Chemical potential mass density

Find the number density of positrons resulting from pair production by y-rays in thermal equilibrium in oxygen at a temperature of 109 K and a density of 1000 gmcm-3, using the twin conditions that the gas is electrically neutral and that the chemical potentials of positrons and electrons are equal and opposite. (At this temperature, the electrons can be taken as non-relativistic.) The quantum concentration for positrons and electrons is 8.1 x 1028 T93/2 cm-3, the electron mass is 511 keV and kT = 86.2 T9 keV. 

In thermal equilibrium, within a quantum statistical approach a mass action law can be derived, see [12], The densities of the different components are determined by the chemical potentials ftp and fin and temperature T. The densities of the free protons and neutrons as well as of the bound states follow in the non-relativistic case as... 

It is quite likely to find dense quark matter inside compact stars like neutron stars. However, when we study the quark matter in compact stars, we need to take into account not only the charge and color neutrality of compact stars and but also the mass of the strange quark, which is not negligible at the intermediate density. By the neutrality condition and the strange quark mass, the quarks with different quantum numbers in general have different chemical potentials and different Fermi momenta. When the difference in the chemical potential becomes too large the Cooper-pairs breaks or other exotic phases like kaon condensation or crystalline phase is more preferred to the BCS phase. 

As a first step in this direction we will discuss here the two flavor color superconducting (2SC) quark matter phase which occurs at lower baryon densities than the color-flavor-locking (CFL) one, see [18, 32], Studies of three-flavor quark models have revealed a very rich phase structure (see [32] and references therein). However, for applications to compact stars the omission of the strange quark flavor within the class of nonlocal chiral quark models considered here may be justified by the fact that central chemical potentials in stable star configurations do barely reach the threshold value at which the mass gap for strange quarks breaks down and they appear in the system [20], Therefore we will not discuss here first applications to calculate compact star configurations with color superconducting quark matter phases that have employed non-dynamical quark models... 

At nonzero temperatures the mass gap decreases as a function of the chemical potential already in the phase with broken chiral symmetry. Hence the model here gives unphysical low-density excitations of quasi-free quarks. A systematic improvement of this situation should be obtained by including the phase transition construction to hadronic matter. However, in the present work we circumvent the confinement problem by considering the quark matter phase only for densities above the nuclear saturation density no, i.e. ub > 0.5 no. 

Energy release due to (anti)neutrino untrapping. The configurations for the quark stars are obtained by solving the Tolman-Oppenheimer-Volkoff equations for a set of central quark number densities nq for which the stars are stable. In Fig. 13 the configurations for different antineutrino chemical potentials are shown. The equations of state with trapped antineutrinos are softer and therefore this allows more compact configurations. The presence of antineutrinos tends to increase the mass for a given central density. 

Figure 13. Quark star configurations for different antineutrino chemical potentials r = 0, 100, 150 MeV. The total mass M in solar masses (MsUn = M in the text) is shown as a function of the radius R (left panel) and of the central number density nq in units of the nuclear saturation density no (right panel). Asterisks denote two different sets of configurations (A,B,f) and (A ,B ,f ) with a fixed total baryon number of the set.

Ca is a comparatively difficult element for the body to absorb and digest. It is essentially only available for consumption associated with various other moieties (e.g., citrate, phosphate, and other anions). Each Ca source has unique physical, structural, and chemical properties such as mass, density, coordination chemistry, and solubility that are largely determined by the anions associated with the Ca +. Aqueous solubility of various Ca salts can vary markedly and comparisons are frequently made under standardized conditions. The water solubility of CCM is moderate when ranked versus other Ca sources frequently used as dietary supplements and food/beverage fortificants. The solubility of CCM (6 2 3 molar ratio) is 1.10-g salt in 100 ml of H2O at 25 °C (Fox et ah, 1993a). Table 6.4 lists the solubility of various Ca sources in water at specific temperatures, and also includes some information on potential sensory characteristics. 

Temperature, Heat capacity. Pressure, Dielectric constant. Density, Boiling point. Viscosity, Concentration, Refractive index. Enthalpy, Entropy, Gibbs free energy. Molar enthalpy. Chemical potential. Molality, Volume, Mass, Specific heat. No. of moles. Free energy per mole. 

