Nonideal Colligative Properties

Let s compute the vapor pressure of a solvent over a polymer solution by using the Flory-Huggins theory. Let component A represent the pohnner and B represent the small molecule. To describe the equilibrium, follow the strategy of Equation (16.2). Use Equation (31.20), and set the chemical potential of B in the vapor phase equal to the chemical potential of B in the polymer solution. The vapor pressure of B over a polymer solution is 

Equation (31.22) resembles Equation (16.2) for small molecule solutions, except that the volume fraction 4 b replaces the mole fraction xb, and a term (1 - ) (1 - Njs/Na ) accounts for nonideality due to the difference in molecular 

The Phase Behavior of Polymers Differs from that of Small Molecules 

Wc now show the basis for the asymmetry in polymer solutions. The coexistence curve is defined by the common tangent, 

The spinodal decomposition curve is given by the values of f that cause the second derivative of Equation (31.21) to equal zero. 

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In Chapters 6, 7, and 8, the thermodynamic framework is successively apphed to phase transformations of single-component systems, chemical reactions, and ideal solutions. Included are discussions of the thermodynamics of open systems, the phase rule, and colligative properties. Chapter 9 gives the framework for discussing nonideal multicomponent systems and describes a... 

The colligative properties are of importance by themselves, but they can also be used to determine the molar mass of a solute, since they all depend on the molar concentration and since the mass concentration generally is known. To this end, the determination of the freezing point often is most convenient. Because of nonideality, determinations should be made at several concentrations and the results extrapolated to zero. For determination of the molar mass of macromolecules, osmotic pressure measurement is to be preferred, since membranes exist that are not permeable for macromolecules, while they are for small-molecule solutes, and even small quantities of the latter have a relatively large effect on the colligative properties. Actually, a difference in osmotic pressure is thus determined, the difference being due to the macromolecules only. 

A substance in solution has a chemical potential, which is the partial molar free energy of the substance, which determines its reactivity. At constant pressure and temperature, reactivity is given by the thermodynamic activity of the substance for a so-called ideal system, this equals the mole fraction. Most food systems are nonideal, and then activity equals mole fraction times an activity coefficient, which may markedly deviate from unity. In many dilute solutions, the solute behaves as if the system were ideal. For such ideally dilute systems, simple relations exist for the solubility of substances, partitioning over phases, and the so-called colligative properties (lowering of vapor pressure, boiling point elevation, freezing point depression, osmotic pressure). 

Nonideality of solutions is discussed in Section 2.2.5. It can be expressed as the deviation of the colligative properties from that of an ideal, i.e., very dilute, solution. Here we will consider the virial expansion of osmotic pressure. Equation (2.18) can conveniently be written for a neutral and flexible polymer as... 

The interactions of ions with water molecules and other ions affect the concentration-dependent (colligative) properties of solutions. Colligative properties include osmotic pressure, boiling point elevation, freezing point depression, and the chemical potential, or activity, of the water and the ions. The activity is the driving force of reactions. Colligative properties and activities of solutions vary nonlinearly with concentration in the real world of nonideal solutions. 

The colligative properties of an ideal solution are equal to the concentrations of the components, and their activity coefficients equal one. The deviation of the activity coefficient from one expresses the degree of nonideality. Figure 3.2 shows the change in the aqueous activity coefficients of several ions over the concentrations found in soil solutions and groundwater. 

Colligative properties are related to the number of dissolved solute particles, not their chemical nature. Compared with the pure solvent, a solution of a nonvolatile nonelectrolyte has a lower vapor pressure (Raoult s law), an elevated boiling point, a depressed freezing point, and an osmotic pressure. Colligative properties can be used to determine the solute molar mass. When solute and solvent are volatile, the vapor pressure of each is lowered by the presence of the other. The vapor pressure of the more volatile component is always higher. Electrolyte solutions exhibit nonideal behavior because ionic interactions reduce the effective concentration of the ions. 

APPENDIX 2.A REVIEW OF THERMODYNAMICS FOR COLLIGATIVE PROPERTIES IN NONIDEAL SOLUTIONS... 

When solute and solvent are volatile, each lowers the vapor pressure of the other, with the vapor pressure of the more volatile component greater. When the vapor is condensed, the new solution is richer in that component than the original solution. Calculating colligative properties of electrolyte solutions requires a factor (/) that adjusts for the number of ions per formula unit. These solutions exhibit nonideal behavior because charge attractions effectively reduce the concentration of ions. 

Colligative properties reflect this difference. Figure 31.3 shows the vapor pressure of the solvent benzene over a solution containing rubber, which is a polymer (a) as a function of mole fraction, and (b) as a function of volume fraction. The vapor pressure of a small-molecule solvent over a polymer solution shows nonideal behavior when plotted versus mole fraction x. The Flory-Huggins theory described in the next section shows that a better measure of concentration in polymer solutions is the volume fraction cf>. 

Second Virial Coefficient The coefficient of the most important term of the virial equation that accounts for the nonideality of behavior of a system, in particular of the colligative and other properties of dilute solutions. Generally, the virial equation is of the form. 

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