Osmotic coefficient from freezing point

To determine the osmotic coefficient of aqueous solutions of hydrogen chloride from measurements of freezing point. 

The freezing point depression 6 of aqueous solutions of hydrogen chloride at various molalities m was measured by Randall and Vanselow (J. Amer. Chem. Soc. 1924, 46, 2418). Their results are given in table 1. 

9 osmotic coefficient (on molality scale) at the freezing point of the solution, 

In principle we could calculate cf by formula (1) at each molality, but we should have to smooth the values so obtained bearing in mind that the smaller the value of m the less accurate the value of It is more convenient to evaluate the quantity 

Since we expect (y — q ) to tend to zero like m, we plot the left-hand side of (6) against m  

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The Osmotic Coefficient.—Instead of calculating activity coefficients from freezing-point and other so-called osmotic measurements, the data may be used directly to test the validity of the Debye-Hiickel treatment. If 6 is the depression of the freezing point of a solution of molality m of an electrolyte which dissociates into v ions, and X is the molal freezing-point depression, viz., 1.858° for water, a quantity , called the osmotic coefficient, may be defined by the expression... 

To evaluate from freezing point measurements it is thus necessary to have a knowledge of e. If this is not known then we can do no more than calculate an apparent osmotic coefficient (j>a, which is calculated as though the substance 2 were not dissociated (e = 0) or a coefficient cf)a by assuming complete dissociation (e = 1). These are related by the equations... 

Osmotic coefficients can also be evaluated from freezing point and osmotic pressure measurements that will be described in Sec. 12.2. 

The changes in osmotic coefficients with temperature and concentration make it difficult to solve the above equations accurately, but accurate determinations of the composition and relative amounts of the concentrated liquid and ice can be made from phase diagrams which are plots of the freezing points of solutions versus their concentration. From these, it is possible to determine the exact NaCl concentration at any temperature. Examples are shown in Figure 9 for solutions of 0 to 2.0 M glycerol in 0.15 M NaCl. This figure nicely illustrates how the presence of glycerol reduces the concentration of NaCl in the residual unfrozen solution. 

As we saw in Section 17.5, the activity coefficient of a nonelectrolyte solute can be calculated from the activity coefficient of the solvent, which, in turn, can be obtained from the measurement of colligative properties such as vapor pressure lowering, freezing point depression, or osmotic pressure. We used the Gibbs-Duhem equation in the form [Equation (17.33)]... 

Just as we discussed in Chapter 9, we can use measured activities of solvents (determined from vapor pressure, freezing-point depression, boiling-point elevation, or osmotic pressure) to determine activity coefficients of electrolytes in solution. For an ionic substance, the Gibbs-Duhem equation is... 

We then lose some of the formal resemblance to Eq. 8.1 of ideal solutions, but on the other hand the use of (j) is advantageous in that it is much more sensitive to characterize the deviation from ideality than y,. The osmotic coefficient < > is, in fact, the same coefficient as what is called the boiling or freezing point coefficient. 

There are many measurement techniques for activity coefficients. These include measuring the colligative property (osmotic coefficients) relationship, the junction potentials, the freezing point depression, or deviations from ideal solution theory of only one electrolyte. The osmotic coefficient method presented here can be used to determine activity coefficients of a 1 1 electrolyte in water. A vapor pressure osmometer (i.e., dew point osmometer) measures vapor pressure depression. 

This expression is seen to resemble the first two terms of equation (39.24) with hf defined by (39.51), replacing j, defined by (39.20). Although h and j become identical at infinite dilution, as will be shown below, there is an important difference between these two functions and hence between the activity coefficients derived from them. Whereas j applies to the freezing point of the solution, h refers to the particular temperature, e.g., 25 C, at which the osmotic coefficient is determined, e.g., from vapor pressure measurements. 

Solution of the Equations—The object before us is to calculate, m the case of a strong electrolyte such as KC1, the osmotic pressure exerted by the undissociated molecules and by the 10ns (as distinct from the molecules) m older to see whether one sort or both sorts deviate from van t Hoff s law, and thereby cause the law of mass action to be inapplicable It is assumed that the concentration of the 10ns Ct and of the undissociated molecules C can be obtained accurately from the conductivity expression, the symbol y being used to denote the ionisation coefficient determined by this method When the terms C, and C occur, it is to be understood that they are determined directly by conductivity At the same time we require to know the total osmotic pressure tr of the solution and this is to be understood as directly obtainable from the freezing point data by means of equation (4)... 

As already pointed out the apparent molecular weights of dissolved substances—and consequently the thermodynamic degree of ionisation or activity coefficient a in the case of an electrolyte—as determined by freezing point data are necessarily those which would be obtained from direct measurements of osmotic pressure or from emf measurements, since these different modes of measurement are related thermodynamically It will be recalled that the activity coefficient a for an ion is less than the y value over a wide range of concentration... 

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). 

To estimate the freezing point of blood, we will use the estimate of the osmotic pressure of blood at 25°C, 7.498 bar, computed in Illustration 11.5-4. While the freezing point of blood or the aqueous sodium chloride solution is not known, let us assume it is about the same as the freezing point of water, 273.15 K. The osmotic pressure of the sodium chloride solution at this temperature can be simply calculated from the value at 298.15 K by assuming the activity coefficient is independent of temperature, so that... 

Several approaches exist for evaluating activity coefficients. For non-aqueous systems the most common method has been from electrochemical cells, (sects. 2.5-2.7). Of the remaining approaches available, the freezing point technique is most commonly employed and is considered in sects. 2.8-2.10. This gives osmotic coefficients and activity... 

Even in 1928, Harman (34) concluded from conductivity, transfer numbers, activity coefficients, hydrolysis, osmotic activity, freezing point data, phase relations, and diffusion experiments that there are only two simple silicates, NajSiOj and NaHSiOa, and that silicates in the SiO rNajO ratio range of 2 1 to 4 1 become increasingly colloidal. ... 

In Chapter 8 various equations were derived relating to the partifid pressures, freezing-point depression, osmotic pressure, etc., of an ideal solution. The corresponding expressions which are applicable to non-ideal solutions may be obtained simply by substituting in place of The coefficient has, in fact, been constructed to have this property. It foUows that an activity coefficient may be determined from experiment by application of these equations. As soon as a value of y hai been calculated from a measured property of the solution, for example a partial pressure, it may be used immediately to calculate the value of some other property of the same solution, e.g. its osmotic pressure, at the a ime temperature and pressure. 

In a solution of an electrolyte the activity coefi cientf of the eolverU can be determined by measiirement of its partial pressure, or from the freezing-point depression or the osmotic pressure. The relevant equations are the same as those developed in 9 6. Provided that values of the activity or osmotic coefficient have been determined over a wide range of concentrations, including some solutions which are very dilute, it is possible to calculate the activity coefficient of the solute in some particular solution by application of the Gibbs-Duhem equation. This procedure, as applied to solutions of nonelectrolytes, was described in 9 7 and 9 8. 

The general procedure described in this section for evaluating y requires knowledge of the osmotic coefficient (pm as a function of molality, (pm is commonly evaluated by the isopiestic method (Sec. 9.6.4) or from measurements of freezing-point depression (Sec. 12.2). 

The osmotic coefficient (j) is thermodynamically a well defined property. It can be evaluated from the freezing point depression according to the relation... 

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