As can be seen from the expression for the driving force in terms of the chemical potential differences, which are related to the differences in temperature and concentration, the two transporting processes, heat transfer and mass transfer, are coupled in crystal growth. The degree of contribution from the respective transport process is determined by the degree of condensation of the environmental (ambient) phase. To grow crystals in a diluted ambient phase, a condensation process is required, and so mass transfer plays an essential role. The contribution of heat generated by crystallization in this case is small compared with that of the mass transfer. However, for crystallization in a condensed phase, such as a melt phase, heat transfer plays the essential role, and the contribution from the mass transfer will be very small, because the difference in concentration (density) between the solid and liquid phases is very small, smaller, say, than 1 or 2%. It is therefore necessary to classify the types of ambient phases and to be familiar with their respective characteristics from this standpoint. 

In these equations is the partial molal free energy (chemical potential) and Vj the partial molal volume. The Mj are the molecular weights, c is the concentration in moles per liter, p is the mass density, and z, is the mole fraction of species i. The D are the multicomponent diffusion coefficients, and the are the multicomponent thermal diffusion coefficients. The first contribution to the mass flux—that due to the concentration gradients—is seen to depend in a complicated way on the chemical potentials of all the components present. It is shown in the next section how this expression reduces to the usual expressions for the mass flux in two-component systems. The pressure diffusion contribution to the mass flux is quite small and has thus far been studied only slightly it is considered in Sec. IV,A,6. The forced diffusion term is important in ionic systems (C3, Chapter 18 K4) if gravity is the only external force, then this term vanishes identically. The thermal diffusion term is impor-... 

Table 1. Relationship between X and the physical solute properties using different FFF techniques [27,109] with R=gas constant, p=solvent density, ps=solute density, co2r=centrifugal acceleration, V0=volume of the fractionation channel, Vc=cross-flow rate, E=electrical field strength, dT/dx=temperature gradient, M=molecular mass, dH=hydrodynamic diameter, DT=thermal diffusion coefficient, pe=electrophoretic mobility, %M=molar magnetic susceptibility, Hm=intensity of magnetic field, AHm=gradient of the intensity of the magnetic field, Ap = total increment of the chemical potential across the channel...

Clearly, Eq. (13) concerns the electrostatic interactions only, so that a suitably chosen hard-core contribution, e.g. of Camahan-Starling type [25] must be added to the free energy densities. Differentiation with respect to the densities of the species finally yields the chemical potential and the activity coefficients required for evaluating the mass action law determining the concentrations of free ions and ion pairs. 

In the end of thermonuclear evolution, the core of a massive star can lose mechanical stability for various reasons. In the stellar mass range 8M < M < 20M a partially degenerate core with mass close to the Chandrasekhar limit Mcore Men and high density (p 109 — 1010 g/cm3) appears. Under these physical conditions, the chemical potential of degenerate electrons becomes so high that neutronisation reactions e + (A, Z) (A, Z — 1) + ve... 

The driving force in these processes is the deviation from the equilibrium and thus the chemical potential //. The mass transfer ceases, when a thermodynamic equilibrium is realized at the interface. The possible density or concentration... 

Therefore, each realisable reaction is comparable to a kind of scale which allows the comparison of chemical potentials or their sums, respectively. But the measurement is often impossible due to any inhibitions, i.e., the scale is jammed. If there is a decline in potential from the left to the right side, that only means that the process can proceed in this direction in principle however, it does not mean that the process will actually run. Therefore, a potential drop is a necessary but not sufficient condition for the reaction considered. The problem of inhibitions can be overcome if appropriate catalysts are available or indirect methods including chemical (using the mass action law), calorimetric, electrochemical and others can be used. Because we are interested in a first knowledge of the chemical potential, we assume for the moment that all these difficulties have been overcome and consider the values as given, just as we would consult a table when we are interested in the mass density or the electric conductivity of a substance. ... 

The role of the metal in double layer properties can be understood in greater detail when the system is examined on the basis of the jellium model. This model was developed to describe the electron gas within sp metals. It can be used to estimate several properties of interest, including the chemical potential of an electron in the metal, the extent of electron overspill, and the work function of the metal. More recently, it has been extended to describe metal surfaces in contact with polar solvents [26]. In its simplest form, the metal atoms in the metal are modeled as a uniform positive background for the electron gas, no consideration being given to their discrete nature and position in the metal lattice. The most important property of the system is the average electron density, N ), which depends on the number of metal atoms per unit volume and the number of valence electrons per atom, n. Thus, if pjj, is the mass density of the metal, and M, its atomic mass... 

Thermal radiation may be characterized as a photon gas, consisting of zero-mass bosons with zero chemical potential. Without proof, the photon number density may be written [6, Sect. 2.3],... 

A = difference in full energy derivative, Equation 4 = dimensionless temperature = generic critical exponent p. = chemical potential p = mass density a, = rotor speed... 

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