Partial mole density

At any time during the deposition, the mole fraction of Mn in the electrodeposit, Mft Mn, can be expressed as the ratio of the partial current densities,... 

Thus, in terms of partial current densities, let the reaction be proceeding in the forward direction under equilibrium conditions and let its velocity be represented by j. The arrow represents direction. If you like, you can say, From left to right. Now we choose to use electrochemical terminology, and then the full definition of j would be the partial current density in one direction. (The density refers to the standardization with respect to the area one is considering, a square centimeter [cm2] or square meter [m2].) If we were talking about a chemical reaction, then we should use the symbol v and the definition then would be the rate of reaction in one direction. In a chemical reaction, v would be the number of gram moles per square centimeter per second (g mol cm-2 s-1) (or per square meter per second [m-2 s-1]). 

The number of independent equilibrium relationships among the partial mole numbers, N, is equal to the number of independent steps in the fast equimolecular mechanism, which we call R. A ere changes in N are possible, R may not exceed (S — C — 1), where S is the total number of species being considered (the range of the index i), and C is the number of stoichiometric components of these species (the independent chemical elements). When R = S — C — 1), the composition to which the system tends through the fast reactions alone is a imique function of N, determined by the temperature and the initial composition. This function is conspicuously independent of density. 

The enrichment of the slow dissolving component, B, in an alloy surface under simultaneous dissolution conditions may be rationalized by a model of alloy dissolution that is based on the simplifying assumptions (1) that a homogeneous solid solution may be described as a heterogeneous dispersion of atomic dimensions with area fraction (surface mole fraction) X j for component j, and (2) that the alloy components dissolve independently. The partial current density ij of an alloy component j will then be given by ij = i -X j, where i is the current density of the pure metal, j, and for a binary alloy A-B, the total current density of alloy dissolution. 

The OZ equation contains the partial number densities that are defined as p, = Ni/V, where N, is the number of molecules of species i within the volume V. One can also define the mole fraction of species i through p = xfi. The OZ equation can be conveniently cast into the following matrix form. 

At pressures to a few bars, the vapor phase is at a relatively low density, i.e., on the average, the molecules interact with one another less strongly than do the molecules in the much denser liquid phase. It is therefore a common simplification to assume that all the nonideality in vapor-liquid systems exist in the liquid phase and that the vapor phase can be treated as an ideal gas. This leads to the simple result that the fugacity of component i is given by its partial pressure, i.e. the product of y, the mole fraction of i in the vapor, and P, the total pressure. A somewhat less restrictive simplification is the Lewis fugacity rule which sets the fugacity of i in the vapor mixture proportional to its mole fraction in the vapor phase the constant of proportionality is the fugacity of pure i vapor at the temperature and pressure of the mixture. These simplifications are attractive because they make the calculation of vapor-liquid equilibria much easier the K factors = i i ... 

Relative Humidity (rh). Relative humidity is the ratio of the mole fraction of water vapor present in the air to the mole fraction of water vapor present in saturated air at the same temperature and barometric pressure it approximately equals the ratio of the partial pressure (or density) of the water vapor in the air to the saturation pressure (or density) of water vapor at the same temperature. 

Let us now ask how this value could be used as a basis from which to measure the local disturbance of the water structure that will be caused by each ionic field. The electrostriction round each ion may lead to a local increase in the density of the solvent. As an example, let us first consider the following imaginary case let us suppose that in the neighborhood of each ion the density is such that 101 water molecules occupy the volume initially occupied by 100 molecules and that more distant molecules are not appreciably affected. In this case the local increase in density would exactly compensate for the 36.0 cm1 increment in volume per mole of KF. The volume of the solution would be the same as that of the initial pure solvent, and the partial molal volume of KF at infinite dilution would be zero. Moreover, if we had supposed that in the vicinity of each ion 101 molecules occupy rather less than the volume initially occupied by 100 molecules, the partial molal volume of the solute would in this case have a negative value. 

Whereas the Mg atoms are in contact with each other and the Cu atoms are in contact with each other, the Cu partial structure floats inside the Mg partial structure. The hard sphere model proves to be insufficient to account for the real situation atoms are not really hard. The principle of the most efficient filling space should rather be stated as the principle of achieving the highest possible density. Indeed, this shows up in the actual densities of the Laves phases they are greater than the densities of the components (in some cases up to 50 % more). For example, the density of MgCu2 is 5.75 g cm-3, which is 1% more than the mean density of 5.37 g cm-3 for 1 mole Mg + 2 moles Cu. Therefore,... 

Fig. 5. Temperature dependence of the stoichiometry coefficient v for the homogeneous decomposition of hydrazine from shock tube data. , ps2h4 = 11.6, pt = 0.7 9, Fn2h4 = 5.0, Px 1-2 , Pn2h4 = 2.2, pt = 2.5 O, Pn2h4 = 1-2, pt = 1.2 Pn2h4 = 11 Pt = 2.5 Q, Pn2h4 = 11, Pt = 5.7, Q, pm2h4 = 7.4 px = 7.5. pn2h4 = partial density of N2H4 in 10 5 mole.l-1 pt — total density at reaction conditions in 10-2 mole.l". (From Michel and Wagner28.)...

