Molar volume partial

Partial Molar Volume 1 0. Cryoscopic Determination of Molar Mass 1 1. Freezing-Point Depression of Strong and Weak Electrolytes 12. Chemical Equilibrium in Solution 

In this experiment the partial molar volumes of sodium chloride solutions will be calculated as a function of concentration from densities measured with a pycnometer. 

Most thermodynamic variables fall into two types. Those representing extensive properties of a phase are proportional to the amount of the phase under consideration they are exemplified by the thermodynamic functions V, E, H, S, A, G. Those representing intensive properties are independent of the amount of the phase they include p and T. Variables of both types may be regarded as examples of homogeneous functions of degree 1 that is, functions having the property 

Among intensive variables important in thermodynamics are partial molar quantities, defined by the equation 

A property of great usefulness possessed by partial molar quantities derives from Euler s theorem for homogeneous functions, which states that, for a homogeneous function f n-[..Dj.) of degree 1, 

Measure the change in total volume V of the solution when one mole of solute is added to a very large quantity (strictly speaking, an infinite quantity) of the solution at the desired concentration. Because very large quantities of solution are used, the addition of one mole of solute does not change the concentration of the solution appreciably. As in the description of equrhbrium in Chapter 10, we may refer to the infinite copy model.  

Measure the change in total volume V of the solution when a small quantity of the solute is added to the solution. Then calculate the change for one mole (that is, divide AV by Ari2) as if no change in composition occurred when a whole mole of solute was added. Repeat this procedure, but add a smaller quantity A 2 of solute and compute Ak /Aw - Repeat again with a still smaller quantity 

CALCULATION OF PARTIAL MOLAR OUANTITIES AND EXCESS MOLAR OUANTITIES 

Most frequently, volume data for solutions are tabulated as density p as a function of composition. The procedure for obtaining y i2 is illustrated by reference to the densities and weight percent concentrations of ethanol-water mixtures (Table 18.1, Columns 1 and 4 at 25°C). 

To obtain y i2, that is, (dV/dn2)nij,p, we need values of V for a fixed quantity of water and for variable quantities 2 of ethanol. For this purpose we convert the relative weights given in Column 1 to relative numbers of moles, that is, to 2/wi in Column 2. The numbers in Column 2 also are the moles of ethanol accompanying one mole of water in each of the solutions listed in Column 1. 

Let us consider a two-component melt. The volume of this system can be expressed as the function of temperature, pressure, and the amount of substances of both the components 

The partial molar volume depends on temperature, pressure, and the composition of the system. The physical sense of the partial molar volume is the volume increase caused by the addition of 1 mol of component to an amount of solution such that the composition of the solution at constant temperature and pressure does not change. The partial molar volume can thus attain a negative value also. 

showing total volume of fluid / AV is partial molar volume of NaCl 1 mole NaCl added 

AV seen there is evidently the volume occupied by 1 mole of NaCl in a 1 molal NaCl 

Clearly we could also add a mole of water (18.01 g occupying 18.0 cm ) to our roomful of salt solution and determine its partial molar volume (t o) as well. This 

This will seem like a reasonable conclusion to anyone who recalls our discussion of Euler s Theorem for homogeneous functions in Chapter 2, since V is homogeneous in the first degree in the masses (or mole numbers) of the components NaCl and H2O. It is, in other words, an extensive state variable. 

Equation (9.1) can be seen to be a reasonable conclusion from another point of view as well. This time let s consider not the total volume of the system but the molar volume, which is 

In order to gain insight into the significance of a partial molar quantity as defined by Eq. 9.2.1, let us first apply the concept to the volume of an open single-phase system. Volume has the advantage for our example of being an extensive property that is easily visualized. Let the system be a binary mixture of water (substance A) and methanol (substance B), two liquids that mix in all proportions. The partial molar volume of the methanol, then, is the rate at which the system volume changes with the amount of methanol added to the mixture at constant temperature and pressure Fb = dV/dn )T,p,riA- 

Let us calculate the mole fraction composition of this mixture  

Now suppose we prepare a large volume of a mixture of this composition (xb = 0.307) and add an additional 40.75 cm (one mole) of pure methanol, as shown in Fig. 9.1(a) on the next page. If the initial volume of the mixture at 25 °C was 10,000.0 cm, we find 

The volume of the mixture to which we add the methanol does not matter as long as it is large. We would have observed practically the same volume increase, 38.8 cm, if we had mixed one mole of pure methanol with 100,000.0 cm of the mixture instead of only 10,000.0 em.  

we may interpret the partial molar volume of B as the volume change per amount of B added at eonstant T and p when B is mixed with sueh a large volume of mixture that the eomposition is not appreeiably affeeted. We may also interpret the partial molar volume as the volume ehange per amount when an infinitesimal amount is mixed with a finite volume of mixture. 

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Since we make the simplifying assumption that the partial molar volumes are functions only of temperature, we assume that, for our purposes, pressure has no effect on liquid-liquid equilibria. Therefore, in Equation (23), pressure is not a variable. The activity coefficients depend only on temperature and composition. As for vapor-liquid equilibria, the activity coefficients used here are given by the UNIQUAC equation. Equation (15). ... 

