Random relative motion

The solution of the partial differential equation (3.3.43) written for P(r", t + t) subject to the initial condition (3.3.44) and boundary conditions (3.3.45) and (3.3.46) is readily obtained. However, since the time scale of random relative motion may be viewed as being considerably smaller than that of aggregation, the preceding diffusion process may be construed to have reached steady state. The steady-state version of (3.3.43) written for... 

Taking into account the modulation of AE(R) by the random relative motion of the molecules A and B on which are located the potential wells and following the procedure of Kloffler and Brickmann, the average transition probability of the proton transfer from molecule A to molecule B at time t is given by... 

Outer sphere relaxation arises from the dipolar intermolecular interaction between the water proton nuclear spins and the gadolinium electron spin whose fluctuations are governed by random translational motion of the molecules (106). The outer sphere relaxation rate depends on several parameters, such as the closest approach of the solvent water protons and the Gdm complex, their relative diffusion coefficient, and the electron spin relaxation rate (107-109). Freed and others (110-112) developed an analytical expression for the outer sphere longitudinal relaxation rate, (l/Ti)os, for the simplest case of a force-free model. The force-free model is only a rough approximation for the interaction of outer sphere water molecules with Gdm complexes. 

Doppler broadening arises from the random thermal motion of the atoms relative to the observer. The velocity V, of an atom in the line of sight will vary according to the Maxwell distribution, the atoms moving in all directions relative to the observer. The frequency will be displaced by... 

Atoms taking part in diffusive transport perform more or less random thermal motions superposed on a drift resulting from field forces (V//,-, Vrj VT, etc.). Since these forces are small on the atomic length scale, kinetic parameters established under equilibrium conditions (i.e., vanishing forces) can be used to describe the atomic drift and transport, The movements of atomic particles under equilibrium conditions are Brownian motions. We can measure them by mean square displacements of tagged atoms (often radioactive isotopes) which are chemically identical but different in mass. If this difference is relatively small, the kinetic behavior is... 

For dilute suspensions, particle-particle interactions can be neglected. The extent of transfer of particles by the gradient in the particle phase density or volume fraction of particles is proportional to the diffusivity of particles Dp. Here Dp accounts for the random motion of particles in the flow field induced by various factors, including the diffusivity of the fluid whether laminar or turbulent, the wake of the particles in their relative motion to the fluid, the Brownian motion of particles, the particle-wall interaction, and the perturbation of the flow field by the particles. 

Thus, the relative motion of the molecules (atoms) is described by a series of wave packets that differ by a random lateral displacement vector b (with the polar coordinates b, ) corresponding to different impact points perpendicular to the momentum p0. As shown in Appendix F, such a displacement of ) can be generated by the operator exp(-ib-P/h), where P is the momentum operator with eigenstates p) and associated eigenvalues p, i.e.,... 

As can be seen, the flux for heat conduction across the air boundary layer is proportional to / au(7 surf 7"ta) for all three shapes considered (pea = 7s111 for Eq. 7.14).2 Because the conduction of heat in a gas phase is based on the random thermal motion of the molecules, the composition of air, such as its content of water vapor, can influence Kau. Air can hold more water vapor as the temperature increases in that regard, decreases as the water vapor content increases because H2O has a lower molecular weight (18) than is the average for air, which is mainly N2 and O2 (molecular weights of 28 and 32, respectively). For instance, at 20°C Kait at a pressure of 1 atm and 100% relative humidity is 1% less for than it is for dry air, and at 40°C, Km is then 2% lower (Appendix I). 

The simplest of the various conservation principles to apply is conservation of mass. It is instructive to consider its application relative to two different, but equivalent, descriptions of our fluid system. In both cases, we begin by identifying a specific macroscopic body of fluid that lies within an arbitrarily chosen volume element at some initial instant of time. Because we have adopted the continuum mechanics point of view, this volume element will be large enough that any flux of mass across its surface that is due to random molecular motions can be neglected completely. Indeed, in this continuum description of our system, we can resolve only the molecular average (or continuum point) velocities, and it is convenient to drop any reference to the averaging symbol (). The continuum point velocity vector is denoted as u.4... 

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