Mass-velocity effect

For a micro-channel connected to a 100 pm T-junction the Lockhart-Martinelli model correlated well with the data, however, different C-values were needed to correlate well with all the data for the conventional size channels. In contrast, when the 100 pm micro-channel was connected to a reducing inlet section, the data could be fit by a single value of C = 0.24, and no mass velocity effect could be observed. When the T-junction diameter was increased to 500 pm, the best-fit C-value for the 100 pm micro-channel again dropped to a value of 0.24. Thus, as in the void fraction data, the friction pressure drop data in micro-channels and conventional size channels are similar, but for micro-channels, significantly different data can be obtained depending on the inlet geometry. 

For heavy elements, all of the above non-relativistic methods become increasingly in error with increasing nuclear charge. Dirac 47) developed a relativistic Hamiltonian that is exact for a one-electron atom. It includes relativistic mass-velocity effects, an effect named after Darwin, and the very important interaction that arises between the magnetic moments of spin and orbital motion of the electron (called spin-orbit interaction). A completely correct form of the relativistic Hamiltonian for a many-electron atom has not yet been found. However, excellent results can be obtained by simply adding an electrostatic interaction potential of the form used in the non-relativistic method. This relativistic Hamiltonian has the form... 

Thus, the main relativistic effects are (1) the radical contraction and energetic stabilization of the s and p orbitals which in turn induce the radial expansion and energetic destabilization of the outer d and f orbitals, and (2) the well-known spin-orbit splitting. These effects will be pronounced upon going from As to Sb to Bi. Associated with effect (1), it is interesting to note that the Bi atom has a tendency to form compounds in which Bi is trivalent with the 6s 6p valence configuration. For this tendency of the 6s electron pair to remain formally unoxidized in bismuth compounds (i.e. core-like nature of the 6s electrons), the term inert pair effect or nonhybridization effect has been often used for a reasonable explanation. In this context, the relatively inert 4s pair of the As atom (compared with the 5s pair of Sb) may be ascribed to the stabilization due to the d-block contraction , rather than effect (1) . On the other hand, effect (2) plays an important role in the electronic and spectroscopic properties of atoms and molecules especially in the open-shell states. It not only splits the electronic states but also mixes the states which would not mix in the absence of spin-orbit interaction. As an example, it was calculated that even the ground state ( 2 " ) of Bij is 25% contaminated by Hg. In the Pauli Hamiltonian approximation there is one more relativistic effect called the Dawin term. This will tend to counteract partially the mass-velocity effect. 

There are several effects of relativity on chemical bonding and spectroscopic properties. We first focus on a relativistic effect called the mass-velocity effect. As the electron approaches the speed of light, its mass increases with velocity since in special theory of relativity mass is no longer a constant and varies as... 

In recent years, there has been an increasing interest in the inclusion of relativistic effects for molecules containing heavy atoms. One of the most practical yet reliable methods is to use relativistically derived effective core potentials. Major relativistic effects such as the Darwin and mass-velocity effects are easily taken into account in the form of a spin-free (SF) one-electron operator. The spin-orbit (SO) interaction is in general too strong to be considered as a small perturbation, and therefore should be treated explicitly as a part of the total Hamiltonian. 

The first term, Hj, is the spin-orbit (one electron term) and spin-other-orbit (two electron term) couplings, which are the topic of the following subsection. The second term Hf contains the spin-spin coupling term and Fermi contact interaction. Both the Hj and f/ can lift degeneracy in multiplets. The parameter Hf is the Dirac correction term for electron spin and Ff is the classical relativistic correction to the interaction between electrons due to retardation of the electromagnetic field produced by an electron. The parameter H is the so-called mass-velocity effect, due to the variation of electron mass with velocity. Finally, H is the effect of external electric and magnetic fields. 

Summarizing then, it has been shown that for at least three reactions, that under strong mass transfer control with a limiting gas phase conponent, liquid mass velocity effects on the global reaction rates showing a minimum can be obtained due to the countering effects of liquid velocity on gas/catalyst and liquid/cata-lyst contacting. 

To quantify this model further a more thorough knowledge of the physics of the flow in trickle beds is needed. However, at this time it can be used as a criteria for when to expect a liquid mass velocity effect and as a vehicle for checking the consistency of experimental data. 

All sextet states of lr3 have smaller 6s populations, especially at the apex atoms compared to the octet states. Since relativistic mass-velocity effect stabilizes the 6s orbitals, the octet electronic states with enhanced 6s populations are favored for lr3 over the sextet states. The 6p populations of all electronic states in this table are between 0.13 and 0.20. The apex atoms of the octet states exhibit smaller 5d population compared to the base atoms, while for all sextet, quartet and doublet states the apex atoms exhibit more 5d populations than the base atoms. The 6s Mulliken populations of all of the electronic states of Ir3 are larger compared to the corresponding 5s Mulliken populations of Rh3. This is compensated by slightly smaller 5d and 6p populations in the case of lr3. The enhanced 6s populations for all of the electronic states of lr3 can be explained based on the relativistic mass-velocity contraction of the 6s orbital of the Ir atom, which stabilizes this orbital thus leading to enhanced 6s populations. 

---