The colligative properties

We have seen in Chapter 3 that measurements of the colligative properties are of great value in obtaining molecular weights. The colligative properties of solutions depend largely on the relative amounts of solvent and solute present, and only to a small extent on the nature of the solute species. We can derive the relationship between the colligative properties from thermodynamics. 

We shall start with Raoult s law as an empirical relationship. For a system of two components, solvent and solute, the partial vapor pressure pi of the solvent in a solution of solvent mole fraction Xi is related to the vapor pressure of the pure solution, 

In 1886 the French chemist Francois Marie Raoult 1830-1907 reported extensive vapor-pressure results on solutions, and found them to agree very satisfactorily with this equation. Solutions that obey this law are said to be ideal For solutions that do not, the mole fraction is corrected by multiplying it by an activity coefficient. 

The law can be understood easily if we consider that the tendency of solvent molecules to escape from the surface of the solution is proportional to the fraction of solvent molecules present at the surface, i.e, to the mole fraction. 

The equilibrium between a solution of mole fraction. v and its vapor at 1 atrh pressure can be expressed as 

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Osmotic pressure is one of four closely related properties of solutions that are collectively known as colligative properties. In all four, a difference in the behavior of the solution and the pure solvent is related to the thermodynamic activity of the solvent in the solution. In ideal solutions the activity equals the mole fraction, and the mole fractions of the solvent (subscript 1) and the solute (subscript 2) add up to unity in two-component systems. Therefore the colligative properties can easily be related to the mole fraction of the solute in an ideal solution. The following review of the other three colligative properties indicates the similarity which underlies the analysis of all the colligative properties ... 

One way to describe this situation is to say that the colligative properties provide a method for counting the number of solute molecules in a solution. In these ideal solutions this is done without regard to the chemical identity of the species. Therefore if the solute consists of several different components which we index i, then nj = S nj j is the number of moles counted. Of course, the total mass of solute in this case is given by mj = Sjnj jMj j, so the molecular weight obtained for such a mixture is given by... 

As noted above, all of the colligative properties are very similar in their thermodynamics if not their experimental behavior. This similarity also extends to an application like molecular weight determination and the kind of average obtained for nonhomogeneous samples. All of these statements are also true of osmotic pressure. In the remainder of this section we describe osmotic pressure experiments in general and examine the thermodynamic origin of this behavior. 

The lowering of freezing point and the generation of osmotic pressure both depend on the total concentration of solute particles. Therefore, by using the colligative property to determine the amount of solute present, and knowing its mass, we can infer its molar mass. 

For practical purposes, the colligative property that is most useful for measuring relative molar masses of polymers is osmotic pressure. As Table 6.2 shows, all other properties take such small values that their measurement is impractical. 

The easiest of the colligative properties to visualize is the effect of solute molecules on the vapor pressure exerted by a liquid. In a closed system, the solvent and its vapor reach dynamic equilibrium at a partial pressure of solvent equal to the vapor pressure. At this pressure, the rate of condensation of solvent vapor equals the rate of evaporation from the liquid. 

The dissolution of a solute into a solvent perturbs the colligative properties of the solvent, affecting the freezing point, boiling point, vapor pressure, and osmotic pressure. The dissolution of solutes into a volatile solvent system will affect the vapor pressure of that solvent, and an ideal solution is one for which the degree of vapor pressure change is proportional to the concentration of solute. It was established by Raoult in 1888 that the effect on vapor pressure would be proportional to the mole fraction of solute, and independent of temperature. This behavior is illustrated in Fig. 10A, where individual vapor pressure curves are... 

Osmotic pressure is one of the colligative properties of solutions containing both low-Molecular weight compounds and high polymers. The major difficulty faced in the study of the behaviour of low Molecular weight compounds in solution by the Osmotic pressure measurement method is the selection of a suitable semi-permeable membrane. 

The isopiestic method measures a difference in vapour pressure while the isothermal distillation technique depends upon a difference in volume. Despite the specific changes being measured in the techniques each change is proportional to the colligative property of the solution - the lowering of the vapour pressure. 

Of the preponderance of small ions, the colligative properties of polyelectrolytes in ionising solvents measure counterion activities rather than Molecular weight. In the presence of added salt, however, correct Molecular weights of polyelectrolytes can be measured by membrane osmometry, since the small ions can move across the membrane. The second virial coefficient differs from that previously defined, since it is determined by both ionic and non-ionic polymer-solvent interactions. 

In the next example, we will examine the colligative property of osmotic pressure. This will require us to use the relationship 1r = i(nRT/V). 

The second period, from 1890 to around 1920, was characterized by the idea of ionic dissociation and the equilibrium between neutral and ionic species. This model was used by Arrhenius to account for the concentration dependence of electrical conductivity and certain other properties of aqueous electrolytes. It was reinforced by the research of Van t Hoff on the colligative properties of solutions. However, the inability of ionic dissociation to explain quantitatively the properties of electrolyte solutions was soon recognized. 

