Solutions, perfect vapour pressure

Lewis and Randall give an example of calculating the activity of a solute from its vapour pressure. When a solution is in equilibrium with the vapour of the solute x2, we may measure the vapour pressure of x2 over a range of concentrations, and by knowing the fugacity of the vapour at each pressure we may obtain the activity of the solute in the solution. When we may assume that the vapour is a perfect gas, the activity a2 in the solution may be taken as proportional to p2, the vapour pressure of the solute. Hence, as we pass from the mole fraction N2 to an infinitely dilute solution of mole fraction Nx2... 

The relevant parameter is not only the water content but also the water activity. Water activity is a thermodynamic concept which accounts for the fact that materials containing different water contents do not behave in the same way, either chemically or biologically. It reflects the ability of the water to be used in chemical or biological reactions, and it is the concentration corrected for the differences in the ability of the water to undertake chemical reactions. If a non-volatile solute is dissolved in water then the vapour pressure decreases in a specific way for a perfect mixture. A thermodynamically ideal substance always has an activity of unity. 

In very dilute solutions the heat of dilution is found by experiment to be very small, and may be neglected in practice (just as the dilution of perfect gases involves no change in their energy). For such a solution the relative lowering of the vapour pressure is the same at all temperatures. This law has been confirmed by V. Babo in a large number of cases. 

Dilute solutions. As has already been stated (p. 266), the relationship between the osmotic pressure of a solution and the concentration and chemical character of solvent and solute cannot be derived from purely thermodynamical considerations. There are several ways of attaining this end. In the first instance, the variation of the osmotic pressure with the concentration can be determined experimentally, and the results embodied in an empirical equation of the form p=/(c). Or we may deduce relationships from kinetic conceptions of the nature of solutions, in much the same way as the gas laws were deduced. Or, finally, we may deduce the osmotic pressure laws, with the aid of the thermodynamical equations of the previous paragraph, from empirical or theoretical researches on the vapour pressure of solutions. These methods all lead to the same result, that the osmotic pressure of dilute solutions obeys the same laws as the pressure of a perfect gas. In other words, the osmotic pressure of a substance in solution is equal to the pressure which the substance would exert in the form of a perfect gas occupying, at the same temperature, the volume of the solution. 

Deviations from the simple laws. The exact proportionality between the osmotic pressure and the concentration can only hold in dilute solutions. No matter how we account for the osmotic pressure laws, whether by an attraction between the solvent and the solute, or by the impacts of the dissolved molecules, or whether we deduce them from the lowering of the vapour pressure of the solution, we are always forced to restrict the applicability of the simple laws of van t Holf to the region of very dilute solutions. Similarly, the laws of perfect gases can only be regarded as valid in the limiting case of very great... 

The vapour pressure of each component in a perfect solution is thus proportional to its mole fraction. This is Raoulfs latv. 

Fig. 21.1 shows the partial vapour pressures and the total pressure of a perfect solution as a function of composition. In the case of a binary solution (21.8) may be written... 

Fig. 21.1. Total and partial vapour pressure of a perfect solution at constant T.

We note finally that if equation (21.7) is satisfied at a series of different temperatures, then the chemical potentials are necessarily of the form (20.1) and the solution is perfect. Vapour pressure studies thus provide a criterion for deciding whether or not a solution is perfect. 

The case of benzene-toluene mixture discussed above is an example of a solution which follows Raoult s law sufficiently closely. Solutions which obey Raoult s law are called ideal or perfect solutions. According to this law, the total vapour pressure, P, of an ideal solution gets expressed by the relation ... 

In general, the low thermal stability and high vapour pressure for the anticipated vehicle molecule make plausible a conduction mechanism similar to the one in acid solutions for this family of compounds. If we assume the mechanism presented above, then at least some amplification for proton conduction in the solid hydrate TSA.28H2O is observed A = 1.8) (Fig. 26.3). This is in perfect agreement with a ratio [HjO" ]/ [H2O] = 6 for this compound (Fig. 31.5). A is indeed independent of temperature and appears to be a structural feature as suggested by the model. 

The value of vapour pressure osmometry is that the technique is particularly good for low molecular weight polymers, i.e. for molecular weights less than approximately 20000. Membrane osmometry is subject to error at low molecular weight because of solute diffusion through membranes, which are not perfectly semi-permeable as required by the theory. This makes vapour pressure osmometry a convenient method for obtaining M for low molecular weight polymers. 

From (1.5.6) important thermodynamic properties of perfect solutions may be directly deduced e.g. Raoult s law for the vapour pressure, osmotic pressure, equilibrium between a liquid solution and a solid phase. 

So long as the four phases hydrate, two liquid phases, and vapour are present, the condition of the system is perfectly defined. By altering the conditions, however, one of the phases can be made to disappear and a univariant system will then be obtained. Thus, if the vapour phase is made to disappear, the univariant system solution I.—solution II.—hydrate will be left, and the temperature at which this system is in equilibrium will vary with the pressure. This is represented by the curve FI under a pressure of 225 atm. the temperature of equilibrium is 17 1°. Increase of pressure, therefore, raises the temperature at which the three phases can co-exist. 

Greater physical insight into a particular problem may often be obtained by making direct application, not of the phase rule itself, but of the conditions of equilibrium on which it is based. As an example consider the system discussed under case (c) above. Let T and p be the temperature and total pressure respectively and let partial pressures in the vapour and let etc., be the mole fractions in the liquid phase. If the gas phase may be assumed for simplicity to be a perfect mixture and if the liquid phase is an ideal solution, then the eight variables are related by the following six equations Pa+Pb+Pc=P. 

As mentioned in 7 3 it is always possible to discuss the equilibrium of a reaction in solution in terms of the partial pressures in the saturated vapour above the solution (provided that this vapour is a perfect mixture). However, for many purposes it is more useful to express the equilibrium constant of a liquid phase reaction directly in terms of the composition of the liquid. This is done by substituting in equation (10 1) any of the appropriate expressions for the chemical potential of a component of a solution which have been developed in the last two chapters. It will save space if the equations are developed in a general form applicable to a non-ideal solution. The limiting forms of these expressions appljdng to reaction equilibrium in an ideal solution may be obtained by putting the activity coefficients equal to unity. 

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