Partial volume per mass

Let V be the volume of a solution, M the mass of solvent, My the mass of solute. The partial volumes per mass for the solvent and the solute are the quantities 

The partial volumes per mass are proportional to the partial molecular volumes defined by (5.1.26) 

Let us divide this equation by Ma + Mx). Thus, the volume per mass of solution 

The directly measurable quantities are ws, and °w0 which is the partial volume per mass for pure solvent. Starting from these quantities, we can define the apparent volume per mass of solute by writing 

It is thus possible to obtain the partial volume wt from the apparent volume per mass wt and from the mass fraction iji. 

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The volume changes which occur during dissolution can be directly obtained, by subtraction, from measurements of the volumes of the constituents before mixing and after mixing. However, these changes of volume can be observed in a more indirect manner, which turns out to be more precise. This method consists in measuring the mass per volume of solution as a function of composition. We shall describe this measurement method in the following sections, but first we must establish the relations which exist between masses per volume, partial volumes per mass for the solvent and the solute, and volume of solution. 

With the help of the above-described apparatus, Francois et al.16 discovered a characteristic effect in linear polymer solutions. A preliminary study had already shown to these authors that the partial volume per mass and the branching ratio are related. A measurement of partial volumes per mass should in principle show a chain-end effect. This effect was observed however, another unexpected effect appeared. 

The quantities measured in the experiment are the mass per volume of solution pt and the mass per volume of pure solvent °p0. The apparent partial volume per mass of polymer is deduced from these (measured) quantities by applying eqn (5.2.6). The partial volume per mass is itself a function of w, and... 

The authors quoted above determined the apparent partial volume per mass of polymer wt. However, they did not obtain the real volume per mass w,. Therefore, we shall only consider the apparent volume of the chain... 

Table 7.1. Constants concerning a few usual monomers M = molecular mass, b = collision length, w = volume per mass, b = collision length per unit volume, b j. is calculated with the help of formula (7.2.3) by using as coherent collision lengths for the nuclei, the values of b 11, bQ0h, b 11 given by Table 6.12 in Chapter 6. (non-polarized sample). The partial volumes (per mass) wrf are extracted from ref. 9, and we have v M j(A) (see Chapter 5, Section 2.1).

Another interesting effect has been mentioned by Francois, Candau, and Benoit,16 who observed an anomalous decrease of the partial volume per unit mass of solute when the polymerization degree increases. 

The volume of solute is the product of its mass Jt by the (average) partial volume per unit mass Wj (see Chapter 5, Section 2.1). Thus,... 

Partial volume per unit mass associated with spedes si For a self-avoiding walk on a lattice, factor defining the attraction of adjacent points on the lattice [w = exp( — /U)] Hypervolume representing a three-body interaction In field theory, quantity defining the variation of the effective interaction with respect to the real interaction... 

The partial molar volumes are obtained from the partial volumes per gram by multiplication by the molar masses. 

The main purpose of magnesia addition to a limestone wet scrubbing system is to facilitate high S02 removal. For a wet scrubber that cleans flue gas from a utility coal-fired boiler, the scrubber inlet gas S02 concentration is typically about 700 ppm by volume per one weight percent sulfur in the fired coal. For sub-bituminous coal having only 0.7 weight percent sulfur content, the inlet S02 concentration is about 500 ppm, and, for example, the outlet S02 has to be less than 50 ppm to achieve 90 percent removal. In order to avoid serious inhibition of mass transfer because of S02 back-pressure, the equilibrium S02 partial pressure should be about four or more times lower than the actual S02 partial pressure in the gas. Thus, Figure 3 indicates that for this low-sulfur coal system, the scrubber inlet pH should be at least 5.5, and the outlet pH at least... 

Concentrations will be expressed as mole fraction of a component or species /, x = nj/E nj as molality, mole per mass of solvent, mol kg or molarity, mole per volume of solution. The concentration scale will depend on the properties of the solutes (i.e., ionic, polar, nonpolar, etc.). Pressure, p, and the gas phase partial pressure of species i, / , will be expressed in bars (approximately equal to atmospheres). 

