Solutions mole fraction

Fig. Ill-13. (a) Plots of molecular density versus distance normal to the interface a is molecular diameter. Upper plot a dielectric liquid. Lower plot as calculated for liquid mercury. (From Ref. 122.) (b) Equilibrium density profiles for atoms A and B in a rare-gas-like mixmre for which o,bb/ o,aa = 0.4 and q,ab is given by Eq. III-56. Atoms A and B have the same a (of Eq. m-46) and the same molecular weight of SO g/mol the solution mole fraction is jcb = 0.047. Note the strong adsorption of B at the interface. [Reprinted with permission from D. J. Lee, M. M. Telo de Gama, and K. E. Gubbins, J. Phys. Chem., 89, 1514 (1985) (Ref. 88). Copyright 1985, American Chemical Society.]...

For linear equiHbrium and operating lines, an expHcit expression for the number of theoretical plates required for reducing the solute mole fraction... 

FIG. 4-11 Plot of solute fiigacity i vs. solute mole fraction. 

Here Q is the solute concentration and R the gas constant. This is in fact obeyed over a rather wide range of concentrations, almost up to solute mole fractions of 0.61, with an error of only 25 percent. This is remarkable, since the van t Hoff equation is rigorous only in the infinitely dilute limit. Even in the case of highly nonideal solutions, for example a solution with a ratios of 1.5 and e ratios of 4, the van t Hoff equation is still obeyed quite well for concentrations up to about 6 mole percent. It appears from these results that the van t Hoff approximation is much more sensitive to the nonideality of the solutions, and not that sensitive... 

Vapor pressure lowenng is direcdy proportional to solute mole fraction. 

Molality Mass Percent Ppm Solvent Solute Mole Fraction Solvent... 

For those dilute mixtures where the solute and the solvent are chemically very different, the activity coefficient of the solute soon becomes a function of solute mole fraction even when that mole fraction is small. That is, if solute and solvent are strongly dissimilar, the relations valid for an infinitely dilute solution rapidly become poor approximations as the concentration of solute rises. In such cases, it is necessary to relax the assumption (made by Krichevsky and Kasarnovsky) that at constant temperature the activity coefficient of the solute is a function of pressure but not of solute mole fraction. For those moderately dilute mixtures where the solute-solute interactions are very much different from the solute-solvent interactions, we can write the constant-pressure activity coefficients as Margules expansions in the mole fractions for the solvent (component 1), we write at constant temperature and at reference pressure Pr ... 

Point c is a critical point known as the upper critical end point (UCEP).y The temperature, Tc, where this occurs is known as the upper critical solution temperature (UCST) and the composition as the critical solution mole fraction, JC2,C- The phenomenon that occurs at the UCEP is in many ways similar to that which happens at the (liquid + vapor) critical point of a pure substance. For example, at a temperature just above Tc. critical opalescence occurs, and at point c, the coefficient of expansion, compressibility, and heat capacity become infinite. 

Calcite mole fraction X, solid state activity coefficients X, and Yp, the solute mole fractions of calcium at equilibrium in seawater. 

Polyfmethyl methacrylate), initiated and polymerized at 250 by t-butylmagnesium bromide in toluene-THF solution (—). Mole fraction of monomer, X.VJM = 0.1 OM. XTHF is indicated in each case. A mixture of standard polystyrene samples of indicated molar mass (------). All traces are aligned so that the elution volumes correspond. 

FIG. 7.18 Adsorption on carbon from the ethanol-benzene system. The ordinate equals the total number of moles of solution times the change in solution mole fraction per unit weight of carbon. (Data from F. E. Bartell and C. K. Sloan, J. Am. Chem. Soc., 51, 1643 (1929).)... 

The concentration of a solution can be expressed in many ways, including molarity (moles of solute per liter of solution), mole fraction (moles of solute per mole of solution), mass percent (mass of solute per mass of solution times 100%), and molality (moles of solute per kilogram of solvent). When equilibrium is reached and no further solute dissolves in a given amount of solvent, a solution is said to be saturated. The concentration at this point represents the... 

This system of equations (equations 23-26) involves the three solid-phase-composition variables x, y, and xAC. The other three solid-solution mole fractions are related to x, t/, and xAC by the following relations ... 

Determination of T y. In the formulation of the phase equilibrium problem presented earlier, component chemical potentials were separated into three terms (1) 0, which expresses the primary temperature dependence, (2) solution mole fractions, which represent the primary composition dependence (ideal entropic contribution), and (3) 1, which accounts for relative mixture nonidealities. Because little data about the experimental properties of solutions exist, Tg is usually evaluated by imposing a model to describe the behavior of the liquid and solid mixtures and estimating model parameters by semiempirical methods or fitting limited segments of the phase diagram. Various solution models used to describe the liquid and solid mixtures are discussed in the following sections, and the behavior of T % is presented. 

In the ideally dilute limit, colligative properties depend on the total solute mole fraction or molality. If there are a number of different solutes present, this can be expressed as (for the case of freezing-point depression)... 

Dalton s Law of Partial Pressures for Gas Mixtures (Solutions). Mole Fractions and Partial Pressures... 

The activity of a solvent is defined differently from that of a solute. The solvent is the species having the highest mole fraction in a solution mole fraction indicates the fraction of the total number of moles in a system contributed by that species. For a solvent, is yyiVy, where N - is its mole fraction. An ideal solvent has ysoivent equal to 1, meaning that the interactions of solvent molecules with the surrounding molecules are indistinguishable from their interactions in the pure solvent. An ideal solution has all activity coefficients equal to 1. 

Stewart and Munjal (20) determined the solubility of CO2 in distilled water, a S3mthetic sea water, and in three and five-fold concentrations of the synthetic sea water. The Henry s Law constants for CO2 computed for these systems at different temperatures were found to be linearly related to temperature and solution molality. The concentration of CO2 (aq) in moles per total moles of solution (mole fraction) may be written... 

It is useful to express Eq. (2-53) in terms of the solute mole fraction X2- For the arbitrary variable y in general,... 

Consider a solution in which the solute mole fraction is x and the vapor pressure of the pure solvent at the solution temperature is p. Applying Raoult s law (Equation 6.4-1) to the solution, we obtain for the partial pressure of the solvent... 

Since we know how the solution vapor pressure varies with concentration (the relationship being given by Equation 6.5-2) and temperature (through the Clausius-Clapeyron equation. Equation 6.1-3), we can determine the relationships between concentration and both boiling point elevation and freezing point depression. The relationships are particularly simple for dilute solutions x — 0, where x is solute mole fraction). 

Since the coefficients of x in these two equations are constant, it follows that for dilute solutions of nonvolatile, nonreactive, nondissociative solutes, both boiling point elevation and freezing point depression vary linearly with solute mole fraction. 

Use the heat of solution data in Table B.IO and solution heat capacity data to (a) calculate the enthalpy of a hydrochloric acid, sulfuric acid, or sodium hydroxide solution of a known composition (solute mole fraction) relative to the pure solute and water at 25 C (b) calculate the required rate of heat transfer to or from a process in which an aqueous solution of HCl, H2SO4, or NaOH is formed, diluted, or combined with another solution of the same species and (c) calculate the final temperature if an aqueous solution of HCl, H2SO4, or NaOH is formed, diluted, or combined with another solution of the same species adiabatically. Perform material and energy balance calculations for a process that involves solutions for which enthalpy-concentration charts are available. 

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