Translational motion equations

To simulate the particle-particle collision, the hard-sphere model, which is based on the conservation law for linear momentum and angular momentum, is used. Two empirical parameters, a restitution coefficient of 0.9 and a friction coefficient of 0.3, are utilized in the simulation. In this study, collisions between spherical particles are assumed to be binary and quasi-instantaneous. The equations, which follow those of molecular dynamic simulation, are used to locate the minimum flight time of particles before any collision. Compared with the soft-sphere particle-particle collision model, the hard-sphere model accounts for the rotational particle motion in the collision dynamics calculation thus, only the translational motion equation is required to describe the fluid induced particle motion. In addition, the hard-sphere model also permits larger time steps in the calculation therefore, the simulation of a sequence of collisions can be more computationally effective. The details of this approach can be found in the literature (Hoomans et al., 1996 Crowe et al., 1998). 

By far the largest contribution to a gas-phase species entropy comes from translational motion. Equation 8.98 provided a means to calculate this contribution ... 

Equation (1) holds for a particle or a cell of any shape, even when the shape changes with the distance x, provided that it is reaching the surface mainly by a translational motion. Equation (1) also assumes a quasi-steady state of the motion of particles over the potential barrier. This approximation can be made because the region over which the potential acts is very thin and consequently the flux of particles through it can be considered practically constant with respect to the distance a at a given time t. 

Although the above equations are applicable only when is unity, they may be regarded as giving those parts of the internal energy, entropy, etc., of a polyatomic gas which are due to the translational motion. Equations analogous to those above may be readily obtained for the more general case. For example, the equation for the chemical potential is... 

We now have equations for the partition functions for the ideal gas and equations for relating the partition functions to the thermodynamic properties. We are ready to derive the equations for calculating the thermodynamic properties from the molecular parameters. As an example, let us calculate Um - t/o.m for the translational motion of the ideal gas. We start with... 

Equation (6.11) is the Schrodinger equation for the translational motion of a free particle of mass M, while equation (6.12) is the Schrodinger equation for a hypothetical particle of mass fi. moving in a potential field F(r). Since the energy Er of the translational motion is a positive constant (Er > 0), the solutions of equation (6.11) are not relevant to the structure of the two-particle system and we do not consider this equation any further. 

Equation (10.28b) describes the internal motions of the two nuclei and the electrons relative to the center of mass. Our next goal is to solve this equation using the method described in Section 10.1. Equation (10.28a), on the other hand, describes the translational motion of the center of mass of the molecule and is not considered any further here. 

The strain in electric field-associated bending of a PVA-PAA gel is given by the equation g = 6DY/L2 (see Eq. 21). The strain depends on the electric power applied to the gel. Thus, the deflection increases as the thickness becomes small even if the electric power remains constant. The PVA-PAA gel rod of 1 mm diameter bends semicircularly within 1 s under both dc and ac excitation. An artificial fish with a PVA-PAA gel tail 0.7 mm thick has been designed, and it has been demonstrated that the fish swims forward at a velocity of 2 cm/sec as the gel flaps back and forth under sinusoidally varied electric fields (Fig. 13b). This prototype of a biomimetic actuator shows that translational motion may be produced using bending deformation [74],... 

It is now proposed to obtain an expression for ka using Equation 14.11. This equation requires knowledge of the rate constant k3 in the RRKM scheme. The rate constant k3 is the inverse of the time required for the particle of mass p, to pass through the transition state, a one-dimensional box of length l. This time is calculated by classical mechanics. For a translational motion with kinetic energy x = (1/2) p, v2, where v is the velocity, the passage time is t/ and k3 is... 

There is a probability N 8R do for a collision specified by energy transfer T to the translational motion of the struck atom. N is the volume density of atoms, and one approximation for do is given by Equation 9. If a collision occurs, the particle has a probability F(x — 5x, E — T,ri ) of obtaining a total projected range, x. Therefore, N 5R do F(x — 5x, E — T,r ) is the contribution from this specified collision to the total probability for the projected range, x. When this term space is integrated over all collisions, the total contribution becomes ... 

For the rotovibrational spectra of molecules interacting through purely isotropic forces, the Hamiltonian may be written as the sum of two independent terms. One term describes the rotational motion of the molecules, the other the translational motion of the pair. The total energy of the system is then equal to the sum of the rotovibrational and the translational energies. At the same time, the supermolecular wavefunctions are products of rotovibrational and translational functions. Let r designate the set of the rotovibrational quantum numbers and t the set of translational quantum numbers, the equation for yo may be written [314]... 

Using the previously derived expressions for q, we can now obtain expressions for each of the entropy terms. Equation 8.59 gives the molecular partition function for three-dimensional translational motion of a gas. Substituting this qtrans into Eq. 8.102, we obtain... 

This equation is not exact its derivation involved several approximations. If we include the translational motion of the molecule, then (4.3) gives for the total energy of the molecule... 

