Based on Colligative Properties

Colligative1 properties of dilute polymer solutions depend only on the number of dissolved molecules and not on properties of the molecules themselves, such as mass or size. Osmotic pressure, freezing point depression, boiling point elevation, and vapour pressure lowering are the most prominent examples. These methods essentially allow one to count the number n of solute molecules. From n and the known total mass m of the solute the molar mass M is readily obtained as 

In the case of a polydisperse polymer it is still the total number n of solute molecules that is measured and the total mass m of solute molecules that is known from sample preparation, resulting in the number average molar mass M = M  

1 Derived from the Latin word colligere = to collect, assemble. 

V/1 is the molar volume of the solvent, ns and np the molar concentrations of the solvent and the polymer, respectively, and AGm the Gibbs free energy of mixing. Equation (26) reduces in the limit of infinite dilution to the well-known Van t Hoff equation 

Finite concentrations are treated in terms of a virial expansion 

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Vapour pressure osmometry is the second experimental technique based on colligative properties with importance for molar mass determination. The vapour pressure of the solvent above a (polymer) solution is determined by the requirement that the chemical potential of the solvent in the vapour and in the liquid phase must be identical. For ideal solutions the change of the vapour pressure p of the solvent due to the presence of the solute with molar volume V/1 is given by... 

Methods for the determination of Molecular weight based on colligative property are vapour-pressure lowering, boiling point elevation (ebulliometry), freezing-point depression (cryoscopy), and the Osmotic pressure (osmometry). 

What type of molecular weight average, or M , is based on colligative properties ... 

Hydroxyl number and molecular weight are normally determined by end-group analysis, by titration with acetic, phthalic, or pyromellitic anhydride (264). For lower molecular weights (higher hydroxyl numbers), 19F- and 13C-nmr methods have been developed (265). Molecular weight determinations based on colligative properties, eg, vapor-phase osmometry, or on molecular size, eg, size exclusion chromatography, are less useful because they do not measure the hydroxyl content. 

Colligative properties are solution properties that vary in proportion to the solute concentration and depend only on the number of solute particles. This section covers a few solubility laws based on colligative properties. 

All these techniques use different principles of measurement. Here, only two methods based on colligative properties are described. 

The number-average molecular weight (MJ of polymers can be easily determined from methods based on colligative properties, which are dependent on the number of molecules in the solution [28]. Thus, the addition of a number of solute molecules to a solvent produces a change in the chemical potential (A/Ui) of the solvent from which the molecular and interactional parameters can be deduced. 

Among the different techniques based on colligative properties, the most practical ones to determine Mjj (number-average molecular weight) are the MO [17, 28] and the VPO [88], both performed in dilute solution. 

Completely ah initio predictions can be more accurate than any experimental result currently available. This is only true of properties that depend on the behavior of isolated molecules. Colligative properties, which are due to the interaction between molecules, can be computed more reliably with methods based on thermodynamics, statistical mechanics, structure-activity relationships, or completely empirical group additivity methods. 

Here we shall confine ourselves to the solvents benzene and 1,2-dichloroethane (class 8). Considering benzene, many investigators have demonstrated since the 1930s the feasibility of titrations in this solvent using both potentiometric and spectrophotometric methods, paying much attention to acid-base indicator reactions under the influence of primary, secondary and tertiary amines. Association of carboxylic acids in benzene was studied at a later stage, mainly on the basis of colligative properties, IR spectroscopy and solvent extraction. ... 

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... 

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. 

The most dramatic effects of Lewis bases in organolithium chemistry are observed in polymerization reactions. Aside from colligative property measurements, there is little direct quantitative information on the nature of the organolithium-base interactions responsible for the observed effects. The calorimetric method has been used also to examine the fundamental nature of the interaction of bases with polymeric organolithium compounds 83,88,89). Information is now available on the ground-state interaction of bases with poly(styryl)lithium (PSLi), poly(isoprenyl)lithium (PILi) and poly(butadienyl)lithium (PBDLi). 

The up to now most frequently used techniques as, for example, vapour pressure osmometry (VPO) or freezing point depression (with its limitation regarding the solvent dependent measuring temperature) are based upon the colligative properties of the system the classical absolute light-scattering and ultracentrifugation techniques are only occasionally and approximately applicable with respect to the determination of CMC values. Evaluation of critical micelle concentrations which are based on these latter methods suffer considerably from the insensitivity of these techniques if measurements below the CMC, i.e., below about 10-3 mol dm-3, are carried out. More sensitive methods will be discussed below. 

Unless the nature and number of the liaisons in the initial and final states are known with certainty, the reliability of the x-Parameter (based on Eq. 7 and relationships derived therefrom) suffers accordingly, even with the most accurate thermodynamic methods for measuring colligative physical properties of polymer-liquid systems. It would be well, therefore, to develop methods for defining the mode of complexation at the initial and final states on a molecular basis. Elucidation of the molecular nature of these complexations at gel-saturation (or in true solution) is an end-objective of the work described in Sect. 3 of this review. 

Two other techniques that are also used to measure M are not colligative properties in the strict sense. These are based on end-group analysis and on vapor phase osmometry. Both methods, which are limited to lower molecular weight polymers, are described later in this chapter. Some general details of the various procedures for measuring M directly are reviewed in this section. 

The four colligative properties that are of importance are 1) the vapor pressure lowering 2) the elevation of boiling point 3) the freezing-point depression and 4) the osmotic pressure. An attempt is made below to describe qualitatively and quantitatively each colligative property of solutions, with an emphasis on their interrelationship and their application later in measurement and adjustment of the tonicity of solutions, with particular reference to parenteral formulations. Although theoretical derivations based on thermodynamics can be used to show how each of the colligative properties of solution arises and relate to each other, textbooks on physical chemistry for theoretical derivations are recommended. 

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