Dilute occupation

Show that in the dilute occupation case, Eq. (25.2-11) also applies to boson molecules. 

The parameters a and have the same meaning in all cases. If the dilute occupation approximation cannot be used, Eqs. (25.2-26) and (25.2-29) can be rewritten for noninteracting fermions and bosons ... 

Identical molecules are indistinguishable from each other and the sums are not independent of each other, so that Eq. (27.1 -25) is not usable. There are two reasons for this. First, if the molecules are fermions, any term in which two or more molecules are in the same state must be deleted from the sums. If the volume of the gas is very large and if the gas is dilute, the total number of molecular states available to be occupied can be much larger than the number of molecules in the system. We call this the case of dilute occupation of molecule states, and have assumed this case to apply in Chapters 25 and 26. In this case the forbidden sets of quantum numbers for a system of fermions occur in only a small fraction of the terms of the sum, and these terms can be left in the sum without serious error. This approximation also removes the difference between fermion and boson molecules. 

An electric current in a metallic conductor consists of moving electrons. The Drude model pictures a metal as consisting of mobile electrons and positively charged cores, which are the ions produced when the conduction electrons are removed from the atoms. Dilute occupation of the electron states is assumed. The electrons are pictured as colliding with the cores, impeding their motion. We define a period of time x by... 

Ensher J R, Jin D S, Mathews M R, Weman C E and Cornell E A 1996 Bose-Einstein condensation in a dilute gas measurement of energy and ground-state occupation Phys. Rev. Lett. 77 4984-7... 

The logic that leads us to this last result also limits the applicability of the ensuing derivation. Applying the fraction of total lattice sites vacant to the immediate vicinity of the first segment makes the model descriptive of a relatively concentrated solution. This is somewhat novel in itself, since theories of solutions more commonly assume dilute conditions. More to the point, the model is unrealistic for dilute solutions where the site occupancy within the domain of a dissolved polymer coil is greater than that for the solution as a whole. We shall return to a model more appropriate for dilute solutions below. For now we continue with the case of the more concentrated solution, realizing... 

When it approved the New Animal Drug AppHcation (NADA) of formalin, FDA ruled that use of formalin for fisheries was safe for humans and the environment. They ruled that effluents from fish treatments at 250 mg/L should be diluted 10 times and from egg treatments 75 times if 1,000 —2,000 mg/L were used (10,11). Before registering the compound, FDA also addressed carcinogenicity by stating it was not concerned about human exposure from either water or fish treated with formalin. The U.S. Fish and Wildlife Service (USFWS) has procedural guidelines that should protect workers from harm fill levels of formalin. Calculations based on treatment levels demonstrated that a fishery worker is exposed to not more than 0.117 mg/L formalin in the air, well below the levels estabUshed by the U.S. Occupational Safety and Health Administration to protect workers. 

Dilution of fumes in these areas is generally required for one of two reasons either to reduce the level of harmful (toxic or irritant) fumes to a safe level, normally below the OES (Occupational Exposure Standard) or to dilute offensive odors. Care must be taken with the latter to ensure that the problem is not merely passed on to neighbors. If it is, then a local extract with air-cleaning equipment will be preferred if it is practical. Indeed, a local extract system is always preferable, since it removes the problem at source, resulting in a cleaner environment within the building. 

Equation 13 has an important implication a clathrate behaves as an ideally dilute solution insofar as the chemical potential of the solvent is independent of the nature of the solutes and is uniquely determined by the total solute concentrations 2K yK1.. . 2x yKn in the different types of cavities. For a clathrate with one type of cavity the reverse is also true for a given value of fjiq (e.g. given concentration of Q in a liquid solution from which the clathrate is being crystallized) the fraction of cavities occupied 2kVk s uniquely determined by Eq. 13. When there are several types of cavities, however, this is no longer so since the individual occupation numbers 2k2/ki . ..,2k yKn, and hence the total solute concentration... 

Dermal Effects. Humans that were experimentally exposed to 200 ppm of trichloroethylene vapor for 7 hours experienced dry throats (40% of the subjects), begiiming after 30 minutes (Stewart et al. 1970). The subjects experiencing these symptoms did not experience them when exposed in the same manner on 5 other consecutive days. These effects are presumed to be due to direct contact with the vapor. Skin irritation and rashes have resulted from occupational exposure to trichloroethylene (Bauer and Rabens 1974 El Ghawabi et al. 1973). The dermal effects are usually the consequence of direct skin contact with concentrated solutions, but occupational exposure also involves vapor contact. Adverse effects have not been reported from exposure to dilute aqueous solutions. 

For the one-center valence contribution, there are essentially three factors that control its value (a) the radial wavefunction of the 3d orbitals, (b) the covalent dilution of the 3d orbitals with ligand orbitals, and (c) the occupation pattern of the 3d shell. An additional factor may be low-symmetry induced 3d/4p mixing. We will focus on the first three factors here. 

The potential energy function prohibits double occupancy of any site on the 2nnd lattice. In the initial formulation, which was designed for the simulation of infinitely dilute chains in a structureless medium that behaves as a solvent, the remaining part of the potential energy function contains a finite repulsion for sites that are one lattice unit apart, and a finite attraction for sites that are two lattice units apart [153]. The finite interaction energies for these two types of sites were obtained by generalizing the lattice formulation of the second virial coefficient, B2, described by Post and Zimm as [159] ... 

Pressure-composition-temperature and thermodynamic relationships of of the titanium-molybdenum-hydrogen (deuterium) system are reported. 0-TiMo exhibits Sieverts Law behavior only in the very dilute region, with deviations toward decreased solubility thereafter. Data indicate that the presence of Mo in the 0-Ti lattice inhibits hydrogen solubility. This trend may stem from two factors for Mo contents >50 atom %, an electronic factor dominates whereas at lower Mo contents, behavior is controlled by the decrease in lattice parameter with increasing Mo content. Evidence suggests that Mo atoms block adjacent interstitial sites for hydrogen occupation. Thermodynamic data for deuterium absorption indicate that for temperatures below 297°C an inverse isotope effect is exhibited, in that the deuteride is more stable than the hydride. There is evidence for similar behavior in the tritide. 

Thus, for similar values of ns, the entropy at infinite dilution for TiMo/H2 is about 3.4 eu/H more negative than that for 0-Ti/H2. An explanation for this difference might be that the molybdenum atoms in the metal lattice block potentially available interstitial sites for hydrogen occupation, resulting in nonrandom occupation of sites at low hydrogen content. Rudman (24) proposed such a role for aluminum in titanium. We presently are gathering more data for alloys of varying molybdenum composition to test this hypothesis. 

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