Single-Particle Kinetics

Weisz and co-workers organized the kinetic experiments to study the key underlying phenomena of coke burning, independent of other complicating phenomena. Their first step in this direction was to recognize that 

The rate of oxygen utilization is related to the intrinsic rate of carbon burning by the ratio of CO to CO2 produced in the burning reaction. This relation can be expressed in terms of a constant a that is defined as the moles of coke burned per mole of O2 consumed, and varies between 1 and 2. The relation between the two rates is given by 

It was found that, for intrinsic burning, the value of a was a function of the temperature only (Weisz, 1966). 

From the physical evidence it is clear that, as expected, a sharp coke interface is formed at high temperatures between the burned uid the 

The temperature dependency of the rate constant k for the first-order 

---

Flere we distinguish between nuclear coordinates R and electronic coordinates r is the single-particle kinetic energy operator, and Vp is the total pseudopotential operator for the interaction between the valence electrons and the combined nucleus + frozen core electrons. The electron-electron and micleus-micleus Coulomb interactions are easily recognized, and the remaining tenu electronic exchange and correlation... 

The third term on the right hand side of this expression is the single-particle kinetic energy of the noninteracting electrons whereas the functional Exc[p contains the additional contribution to the energy that is needed to make Eq. (8) equal to Eq. (6). 

The first step is endothermic, and the second is exothermic. The (single-particle) kinetics of the overall reaction are relatively complex (Shen, 1996). 

In this case, the performance of the reactor is governed entirely by single-particle kinetics (e.g., as given in Table 9.1). 

In equation 22.2-13, [1 - /B(t)] comes from single-particle kinetics, such as the SCM, for which results for three shapes are summarized in Table 9.1. The following example illustrates the use of the SCM model with equation 22.2-13. 

Now we want to generalize the kinetic equation for free (unbound) particles that is, we want to derive a kinetic equation for free particles that takes into account collisions between free and bound particles as well. For this purpose it is necessary to determine the binary density operator, occurring in the collision integral of the single-particle kinetic equation, at least in the three-particle collision approximation. An approximation of such type was given in Section II.2 for systems without bound states. Thus we have to generalize, for example, the approximation for/12 given by Eq. (2.40), to systems with bound states. 

The Euler equation of the variation problem now involves the functional derivatives of the single-particle kinetic energy density tr corresponding to the exact many-body density p(r), and the exchange and correlation energy density exc. Such a minimization of equation (145) yields immediately... 

As remarked above, considerable progress has resulted from use of the one-body potential of the density description in a one-electron Schrodinger equation approach. In the language of the density description, this is tantamount to treating the single-particle kinetic energy density exactly, as suggested by Kohn... 

If we add and subtract 2/3 of the single-particle kinetic energy from the right-hand-side of this equation, we find... 

As remarked in the main text, since Ts= tr dr is the single-particle kinetic energy, we cannot relate it directly to the total energy E at equilibrium. Rather,... 

Expressing the single-particle kinetic energy Ts as an orbital functional (1.37) prevents direct minimization of the energy functional (1.38) with respect to n. Instead, one commonly employs a scheme suggested by Kohn and Sham [274], which starts by writing... 

March, N. H., Angel Rubio, A., Alonso, J. A. (1999). Lowest excitation energy in atoms in the adiabatic approximation related to the single-particle kinetic energy functional. J. Phys. B At Mol. Opt Phys. 32,2173-2179. 

Similarly, the heat capacity in (5.116) receives a contribution from the properties of the single particle (kinetic and internal energies), and a second contribution due to the existence of interactions among the particles (see Section 5.5 for more details). The latter contribution is viewed within the realm of the MM approach as a relaxation term, i.e., a redistri-... 

---