Chemical potential pressure coefficient

The coefficient of dE is the inverse absolute temperature as identified above. We now define the pressure and chemical potential of the system as... 

Again, the coefficients of the terms within the brackets can be written as appropriate second derivatives of another function, but in this case the function is a new function for which the independent variables are the temperature, the pressure, the chemical potential of the first component, and the mole numbers of all components except the first. If this new function function is given the symbol 4>, then... 

A mathematical model of the pressure transmission - chemical potential experiment indicates that the two diffusivities / /, and Dc and the reflection coefficient 7Z can robustly be identified from the downstream pressure response. The analysis also confirms the experimental practice of identifying 1Z with the membrane efficiency coefficient deduced from the minimum of the downstream pressure ([7, 11]) whose justification is based on simplified considerations. 

Material motion can be driven by a gradient in concentration or pressure or chemical potential. The coefficients mainly used for the three circumstances are D K, and D, as follows. (For K, Kj, and k, see below.)... 

The conditions for phase equilibrium were presented equalities of temperature, pressure and chemical potential of each species in all phases. The evaluation of chemical potentials of mixtures was discussed, and the following methods and approximations were presented ideal mixture, Henry s law, and simple correlations for activity coefficients. 

Here ns is the amount of substance of stationary liquid, pi is the saturated vapour pressure of the solute at temperature r. Bag is the mixture virial coefficient for solute 4- carrier gas interaction, Bcc is the virial coefficient of the carrier gas, Fjj is the partial molar volume of the solute at infinite dilution in the solvent, is the molar volume of pure liquid A, and pi and po are the column inlet and outlet pressures. The chemical potential at infinite dilution can be calculated by measuring the retention volume of an infinitely small sample for various inlet and outlet pressures and extrapolation to zero pressure drop across the column. Everett and Stoddart proposed using equation (33) to determine the mixture second virial coefficients. The precision in Bag from this method is nearly equivalent to the best static methods. The assumptions required to derive the above equation have been examined by a number of authors. - ... 

The osmotic coefficient has its origin, as you might suspect, through its connection with osmotic pressures. The chemical potential of water in an aqueous solution is inherently less than that of the pure solvent, as you can see from Pi— = RTlnxi. If this solution is separated from the pure solvent by a membrane permeable only to the solvent, pure solvent will pass through the membrane into the solution in an attempt to remove this difference. If the... 

A2. In the second interpretation, Eq. (10.27), the driving force dc/dx is identical to a gradient of chemical potential hence, the equation for D involves the activity coefficient y of the polymer. The equation fits into the analysis of an interacting multicomponent system. Of course, it must be remembered that osmotic pressure and chemical potential are closely related, as discussed in Chapter 9. 

An additional technique for determining Margules coefficients should also be examined. When two or more phases coexist in equilibriiom at fixed temperature and pressure, the chemical potentials for each of the several components are equal in all of the phases. If a and b denote two phases coexisting in equilibrium in the binary system 1-2, the equivalence of chemical potentials, yf = leads to equilibriim expressions having a form... 

At finite concentrations this formula needs modifying in two ways. In the first place, diffusion is governed by the osmotic pressure, or chemical potential, gradient (not, strictly, by the concentration gradient), so that the mean activity coefficient of the electrolyte must be taken into account. In the second place, ionic atmosphere effects must be allowed for. In diffusion, unlike conductance, the two ions are moving in the same direction, and the motion causes no disturbance of the symmetries of the ionic atmospheres there is therefore no relaxation effect. There is a small electrophoretic effect, however, the magnitude of which for dilute solutions has been worked out by Onsager, and the most accurate measurements support the extended formula based on these corrections. 

Panagiotopoulos et al. [16] studied only a few ideal LJ mixtures, since their main objective was only to demonstrate the accuracy of the method. Murad et al. [17] have recently studied a wide range of ideal and nonideal LJ mixtures, and compared results obtained for osmotic pressure with the van t Hoff [17a] and other equations. Results for a wide range of other properties such as solvent exchange, chemical potentials and activity coefficients [18] were compared with the van der Waals 1 (vdWl) fluid approximation [19]. The vdWl theory replaces the mixture by one fictitious pure liquid with judiciously chosen potential parameters. It is defined for potentials with only two parameters, see Ref. 19. A summary of their most important conclusions include ... 

For a solution of a non-volatile substance (e.g. a solid) in a liquid the vapour pressure of the solute can be neglected. The reference state for such a substance is usually its very dilute solution—in the limiting case an infinitely dilute solution—which has identical properties with an ideal solution and is thus useful, especially for introducing activity coefficients (see Sections 1.1.4 and 1.3). The standard chemical potential of such a solute is defined as... 