Here N designates the normalization factor. Clearly this equation in integrated form is the product of Gaussian and Lorentzian distribution functions 0mg and 0m define the line-widths of the two components, respectively. Here, the former represents Eq. (17) to a sufficient approximation for 0m 2 G and the latter was introduced to express the coupled rotational and/or the translational motion of proton pairs in the polymer, discussed by Pechhold53. ... 

Before discussing other results it is informative to first consider some correlation and memory functions obtained from a few simple models of rotational and translational motion in liquids. One might expect a fluid molecule to behave in some respects like a Brownian particle. That is, its actual motion is very erratic due to the rapidly varying forces and torques that other molecules exert on it. To a first approximation its motion might then be governed by the Langevin equations for a Brownian particle 61... 

Equation (237) shows that when the coupling constant is larger than the critical value Xc = 1 the ansatz [Eq. (234)] leads to an acceptable solution. This implies that for X < 1, density fluctuations decay to zero for a long time but for X > 1 they decay to a finite value/. The value off increases from/ = 1/2 for X — 1 to / = 1 for X —> oo. Thus the spectrum of density fluctuation exhibits a delta function peak at zero frequency, with strength / which is the characteristic of a glassy phase. Thus in the glass phase the translational motion is frozen in and the vibrational motion around the arrested position is described by (z). 

Equation 2.17 is of the form A = PDP-1. The 9x9 Hessian for a triatomic molecule (three Cartesian coordinates for each atom) is decomposed by diagonalization into a P matrix whose columns are direction vectors for the vibrations whose force constants are given by the k matrix. Actually, columns 1, 2 and 3 of P and the corresponding k, k2 and k3 of k refer to translational motion of the molecule (motion of the whole molecule from one place to another in space) these three force constants are nearly zero. Columns 4, 5 and 6 of P and the corresponding k4, k5 and k6 of k refer to rotational motion about the three principal... 

The possible energy levels are determined by Schrodinger s wave equation [Reif, 1965], For translational motion of a particle, the wave equation takes the form... 

Thermodynamics deals with relations among bulk (macroscopic) properties of matter. Bulk matter, however, is comprised of atoms and molecules and, therefore, its properties must result from the nature and behavior of these microscopic particles. An explanation of a bulk property based on molecular behavior is a theory for the behavior. Today, we know that the behavior of atoms and molecules is described by quantum mechanics. However, theories for gas properties predate the development of quantum mechanics. An early model of gases found to be very successftd in explaining their equation of state at low pressures was the kinetic model of noninteracting particles, attributed to Bernoulli. In this model, the pressure exerted by n moles of gas confined to a container of volume V at temperature T is explained as due to the incessant collisions of the gas molecules with the walls of the container. Only the translational motion of gas particles contributes to the pressure, and for translational motion Newtonian mechanics is an excellent approximation to quantum mechanics. We will see that ideal gas behavior results when interactions between gas molecules are completely neglected. 

Van der Waals reasoned that when using an ideal-gas-like treatment, the excluded volume should be subtracted from the actual volume of the container to give the free volume in which the molecules could undergo translational motion. As the excluded volume is proportional to the number of molecules (or moles) in the container, it is written as nb, where b is the excluded volume per mole, and the free volume is V — nb. Thus, according to van der Waals, the repulsive force modifies the ideal gas equation to... 

So far we have concentrated on the behavior of particles in translational motion. If the particles are sufficiently small, they will experience an agitation from random molecular bombardment in the gas, which will create a thermal motion analogous to the surrounding gas molecules. The agitation and migration of small colloidal particles has been known since the work of Robert Brown in the early nineteenth century. This thermal motion is likened to the diffusion of gas molecules in a nonuniform gas. The applicability of Fick s equations for the diffusion of particles in a fluid has been accepted widely after the work of Einstein and others in the early 1900s. The rate of diffusion depends on the gradient in particle concentration and the particle diffusivity. The latter is a basic parameter directly... 

In the absence of external fields, we may take axes that move laterally with the molecule, thus eliminating translational motion. In these co-ordinates a diatomic molecule becomes equivalent to a single particle with mass ti=MaMjs/(Ma+Mb), moving in a spherically symmetrical potential U(R), where R is the intranuclear separation. The Schrodinger equation is therefore... 

The Schrodinger equation describing the nulcear motions, Eq. (1-11), contains too many variables, as the translational motion of the center-of-mass (c.o.m.) has not been separated out. When the c.o.m. motion is separated out, and when the origin of the space-fixed coordinate system is located in the c.o.m. of the nuclei, Eq. (1-11)... 

Starting from equation (2.6) with arbitrary origin we first transform to the molecular centre of mass (which ensures that the translational motion is separable), and then to the geometrical centre of the nuclei. The total transformation from arbitrary origin to the new origin is represented by... 

In our subsequent development we shall take the origin of coordinates to be at the centre of mass of the two nuclei, although we could equally well have chosen the molecular centre of mass as origin. Setting aside the translational motion of the molecule, we use equation (2.28) to represent the kinetic energy of the electrons and nuclei. To this we add terms representing the potential energy, electron spin interactions, and nuclear spin interactions. We subdivide the total Hamiltonian Xx into electronic and nuclear Hamiltonians,... 

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