Osmotic pressure nonideal solutions

A theory close to modem concepts was developed by a Swede, Svante Arrhenins. The hrst version of the theory was outlined in his doctoral dissertation of 1883, the hnal version in a classical paper published at the end of 1887. This theory took up van t Hoff s suggeshons, published some years earlier, that ideal gas laws could be used for the osmotic pressure in soluhons. It had been fonnd that anomalously high values of osmotic pressure which cannot be ascribed to nonideality sometimes occur even in highly dilute solutions. To explain the anomaly, van t Hoff had introduced an empirical correchon factor i larger than nnity, called the isotonic coefficient or van t Hoff factor,... 

The thermodynamic preliminaries and concepts needed for defining osmotic pressure are discussed in Sections 3.2a-c. The nonideality of colloidal solutions can be appreciable since the solvent and solute particles are so different in size. Classical thermo-... 

After reaching Equation (20) we abandoned a completely general discussion of osmotic pressure in favor of the simpler assumption of ideality. The ideal result, Equation (25), applies to real solutions in the limit of infinite dilution. The objective of this section is to examine the extension of Equation (20) to nonideal solutions or, more practically, to solutions with concentrations that are greater than infinitely dilute. 

Since both the osmotic pressure of a solution and the pressure-volume-temperature behavior of a gas are described by the same formal relationship of Equation (25), it seems plausible to approach nonideal solutions along the same lines that are used in dealing with nonideal gases. The behavior of real gases may be written as a power series in one of the following forms for n moles of gas ... 

The easiest way to extend these considerations to the osmotic pressure of nonideal solutions is to return to Equation (22), which relates ir to a power series in mole fraction. This equation applies to ideal solutions, however, since ideality is assumed in replacing activity by mole fraction in the first place. To retain the form and yet extend its applicability to nonideal solutions, we formally include in each of the concentration terms a correction factor defined to permit the series to be applied to nonideal solutions as well ... 

As in the analogous case of gases (Section 2.4), corrections for nonideality can be obtained by measurements of osmotic pressure at different solute concentrations, with extrapolation toward the infinite-dilution limit. For electrolytes, the correction for ionic dissociation is important. 

This equation is good for ideal solutions. For an ionic surfactant solution, the solution is nonideal even at very low surfactant concentration and gives a highly nonlinear dependence of osmotic pressure on concentration. This is expected because ionic surfactants have a high affinity for the interfaces of solution-vapor, solution-solid, and solution-membrane as well as for themselves (i.e., micellization). 

For nonideal solutions, the activity and osmotic pressure are related by the expression... 

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... 

Concentrated polymer solutions show strong nonideality. This is, for instance, observed in the osmotic pressure being very much higher than would follow from the molar concentration. The main variables are the [j value and the volume fraction of polymer, and for polyelectrolytes also charge and ionic strength. 

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. 

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. 

What is the direction of the influence of nonideality (for example, positive deviations from Raoult s law) on (a) freezing-point depression, (b) boiling-point elevation, and (c) osmotic pressure compared to the ideal solution case ... 

For nonideal dilute solutions we can study the osmotic pressure using virial expansion. For simplicity, let us specifically assume that the solute is one component. The virial expansion of the osmotic pressure. 

We have not discussed the subject of nonideal polymers in any detail apart from the excluded volume problem. Thus no mention is made of the evaluation of the potential of mean force from the monomer-solvent interaction, and subsequently the evaluation of the osmotic pressure. We refer to the treatment of Yamakawa (Ref. 5, Chapter IV) for this subject and mention only that the osmotic pressure of a polymer solution at finite concentrations is represented as a virial expansion in the polymer concentration. " The second, third, etc., virial coefficients represent the mutual interaction between two, three, etc., polymer chains in solution. Thus the functional integral techniques presented in this review should also be of use in understanding the osmotic pressure of nonideal polymer solutions. We hope that this review will stimulate such studies of this important subject. It should also be mentioned in passing that at the 0-point the second virial coefficient vanishes. In general, the osmotic pressure -n is given by the series... 

Figure 2.8. Osmotic pressure H plotted as a function of polymer concentration c for the ideal solution (dashed line) and nonideal solutions with Aj >0, =0, and <0 (solid lines).

Properties of Theta Solutions Solutions in the theta condition have A2 = 0. When A2 = 0, the second-order term in Il/Ilideai = 1 + A2MC + A Mc + (Eq. 2.20) is absent. The nonideality of the solution does not become apparent until the third-order term A Mc becomes sufficiently large. The osmotic pressure is... 

We briefly review here thermodynamics of a nonideal binary solution. The osmotic pressure Ft is the extra pressure needed to equilibrate the solution with the pure solvent at pressure p across a semipermeable membrane that passes solvent only. The equilibration is attained when the chemical potential of t e pure solvent becomes equal to the chemical potential of the solvent molecule in solute volume fraction at temperature T ... 

Following the same procedure as above, we arrive at the following expression for the osmotic pressure of a nonideal solution ... 

Another approach for nonideal solutions is similar to the virial equation description of real gases. In this case the osmotic pressure is written as... 

In a sufficiently diluted solution, the osmotic pressure (n) is described by the following equation n = w.k. T/m = w.R.T/M, where w = weight concentration of dispersed phase particles, kj = Boltzmann constant, T = absolute temperature, m = average particle weight, R = universal gas constant, M = particle molar weight. Other relationships have been proposed for nonideal solutions. 

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