VSTR is useful for estimating partial molar volumes at infinite dilution but is not used here because of Equation (4-17)... 

We define a partial molar volume Vi such that V = riiVi -I- U2V2... 

We may define, say, partial molar volume, enthalpy, or entropy by analogy with Eq. (8.5) ... 

Remember that Vj is the partial molar volume of the solvent. Therefore a completely general relationship between n and the solvent activity is given by... 

As in osmotic pressure experiments, polymer concentations are usually expressed in mass volume units rather than in the volume fraction units indicated by the Einstein equation. For dilute solutions, however, Eq. (8.100) shows that

molecular weight of the polymer. Substituting this relationship for (pin Eq. (9.9)gives... 

P rtl IMol r Properties. The properties of individual components in a mixture or solution play an important role in solution thermodynamics. These properties, which represent molar derivatives of such extensive quantities as Gibbs free energy and entropy, are called partial molar properties. For example, in a Hquid mixture of ethanol and water, the partial molar volume of ethanol and the partial molar volume of water have values that are, in general, quite different from the volumes of pure ethanol and pure water at the same temperature and pressure (21). If the mixture is an ideal solution, the partial molar volume of a component in solution is the same as the molar volume of the pure material at the same temperature and pressure. 

Vi Partial molar volume of i mVkmol or cmVmol ftVlbmol... 

The dependence of the rate constant on pressure provides another activation parameter of mechanistic utility. From thermodynamics we have (dGldP)T = V, where V is the molar volume (partial molar volume in solutions). We define the free energy of activation by AG = G — SGr. where SGr is the sum of the molar free energies of the reactants. Thus, we obtain... 

Beck, et al. have used the permeation technique to study the effect of uniaxial tensile stresses in the elastic region on hydrogen permeation through pure iron, and have shown that it increases with increase in stress. The partial molar volume of hydrogen (cubic centimetres of hydrogen per mole of iron) in ferrous alloys can be evaluated from the variation of permeation with applied stress, and from the relationship... 

IV. Effect of Pressure on Activity Coefficients Partial Molar Volumes. 160... 

B. Partial Molar Volumes in Saturated Liquids from an Equation of State. 162... 

Thermodynamics gives limited information on each of the three coefficients which appear on the right-hand side of Eq. (1). The first term can be related to the partial molar enthalpy and the second to the partial molar volume the third term cannot be expressed in terms of any fundamental thermodynamic property, but it can be conveniently related to the excess Gibbs energy which, in turn, can be described by a solution model. For a complete description of phase behavior we must say something about each of these three coefficients for each component, in every phase. In high-pressure work, it is important to give particular attention to the second coefficient, which tells us how phase behavior is affected by pressure. 

In order to simplify Eq. (5), it was suggested many years ago by Lewis (L3) that Amagat s law be used, viz., to assume that the partial molar volume of component i at any temperature and pressure is equal to the molar volume... 

It is difficult to measure partial molar volumes, and, unfortunately, many experimental studies of high-pressure vapor-liquid equilibria report no volumetric data at all more often than not, experimental measurements are confined to total pressure, temperature, and phase compositions. Even in those cases where liquid densities are measured along the saturation curve, there is a fundamental difficulty in calculating partial molar volumes as indicated by... 

For dilute solutions, the technical literature contains some direct (dilato-metric) measurements of v2, the partial molar volume of the more volatile component, but the accuracy of these measurements is usually not high. A survey was made by Lyckman and Eckert (L6) and they established the rough correlation shown in Fig. 5. On the ordinate, the partial molar volume is... 

Flo. 5. Partial molar volumes of gases in dilute liquid solutions. 

If we can write an equation of state for liquid mixtures, we can then calculate partial molar volumes directly by differentiation. For a pressure-explicit equation, the most convenient procedure is to use the exact relation... 

FIg. 6. Partial molar volumes of gaseous solutes at infinite dilution in expanded solvents. 

By adopting mixing rules similar to those given in Section II, Chueh showed that Eq. (55) can be used for calculating partial molar volumes in saturated liquid mixtures containing any number of components. Some results for binary systems are given in Figs. 7 and 8, which compare calculated partial molar volumes with those obtained from experimental data. 

Fig. 7. Partial molar volumes in the saturated liquid phase of the n-butane-carbon...

Fig. 8. Partial molar volumes in the saturated liquid phase of the propane-methane system at IOO°F. (O) (0) Data of B. H. Sage and W. N. Lacey, Some Properties of the Lighter Hydrocarbons, Hydrogen Sulfide, and Carbon Dioxide. American Petroleum...

Chueh s method for calculating partial molar volumes is readily generalized to liquid mixtures containing more than two components. Required parameters are and flb (see Table II), the acentric factor, the critical temperature and critical pressure for each component, and a characteristic binary constant ktj (see Table I) for each possible unlike pair in the mixture. At present, this method is restricted to saturated liquid solutions for very precise work in high-pressure thermodynamics, it is also necessary to know how partial molar volumes vary with pressure at constant temperature and composition. An extension of Chueh s treatment may eventually provide estimates of partial compressibilities, but in view of the many uncertainties in our present knowledge of high-pressure phase equilibria, such an extension is not likely to be of major importance for some time. 

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