Binary solution equation

Many substances cannot be obtained in the vapor state at sufficient pressures to make measurements of dielectric constant possible. To overcome this difficulty a large number of dielectric constant measurements of solutions have been carried out. As solvents substances such as benzene, dioxane or hexane which possess no electric moments, were usually chosen. For binary solutions equation (22) may be given the form... 

Table 3 shows results obtained from a five-component, isothermal flash calculation. In this system there are two condensable components (acetone and benzene) and three noncondensable components (hydrogen, carbon monoxide, and methane). Henry s constants for each of the noncondensables were obtained from Equations (18-22) the simplifying assumption for dilute solutions [Equation (17)] was also used for each of the noncondensables. Activity coefficients for both condensable components were calculated with the UNIQUAC equation. For that calculation, all liquid-phase composition variables are on a solute-free basis the only required binary parameters are those for the acetone-benzene system. While no experimental data are available for comparison, the calculated results are probably reliable because all simplifying assumptions are reasonable the... 

The Gibbs-Duhem equation is extremely important in solution chemistry and it can be seen from equation 20.171 that it provides a means of determining the activity of one component in a binary solution providing the activity of the other is known. 

The simplified equation (for the general equations, see Section IV, L) in the case of unsteady-state diffusion with a simultaneous chemical reaction in isothermal, incompressible dilute binary solutions with constant p and D and with coupled phenomena neglected is... 

Navier-Stokes equations, 318, 386-387 Nitrocellulose, 31 Nitroglycerine, 31-32 Normalization binary solutions, 156-157 multicomponent solutions, 157-158 Nusselt number, 118... 

For a binary solution containing 2 = m moles of solute and n = 1 /M moles of solvent (with M in kg-mol 1), the Gibbs-Duhem equation becomes... 

These simple expressions may also be obtained from the chemical potentials according to Eqs. (XII-26) and (XII-32) by appropriately changing subscripts and recalling that x in these equations represents the ratio of the molar volumes, which in the present case is unity. Owing to the identity of volume fractions with mole fractions in this case, Eqs. (18) and (19) are none other than the chemical potentials for a regular binary solution in which the heat of dilution can be expressed in the van Laar form. The critical conditions (see Eqs. 2)... 

It follows that for ionic reactants in binary solutions, the limiting current is given not by Eq. (4.10) but by the equation... 

For binary solutions of symmetric z z electrolytes having a common ion and the same concentration c a = Cma general Henderson equation changes to... 

The thermodynamic properties of real electrolyte solutions can be described by various parameters the solvent s activity Oq, the solute s activity the mean ion activities a+, as well as the corresponding activity coefficients. Two approaches exist for determining the activity of an electrolyte in solution (1) by measuring the solvent s activity and subsequently converting it to electrolyte activity via the thermodynamic Gibbs-Duhem equation, which for binary solutions can be written as... 

The Gibbs-Duhem equation is one of the most extensively used relations in thermodynamics. It is written in the following equivalent forms for a binary solution at constant temperature and pressure ... 

Solution If the gas is assumed to be well mixed, then on the high-pressure (feed-side) side of the membrane, the mole fraction is that of the retentate leaving the membrane. Assuming a binary separation, Equation 10.24 can be written for Component A as ... 

Numerical solution of Chazelviel s equations is hampered by the enormous variation in characteristic lengths, from the cell size (about one cm) to the charge region (100 pm in the binary solution experiments with cell potentials of several volts), to the double layer (100 mn). Bazant treated the full dynamic problem, rather than a static concentration profile, and found a wave solution for transport in the bulk solution [42], The ion-transport equations are taken together with Poisson s equation. The result is a singular perturbative problem with the small parameter A. 

