surface. These steps involve, then, gaseous diffusion, which is determined by the gas-phase diffusion coefficient (Dg) and the gas-surface collision frequency given by kinetic molecular theory. 

EXAM PLE 9.6 Rate of Atomic Collisions as a Function of Pressure. Assuming 1019 atoms per square meter as a reasonable estimate of the density of atoms at a solid surface, estimate the time that elapses between collisions of gas molecules at 10 6 torr and 25°C with surface atoms. Use the kinetic molecular theory result that relates collision frequency to gas pressure through the relationship Z = 1/4 vNIV, for which the mean velocity of the molecules v = (BRTI-kM) 12 and NIV is the number density of molecules in the gas phase and equals pNJRT. Repeat the calculation at 10 8 and 10 10 torr. 

The coefl. of viscosity of hydrogen chloride gas is 0 000141 the free path, 0 0000071 cm. the collision frequency 5670 X102 per second molecular diameter, I 70xl0-7 cm. the coefl. of condensation from gas to liquid, or Dgas/7)uquisquare root of the mean square of the molecular velocity of hydrogen chloride is 4 34xl04 cm. per sec., and the arithmetical mean, 4xl04 cm. per sec. 

These measurements have been carried out in collaboration with de Maeyek[4]). The rate constant was found to be (1 3 0-2)-10n litres/ mol-sec thus the neutralization reaction is the fastest known bi-molecular reaction in aqueous solution. Molecular-kinetic considerations show that the velocity of recombination is solely determined by the collision frequency of the ions. Furthermore, the effective cross section of the proton is so large that the reaction already proceeds spontaneously when ions approach each other within a distance of two to three H-bonds. This means that the motion of the proton within the hydration complex (the diameter of which corresponds to about two to three H-bonds) proceeds rapidly compared to the actual movement of the ions towards each other. 

Fig. 1.3. Collision-induced profile - schematic. Shown is the transition frequency Vo ( line center ), a sum of molecular transition frequencies, (AEA + AEs)/h. At higher frequencies, translational energies of relative motion of the collisional pair are increased, at lower frequencies decreased see Eq. 1.7. The average width is given by Eq. 1.5.

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