If kinetic data are to be used, it is necessary to transform the variables to conform with those of the partial equilibrium model. The units used in the model equations for and nj are moles formed/kg of solution. Thus the mass of solution in the reacting system from which the kinetic data comes must be known. Frequently, one will know the volume and have to approximate the density. A relation between and t is also needed. For this, the mass of solid originally present must be known. The amount of solid reacing, -ANg, for a time interval At can be obtained from rate curves or calculated from an integrated rate equation. The fraction of the original mass reacting in the time interval gives an approximate value of 5, e.g.,... 

In gases the most used quantity for the density of species j is the partial pressure Pj. This can be related to concentration and mole fraction Vj through the relations... 

We always use Cj in moles per liter (or in moles per cubic decimeter or 1 kilomole/m for the SI purist) as the only unit of concentration. The subscript j always signifies species, while the subscript i always signifies reaction. We use j as the species designation and species A as the key reactant. For gases the natural concentration unit is partial pressure Pj, but we always convert this to concentration, Cj = Pj RT, before writing the mass-balance equations. Conversion X means the fiaction of this reactant that is consumed in the reactor, Ca = Cao( 1 — X), but we prefer to use C i rather than X and find the conversion after we have solved the equation in terms of G. We cannot use this unit of density of a species when the density of the fluid varies with conversion, but we prefer to do so whenever possible because the equations are simpler to write and solve. 

Note also that selectivity becomes more complicated in systems where densities and mole numbers change. The above definition is straightforward once a suitable basis is chosen (such as moles of reactant Nao or molar flow rate Fao)> but in terms of concentrations and partial pressures one must be careful in substituting these quantities in the preceding equation. 

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-... 

In obtaining Equation 11 it has been assumed that the partial specific volumes v of the associating species are equal we have also assumed that the specific refractive index increments of the associating solutes are equal. In Equation 12 R is the universal gas constant (8.314 X 107 ergs/deg-mole), p is the density of the solution (gram/ml), and T is the absolute temperature. Equation 11 is also valid for the Archibald experiment but only at rm or rb, the radial positions (in the solution column of the ultracentrifuge cell) of the air-solution meniscus and of the cell... 

For the denaturation of / -Lg in 40% 2-chloroethanol, the presently reported partial specific volume measurements result in a AV value of —590 ml/mole. This value is very close to that previously reported for the denaturation of this protein by 6.4M urea, —610 ml/mole (27). Although this similarity of AV values is striking, it might be the result of a fortuitous compensation of various effects. The change in volume calculated from the difference in partial specific volumes is the sum of a number of contributions (28), such as differences in electrostriction in the two media, changes in the density of solvent components when they interact... 

Now the density of a given weight of ga.s at constant pressure is inversely proportional to the number of moles, and if is taken as the density of the undissociated gas and dj that of the partially dissociated ga.s. then ... 

For ideal solutions, the partial pressure of a component is directly proportional to the mole fraction of that component in solution and depends on the temperature and the vapor pressure of the pure component. The situation with group III-V systems is somewhat more complicated because of polymerization reactions in the gas phase (e.g., the formation of P2 or P4). Maximum evaporation rates can become comparable with deposition rates (0.01-0.1 xm/min) when the partial pressure is in the order of 0.01-1.0 Pa, a situation sometimes encountered in LPE. This problem is analogous to the problem of solute loss during bakeout, and the concentration variation in the melt is given by equation 1, with l replaced by the distance below the gas-liquid interface and z taken from equation 19. The concentration variation will penetrate the liquid solution from the top surface to a depth that is nearly independent of zlDx and comparable with the penetration depth produced by film growth. As result of solute loss at each boundary, the variation in solute concentration will show a maximum located in the melt. The density will show an extremum, and the system could be unstable with respect to natural convection. 

If we center our attention on the second method, the problem becomes one of expressing the partial derivatives of the density with respect to the mole numbers in terms of the partial derivatives of the density with respect to the molarity. The differential of the density is... 

The introduction of the volume fractions motivates the definition of two density functions, the effective density paR = dma/dva and the partial density pa = dma/dv, which relate the mass rna of ipa to its volume va and to the bulk volume v. The density functions are coupled by pa = napaR. Furthermore, p3 = nFCm Mm denotes the partial density of p3 [5], Herein, the concentration d,1, = drim/dvF relates the moles rim to the volume vF, and the molar mass Mm = dm3/drim relates the mass m3 to the moles. 

In Chapter 1, the assumption that gases and gas mixtures behave ideally at low pressures (1 bar and below) was stated. (Deviation from this with large amounts of readily condensable vapours under compression near atmospheric pressure was dealt with in Chapter 3.) The ideal gas equation, expressing the relationship between the variables pressure, volume, temperature and amount (number of moles) of gas, together with the expression of pressure in terms of particle number density (n) and Dalton s law of partial pressures, allow many calculations useful to vacuum technology to be carried out (Examples... 

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