In liquid metal solutions Z is normally of the order of 10, and so this equation gives values of j/S(a+b) which are close to that predicted by the random solution equation. But if it is assumed that the solute atom, for example oxygen, has a significantly lower co-ordination number of metallic atoms than is found in the bulk of the alloy, then Z in the ratio of the activity coefficients of the solutes in the quasi-chemical equation above must be correspondingly decreased to the appropriate value. For example, Jacobs and Alcock (1972) showed that much of the experimental data for oxygen solutions in binary liquid metal alloys could be accounted for by the assumption that the oxygen atom is four co-ordinated in these solutions. 

The variable x is usually the mole fraction of the components. The last expression was first introduced by Guggenheim [5]. Equation (3.60) is a particular case of the considerably more general Taylor series representation of Y as shown by Lupis [6]. Let us apply a Taylor series to the activity coefficient of a solute in a dilute binary solution ... 

The values of d - d0 calculated from equation (53) are compared to the measured values in Table IV. The agreement is quite reasonable considering the binary solution data at high concentrations are only reliable to 10 x 10 ° g cm-3. Calculations at... 

Once the density and compressibilities of mixed electrolyte solutions are known at 1 atm, values at high pressures can be made by using the secant bulk modulus equation of state. The major difficulty, at present, with using additivity methods to estimate the PVT properties of mixed electrolytes is the lack of experimental data for binary solutions over a wide range of concentrations and temperatures. Hopefully, in the near future we will be able to provide some of these data by measurements in our laboratory in Miami. 

Although the description of deviations from ideality in terms of the excess Gibbs function gives us one quantity instead of the two activity coefficients of the two components of a binary solution, we still need to calculate the activity coefficients first, as observed in Equation (16.57). 

The second law of thermodynamics dictates that ct is positive when Vp2 0 therefore, I>0. Hence, based on Equation Al-15, diffusion in binary solutions is always down the chemical potential (or activity) gradient. Comparing J2 in Equation Al-14 with Pick s law and assuming constant molar density p, we have... 

The value of C has to be determined experimentally for a particular system but is typically 0.2-0.3. Pitzer5 presented equations that give activity coefficients for both binary solutions and mixed electrolytes for I up to 6 m, but there are several parameters that must be determined empirically. 

An alternative way of looking at monolayers is to consider them as two-dimensional binary solutions rather than two-dimensional phases of a single component. The advantage of this approach is that it does acknowledge the presence of the substrate and the fact that it plays a role in the overall properties of the monolayer. Although quite an extensive body of thermodynamics applied to two-dimensional solutions has been developed, we consider only one aspect of this. We examine the film pressure as the two-dimensional equivalent of osmotic pressure. It will be recalled that, at least for low osmotic pressures, the relationship among uosm, V, n, and Tis identical to the ideal gas law (Equation (3.25)). Perhaps the interpretation of film pressure in these terms is not too farfetched after all ... 

One isotherm that is both easy to understand theoretically and widely applicable to experimental data is due to Langmuir and is known as the Langmuir isotherm. In Chapter 9, we see that the same function often describes the adsorption of gases at low pressures, with pressure substituted for concentration as the independent variable. We discuss the derivation of Langmuir s equation again in Chapter 9 specifically as it applies to gas adsorption. Now, however, adsorption from solution is our concern. In this section we consider only adsorption from dilute solutions. In Section 7.9c.4 adsorption over the full range of binary solution concentrations is also mentioned. 

A plot of 2 vs. -t2 for symmetrical systems (i.e., ii vo) is shown in Fig. 1 for a series of values of the heat lerm, It shows how the partial vapor pressure of a component of a binary solution deviates positively from Raoult s law more and mure as the components become more unlike in their molecular attractive forces. Second, the place of T in die equation shows that tlic deviation is less die higher the temperature. Third, when the heat term becomes sufficiently large, there are three values of U2 for the same value of ay. This is like the three roots of the van der Waals equation, and corresponds to two liquid phases in equilibrium with each other. The criterion is diat at the critical point the first and second partial differentials of a-i and a are all zero. 

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