Colligative properties of dilute solutions

Since the density p appears in a dimensionless combination here, the concentration dependence of the chemical potential comes with a choice of concentration units. The first term on the right side of Eq. (3.1) expresses the colligative property of dilute solutions that the thermodynamic activity of the solute, is proportional to its concentration, p. The excess chemical potential accounts for intermolecular interactions between the solution molecules, and is given by the potential distribution theorem (Widom, 1963 1982) ... 

Because chemical methods are rather limited, the most widely used techniques for measuring the molar mass of a polymer are physical. Methods that depend on the colligative properties of dilute solutions can be used to determine the molar mass of a substance. These include ... 

The next major step was the enunciation of Raoult s law (Raoult 1887, 1888). In 1887, Francois Raoult published his investigations on the vapor pressure of the solvent in dilute solutions. He studied five solutes in water and 14 solutes in each of 11 organic solvents and found that the diminution of the vapor pressure of the solvent upon addition of a given (small) amount of solute was proportionally the same for all cases. The proportionality factor is the mole fraction of the solute. This may be expressed in the currently accepted notation as p° -pi p°X2 this is known as Raoult s law. Raoult had previously discovered the laws of freezing point depression and boiling point elevation (Raoult 1878, 1882), three of the so-called colligative properties of dilute solutions. 

Colligative properties of dilute solutions—polymer solutions particularly—directly result from the variation of the chemical potential of the solvent into which a solute is added. Such properties can be assessed by measuring the osmotic pressure (membrane osmometry), the decrease of the vapor pressure (vapor phase osmometry) or of the freezing point (cryometry). Contrary to the titration of the terminal functional groups, colligative methods do not require a prior knowledge of the polymer structure and depend exclusively on the number of solute molecules. 

The knowledge of the distribution of molar mass of a polydisperse polymer is of extreme importance. Many of the physical properties of polymers depend upon this and a correct distribution is essential in any polymer which is used in a practical situation. One of the most important points in the distribution is the number average molar mass M , and there are several techniques which can be used to measure it. All of these methods involve determining the number of molecules in a given mass of polymer. The polymer is usually dissolved in a solution of known concentration and hence the mass of polymer per unit volume is defined. The most important methods involve the use of the colligative properties of dilute solutions although for a limited number of polymers it is possible to measure M by end-group analysis. 

Arrhenius (1882) was the first to advance the view that an electrolyte, when dissolved in water, dissociates extensively into free ions. His theory soon found support in the work of van t Hoff on the colligative properties of dilute solutions, since it was found that a salt such as NaCl had almost twice, while CaCU had nearly three times, the effect on the vapour pressure, etc., of a solvent as a normal (undissociated) solute. 

The mentioned phenomena are among the remarkable properties of diluted solutions and are determined solely by the numbor of molos of substance dissolved in one unit of volume of the solution, not by the nature of that substance. Such properties, being functions of molar concentration only, are called colligative properties. 

The theory of electrolytic dissociation, AVhereas the osmotic pressure and the other colligative properties of aqueous solutions of substances, such as cane sugar, obey van t Hoff s laws, marked deviations are met with in aqueous solutions of acids, bases, and salts, even at great dilutions. The osmotic pressure and lowering of the freezing point for these solutions are still found to be approximately proportional to the molecular concentration, but are considerably greater than the theoretical values. To allow for this van t Hoff introduced a new term into his osmotic pressure equation, writing for such solutions... 

Ionic solutes are dissociated in solution into cations and anions. Thus aim aqueous solution of NaCl contains 1 mol of Na ions and 1 mol of Cl ions per kilogram of solvent. There are 2 mol of particles per kilogram of solvent, and therefore the colligative properties of the solution are greater than those of a 1 m solution of nonionic solute. In very dilute solutions, the colligative properties of solutions of ionic solutes are a multiple of the analogous properties of nonionic solutes. For example,... 

The colligative properties of dilute polymer solutions can be used to accurately measure Mn, the number average molecular weight of polymers [524,525]. This class of techniques is... 

Explain each of the following statements (a) The boiling point of seawater is higher than that of pure water, (b) Carbon dioxide escapes from the solution when the cap is removed from a carbonated soft-drink bottle, (c) Molal and molar concentrations of dilute aqueous solutions are approximately equal, (d) In discussing the colligative properties of a solution (other than osmotic pressure), it is preferable to express the concentration in units of molality rather than in molarity, (e) Methanol (b.p. 65°C) is useful as an antifreeze, but it should be removed from the car radiator during the summa- season. 

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. 

Osmotic Pressure The osmotic pressure counts the number of independently moving units per volume of the solution. It is one of the colligative properties of the solution. At low concentrations, the whole chain moves as a unit. The center-of-mass displacement is synonymons to a change in the position of the whole chain (in the small wave vector limit). The dilute polymer solution is an ideal solution of p/N = cNjM solute molecules in a nnit volume. The osmotic pressure nyeai of the solution is therefore... 