The activity coefficient of the solvent remains close to unity up to quite high electrolyte concentrations e.g. the activity coefficient for water in an aqueous solution of 2 m KC1 at 25°C equals y0x = 1.004, while the value for potassium chloride in this solution is y tX = 0.614, indicating a quite large deviation from the ideal behaviour. Thus, the activity coefficient of the solvent is not a suitable characteristic of the real behaviour of solutions of electrolytes. If the deviation from ideal behaviour is to be expressed in terms of quantities connected with the solvent, then the osmotic coefficient is employed. The osmotic pressure of the system is denoted as jz and the hypothetical osmotic pressure of a solution with the same composition that would behave ideally as jt. The equations for the osmotic pressures jt and jt are obtained from the equilibrium condition of the pure solvent and of the solution. Under equilibrium conditions the chemical potential of the pure solvent, which is equal to the standard chemical potential at the pressure p, is equal to the chemical potential of the solvent in the solution under the osmotic pressure jt,... 

Generally, the activity coefficient y depends on the composition of solution. In the ranges of our narrow purposes of investigations of the macromolecules chemical potential conformation term influence on the osmotic pressure of polymeric solutions we will be neglect by the change of y lying y = const in all range of the macromolecules concentrations into solution. This permits to write... 

As seen from Eq. (130) an activity coefficient may deviate significantly from unity at higher salt concentrations. The activity coefficient can therefore also be used as a measure of the deviation of the salt solution from a thermodynamically ideal solution. If the chemical potential of a solute in a (pressure-dependent) standard state of infinite dilution is /x°, we find the standard partial molar volume from... 

In this chapter, we shall consider the methods by which values of partial molar quantities and excess molar quantities can be obtained from experimental data. Most of the methods are applicable to any thermodynamic property J, but special emphasis will be placed on the partial molar volume and the partial molar enthalpy, which are needed to determine the pressure and temperature coefficients of the chemical potential, and on the excess molar volume and the excess molar enthalpy, which are needed to determine the pressure and temperature coefficients of the excess Gibbs function. Furthermore, the volume is tangible and easy to visualize hence, it serves well in an initial exposition of partial molar quantities and excess molar quantities. 

Notation-. T is the temperature, Vi the fluid velocity, II,j the viscous pressure tensor, Jg the heat current density, p its chemical potential, the current density of molecular species a, v J the stoichiometric coefficient (13), and Wp the speed of reaction p. 

The coefficients Lu, L2A, and L34 describe the viscous flow contributions of the transport of all three species in a total pressure gradient totai- Because a pressure gradient also imposes a chemical potential gradient on each species (eq 24), experimentally, there is always a superposition of diffusive and viscous flow e.g., for the description of the water flux in a total pressure gradient, all coefficients must be included, i.e.. 

Instead of the dilute solution approach above, concentrated solution theory can also be used to model liquid-equilibrated membranes. As done by Weber and Newman, the equations for concentrated solution theory are the same for both the one-phase and two-phase cases (eqs 32 and 33) except that chemical potential is replaced by hydraulic pressure and the transport coefficient is related to the permeability through comparison to Darcy s law. Thus, eq 33 becomes... 

The membrane and diffusion-media modeling equations apply to the same variables in the same phase in the catalyst layer. The rate of evaporation or condensation, eq 39, relates the water concentration in the gas and liquid phases. For the water content and chemical potential in the membrane, various approaches can be used, as discussed in section 4.2. If liquid water exists, a supersaturated isotherm can be used, or the liquid pressure can be assumed to be either continuous or related through a mass-transfer coefficient. If there is only water vapor, an isotherm is used. To relate the reactant and product concentrations, potentials, and currents in the phases within the catalyst layer, kinetic expressions (eqs 12 and 13) are used along with zero values for the divergence of the total current (eq 27). 

From Fig. 19.3a-c, and as opposed to purely sorption controlled processes, it can be seen that during pervaporation both sorption and diffusion control the process performance because the membrane is a transport barrier. As a consequence, the flux 7i of solute i across the membrane is expressed as the product of both the sorption (partition) coefficient S, and the membrane diffusion coefficient Di, the so-called membrane permeability U, divided by the membrane thickness f and times the driving force, which maybe expressed as a gradient of partial pressures in place of chemical potentials [6] ... 

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-... 

Here, the quantities jn ° and ji are, respectively, the chemical potentials of pure solvent and of the solvent at a certain biopolymer concentration V is the molar volume of the solvent and n is the biopolymer number density, defined as n C/M, where C is the biopolymer concentration (% wt/wt) and M is the number-averaged molar weight of the biopolymer. The second virial coefficient has (weight-scale) units of cm mol g. Hence, the more positive the second virial coefficient, the larger is the osmotic pressure in the bulk of the biopolymer solution. This has consequences for the fluctuations in the biopolymer concentration in solution, which affects the solubility of the biopolymer in the solvent, and also the stability of colloidal systems, as will be discussed later on in this chapter. 

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