As water swells the membrane, we assume the partial molar volume of water plus the partial molar volume of the dry membrane adds to the total membrane volume per mole. If M w is the equivalent weight of the dry membrane, Mw (18 gm/mol) is the mass per mole of water, is the dry membrane density, the mass per equivalent of the swelled membrane is, and... 

Fig. 11.30. Conceptual overview of MALDI imaging, (a) Freshly cut tissue section (here mouse brain) (b) mounted section after application of matrix, here by a robotic picoliter volume spotter (c) partial series of mass spectra along the x-axis of the sample (d) three-dimensional volumetric plot of the complete dataset with selected m/z ranges per image. Reproduced from Ref. [201] with permission. Nature Publishing Group, 2007.

Figure 6.14 is the same as Figure 6.6, but the mol fraction has been replaced by the mass fraction, and the molar volume has been replaced by the specific volume (volume per unit mass). Show that, for the construction shown, the two tangent intercepts, a and b, are the partial mass volumes. Start with Eq. 6.5 and observe that each mol fraction is the number of mols that species divided by the total number of mols, tij. Cancel the rij-s in the denominators. Then replace each of the values with the mass of that species divided by its molecular weight. Then divide both sides by the total mass of the system. 

FIGURE 6.15 The solution volume per unit mass and partial mass volumes for ethanol-water at 20°C. (From Hougen, O. A., K. M. Watson and R. A. Ragatz, Chemical Process Principles, Part I Material and Energy Balances, ed. 2. 1954, New York Wiley, p. 333. Reprinted by permission of the estate of O. A. Hougen.) This figure uses V2 for Vethanoi, mass, and so forth. 

Humidity (absolute humidity) is defined as the mass of water per mass of dry air with units kg kg . The capacity of air for moisture removal depends on its humidity and temperature. Equilibrium data of solids in contact with humid air provides information about water capacity of solids. Therefore, typically water content of the solid (insulin here) is plotted as a function of the relative humidity of air as shown in Figure 11.3. Curves of these type are nearly independent of temperature. These curves can be obtained using ideal gas law. FVom ideal gas law, the concentration of water in vapor form in moles per volume (c ,) can be related to partial pressure of water, and therefore, to humidity as follows... 

Consider a binary mixture of ethanol, EtOH (CH3CH2OH) and water (H2O). A plot of the partial molar volumes of EtOH and H2O versus mole fraction EtOH (xeioh) is provided in the following figure. Note that the units of partial molar volume are on a per mass basis so that both plots fit on the same scale. Answer the following questions ... 

To prove these assertions, we begin with the volume and mass average velocities. We find it convenient to describe the concentration in these systems as p the mass of species z per volume and as F the partial specific volume. Then... 

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

partial molar volume of the polymer in solution, and M is the molecular weight of the polymer. Substituting this relationship for (pin Eq. (9.9)gives... 

The quantity of a solute adsorbed can be given conveniently in terms of moles or volume (for adsorption) or ion-equivalents (for ion exchange) per unit mass or volume (dry or wet) of sorbent. Common units for adsorption are moV(m of fluid) for the fluid-phase concentration Cj and moV(kg of clean adsorbent) for adsorbed-phase concentration /ij. For gases, partial pressure may replace concentration. 

The plot of the rate of disappearance of CO per volume of liquid in the serum bottles versus partial pressure of CO in the gas phase based on (3.14.4.14) could give the constant slope value of KLa/H. Henry s constant is independent of the acetate concentration but it is only dependent on temperature. The overall volumetric mass transfer coefficient can be calculated based on the above assumption. The data for various acetate concentrations and different parameters were plotted to calculate the mass transfer coefficient. 

Combination of the Perrin function, p often referred as the frictional ratio due to shape with the frictional ratio (f/fo) enables the degree of expansion of the molecule (vh/ w ) to be estimated, where Vh, (cw / g) is the volume of the swollen molecule (Polysaccharide or protein + associated solvent) per unit mass of polysaccharide and v is the partial specific volume (essentially the anhydrous molecule) ... 

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