You might have expected a value of 3, on the basis of the fact that K2SO4 ionizes to give three ions. At first, this was taken as evidence that salts were not completely ionized in solution. In 1923, however, Peter Debye and Erich Hiickel were able to show that the colligative properties of salt solutions could be explained by assuming that the salt is completely ionized in solution but that the activities, or effective concentrations, of the ions are less than their actual concentrations as a result of the electrical interactions of the ions in solution. The Debye-Huckel theory allows us to calculate these activities. When this is done, excellent agreement is obtained for dilute solutions. 

Colligative A colligative property of a solute in a dilute solution is an effect whose magnitude depends only on the concentration of the solute particles (number of particles per unit volume of solution) and not on features such as size, shape or chemical composition. Examples are osmotic pressure, depression of freezing point, elevation of boiling point and the vapour pressure of the solution. If the value of any one of these is known for a particular solution, then the values of the others may readily be calculated. 

Except in very dilute solutions, the values of colligative properties of electrolyte solutions are less than expected because of the attraction between ions in solution. 

The characteristic behavior (6.12.6) of a 1 1 electrolyte merely reflects the fact that at infinite dilution, each solute A produces two free particles, here H" and Cl , in the solution. Similar effects can be observed with respect to any other colligative property of the solution. 

Vapor-Pressure Lowering n One of the colligative properties of a solution and the basis of a method for determination of the molecular mass of a solute. For a dilute solutions, the solvent vapor pressure lowering is determined by (po - p)/po = X2, where po and p are the vapor pressures of the pure solvent and the solution respectively and X2 is the mole fraction of the solute. 

The number-average molar mass, Af , of a polymer is of great importance and there are several methods for its measurement. The common feature underlying each of these methods, is that they measure the number of polymer molecules in a given mass of polymer. The methods which involve measurement of the colligative properties of dilute polymer solutions are applicable to most polymers and will be considered in Sections 3.6-3.8. Additionally, for certain polymers it is possible to measure A/ by end-group analysis and examples of such measurements will be described in Section 3.9. 

Colligative properties are those properties of a solution of a non-volatile solute which in the limit of infinite dilution, depend only upon the number of solute species present in unit volume of the solution and not upon the nature or size of rtose species. Thus the colligative properties of polymer solutions enable Af to be measured for linear and branched homopolymers and copolymers with equal ease. The four important colligative effects are the osmotic pressure of the solution, the lowering of solvent vapour pressure, the elevation of solvent boiling point and the depression of... 

Solutions of Electrolytes—Calculating colligative properties of electrolyte solutions is more difficult than for solutions of nonelectrolytes. The solute particles in electrolyte solutions are ions or ions and molecules. Calculations using equations (14.5) and (14.6) must be based on the total number of particles present, and the van t Hoff factor is introduced into these equations to reflect this number. In all but the most dilute solutions, composition must be in terms of activities— effective concentrations that take into accoimt interionic forces. 

It is important to realise that whilst complete dissociation occurs with strong electrolytes in aqueous solution, this does not mean that the effective concentrations of the ions are identical with their molar concentrations in any solution of the electrolyte if this were the case the variation of the osmotic properties of the solution with dilution could not be accounted for. The variation of colligative, e.g. osmotic, properties with dilution is ascribed to changes in the activity of the ions these are dependent upon the electrical forces between the ions. Expressions for the variations of the activity or of related quantities, applicable to dilute solutions, have also been deduced by the Debye-Hiickel theory. Further consideration of the concept of activity follows in Section 2.5. 

In physical chemistry, we apply the term colligative to those properties that depend upon number of molecules present. The principal colligative properties are boiling point elevation, freezing point depression, vapour pressure lowering, and osmotic pressure. All such methods require extrapolation of experimental data back to infinite dilution. This arises due to the fact that the physical properties of any solute at a reasonable concentration in a solvent are... 

Using measurements of different half-cell combinations, a set of standard reduction potentials may be constructed. This set will be similar to a table of standard reduction potentials. The solutions used in the half-cells must be of known concentration. These solutions are produced by weighing reagents and diluting to volume. The measurements will require a balance and a volumetric flask. It is also possible to produce known concentrations by diluting solutions. This method requires a pipette and a volumetric flask. Review the Solutions and Colligative Properties chapter for solution techniques. 

Freezing-point depression, boiling-point elevation and osmotic pressure are known as colligative properties, because they are dependent on the properties of the solvent and the total mole fraction of all solutes, but are independent of any particular property of the solutes. Equations (61)-(63) are usually written in terms of mB, the sum of the molalities of all the solutes, which for ideally dilute solutions is related to xB by... 

As solutes are added to the liquid phase and the mole fraction of water is thereby lowered, water molecules have less tendency to leave the solution. Hence, the water vapor partial pressure in the gas phase at equilibrium becomes less —this is one of the colligative properties of solutions that we mentioned earlier. In fact, adjacent to dilute solutions P at equilibrium depends linearly on the mole fraction of water (Nw) in the liquid phase. This is Raoult s law (also mentioned in Appendix IV). For pure water, Nw equals 1 and Pwv has its maximum value, namely P w> the saturation vapor pressure. 

---