Measuring The colligative property of freezing point depression can be observed in a simple laboratory investigation. You will measure the temperatures of two beakers and their contents. 

Salt is a strong electrolyte that produces two ions, Na+ and Cl, when it dissociates in water. Why is this important to consider when calculating the colligative property of freezing point depression ... 

A measure of any of the colligative properties involves counting solute (polymer) molecules in a given amount of solvent. The most common technique for polymers is membrane osmometry. The technique is based on the use of a semipermeable membrane through which solvent molecules freely pass, but through which the large polymer molecules are unable... 

The alternative value, which describes the polymer-solvent interaction is the second virial coefficient, A2 from the power series expressing the colligative properties of polymer solutions such as vapor pressure, conventional light scattering, osmotic pressure, etc. The second virial coefficient in [mL moH] assumes the small positive values for coiled macromolecules dissolved in the thermodynamically good solvents. Similar to %, also the tabulated A2 values for the same polymer-solvent systems are often rather different [37]. There exists a direct dependence between A2 and % values [37]. 

There are several basic physical-chemical principles involved in the ability of aerosol particles to act as CCN and hence lead to cloud formation. These are the Kelvin effect (increased vapor pressure over a curved surface) and the lowering of vapor pressure of a solvent by a nonvolatile solute (one of the colligative properties). In Box 14.2, we briefly review these and then apply them to the development of the well-known Kohler curves that determine which particles will grow into cloud droplets by condensation of water vapor and which will not. 

This relationship constitutes the basic definition of the activity. If the solution behaves ideally, a, =x, and Equation (18) define Raoult s law. Those four solution properties that we know as the colligative properties are all based on Equation (12) in each, solvent in solution is in equilibrium with pure solvent in another phase and has the same chemical potential in both phases. This can be solvent vapor in equilibrium with solvent in solution (as in vapor pressure lowering and boiling point elevation) or solvent in solution in equilibrium with pure, solid solvent (as in freezing point depression). Equation (12) also applies to osmotic equilibrium as shown in Figure 3.2. 

Solutes Affect the Colligative Properties of Aqueous Solutions... 

When discussing the colligative properties of a solution (Chapter 21), it is more important to relate the moles of solute to a constant amount of solvent rather than to the volume of the solution, as in the case of molarity. In practice this is accomplished by using a kilogram of solvent instead of a liter of solution as the reference. A solution that contains one mole of solute per kilogram of solvent is known as a one molal solution it is abbreviated 1.00 m. In general,... 

The colligative properties of solutions are those properties that depend upon the number of dissolved molecules or ions, irrespective of their kind. They are the lowering of the vapor pressure, the depression of the freezing point, the elevation of the boiling point, and the osmotic pressure. These properties may be used in determining molecular weights of dissolved substances. 

According to modem theory, many strong electrolytes are completely dissociated in dilute solutions. The freezing-point lowering, however, does not indicate complete dissociation. For NaCl, the depression is not quite twice the amount calculated on the basis of the number of moles of NaCl added. In the solution, the ions attract one another to some extent therefore they do not behave as completely independent particles, as they would if they were nonelectrolytes. From the colligative properties, therefore, we can compute only the "apparent degree of dissociation" of a strong electrolyte in solution. 

Let us assume that a is the fraction of NaCl molecules that appear to dissociate, and that 1 - a is the fraction that act as if they were still combined as NaCl molecules. Remember that we are talking about our apparent degree of dissociation, as measured by the colligative properties. Then we have, if we start with n moles of NaCl,... 

Depression of Freezing Temperature. One of the colligative properties of solutions of nonvolatile solutes is that the freezing temperature is lower than that of the pure solvent. The depression of the freezing temperature is approximately proportional to the mass ratio of solute to solvent—that is,... 

Y" l Sarah F. McDuffie and Cather-MJI ine E. Matthews, "Antifreeze Solutions The Colligative Properties of Antifreeze," The Science Teacher, Vol. 63,1996,41-43. 

We have seen in the preceding sections that the chemical potentials are extremely important functions for the determination of equilibrium relations. Indeed, all of the relations pertaining to the colligative properties of solutions are readily obtained from the conditions of equilibrium involving the chemical potentials. In many of the relations developed in the remainder of this chapter the chemical potentials appear as independent variables. It would therefore be extremely convenient if their values could be determined by direct experimental means. Unfortunately, this is not the case and we must consider them as functions of other variables. 

One-component, two-phase systems are discussed in the first part of this chapter. The major part of the chapter deals with two-component systems with emphasis on the colligative properties of solutions and on the determination of the excess chemical potentials of the components in the solution. In the last part of the chapter three-component systems are discussed briefly. 

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