Kinetic terms, neglect

An alternative way of deriving the BET equation is to express the problem in statistical-mechanical rather than kinetic terms. Adsorption is explicitly assumed to be localized the surface is regarded as an array of identical adsorption sites, and each of these sites is assumed to form the base of a stack of sites extending out from the surface each stack is treated as a separate system, i.e. the occupancy of any site is independent of the occupancy of sites in neighbouring stacks—a condition which corresponds to the neglect of lateral interactions in the BET model. The further postulate that in any stack the site in the ith layer can be occupied only if all the underlying sites are already occupied, corresponds to the BET picture in which condensation of molecules to form the ith layer can only take place on to molecules which are present in the (i — l)th layer. 

The derivative (nonadiabatic) coupling, ffy, is the term neglected in the Bom-Oppenheimer approximation that is responsible for nonadiabatic transitions between different states I and. /. It originates from the nuclear kinetic energy operator operating on the electronic wavefunctions ijf] and is given by... 

Using the first Friedmann equation (recalling that we neglect the curvature K and the kinetic term in the energy density) gives... 

Although the electron wavefunction, (r, ie e)) is primarily a function of the electron co-ordinates, it indirectly depends also on the nuclear co-ordinates because they appear in the potential terms + V f. The Bom-Oppenheimer approximation now neglects the influence of nuclear motion, i.e., the kinetic term with Tn/ on the solutions (p for the electron part of the system, by solving for the latter by the following equation ... 

In practice, the size of the kinetic energy terms neglected will be rather small compared with the enthalpy terms. For example, suppose the conditions at the second stage inlet of a steam turbine are 20 bar and 250°C, giving a local specific enthalpy from steam tables of 2904 x lO J/kg. A typical value of inlet velocity to the second stage is lOOm/s. which implies a specific kinetic eneigy of 5 x 10 J/kg, only... 

Let US consider the simplified Hamiltonian in which the nuclear kinetic energy term is neglected. This also implies that the nuclei are fixed at a certain configuration, and the Hamiltonian describes only the electronic degrees of freedom. This electronic Hamiltonian is... 

If we scale time as t = xr, then the frst term in (5.52) decreases as l/>/, while the other two are independent of friction. Therefore, at large rj the second derivative term in (5.52), as well as the kinetic energy term in the action, can be neglected, and the entire effect of friction is to change the timescale. That is, the solution to (5.52) is Q x) = Q x/ri) where Q is a function independent of rj. The instanton velocity is scaled as Q cc and the action (5.38) grows linearly with r, ... 

Equation 6-10 is the macroscopic energy balance equation, in which potential and kinetic energy terms are neglected. From tliermodynamics, the enthalpy per unit mass is expressed as... 

The metal cluster will be modeled as an infinitely deep spherical potential well with the represented by an infinitely high spherical barrier. Let us place this barrier in the center of the spherical cluster to simplify the calculations. The simple Schrodinger equation, containing only the interaction of the electrons with the static potential and the kinetic energy term and neglecting any electron-electron interaction, can then be solved analytically, the solutions for the radial wave functions being linear combinations of spherical Bessel and Neumann functions. 

The Born-Oppenheimer approximation allows the two parts of the problem to be solved independently, so we can construct an electronic Hamiltonian which neglects the kinetic energy term for the nuclei ... 

This expression is exact within our original approximation, where we have neglected relativistic effects of the electrons and the zero-point motions of the nuclei. The physical interpretation is simple the first term represents the repulsive Coulomb potential between the nuclei, the second the kinetic energy of the electronic cloud, the third the attractive Coulomb potential between the electrons and the nuclei, and the last term the repulsive Coulomb potential between the electrons. 

In the derivation above, we have included the kinetic energy of the nuclei in the Hamiltonian and considered a stationary state. In Eq. II.3, this term has been neglected, and we have instead assumed that the nuclei have given fixed positions. It has been pointed out by Slater34 that, if the nuclei are not situated in the proper equilibrium positions, the virial theorem will appear in a slightly different form. (A variational derivation has been given by Hirschfelder and Kincaid.11)... 

In the last column of Table 4.2, values are given for G /A calculated by ignoring the effect of changes in the kinetic energy of the gas, that is by neglecting the log term in equation 4.57. It will be noted that the effects are small ranging from about 2 per cent for P2a = 9 MN/m2 to about 13 per cent for P2a = 4 MN/m2. 

There are two objectives of setting up a kinetic mathematical model for chiral products. The first is the elucidation of the reaction mechanism with identification of the rate-controlhng step. The second is to derive a mathematical expression for the selectivity in terms of the ratio of the major product to the minor product. Then, based upon this expression, the reaction conditions such as pressure or feed ratio are changed to increase the selectivity. However, when the enantiomeric purity is over 99%, the selectivity is extremely high hence, the reaction mechanism for the major manifold can be neglected to simplify the establishment of the kinetic model. 

In chemical processes, the kinetic and potential energy terms are usually small compared with the heat and work terms, and can normally be neglected. 

The kinetics of the irreversible reaction A -> B + C has been studied in a laboratory scale recycle reactor. A nonporous solid catalyst is being used to obtain an appreciable reaction rate. The data reported by your technician are shown in Figure 6P.6. Each of the sets of data was recorded holding the partial pressures of the other species constant. The homogeneous gas phase reaction may be neglected. The reaction rate is expressed in terms of the... 

For the reactions in question no term may be neglected and it was necessary to carefully plan the experimental program to facilitate evaluation of all five kinetic parameters. On the basis of the data below determine these parameters when the weak acid employed is acetic acid. 

If the logarithmic term in the denominator (which comes from the change in kinetic energy of the gas) is neglected, the resulting equation is called the Weymouth equation. Furthermore, if the average density of the gas is used in the Weymouth equation, i.e.,... 

---