Volumes of Pure Components

Critical Volume of Pure Components.—The critical volume is the most difficult of the three critical constants to measure and consequently critical volumes are rarely measured to an accuracy of better than 0.5 per cent. The best method is to measure orthobaric liquid and gas densities up to within a kelvin or so of the critical temperature. The critical density is obtained using the law of rectilinear diameters, i.e. by extrapolating the mean of these densities to the critical temperature. This technique was originally proposed by Cailletet and Mathias and has been investigated carefully by Schneider and co-workers and discussed in detail by Rowlinson.  

Several techniques have been used to determine the orthobaric densities of the liquid and gas. The simplest method is that proposed by van Eck and recently used by the Reporter. A series of heavy-walled tubes are partially filled with varying but known amounts of the substance under investigation. Each is heated slowly in turn until the meniscus disappears out of either the top or bottom of the tube, i.e. the vapour phase completely condenses or the liquid gas completely evaporates. The temperature at which the tube becomes completely filled with vapour or liquid is noted. The volume of each tube is then found by weighing it in water and then cutting the tube open, emptying it and weighing the pieces in water. This method enables the densities of liquid and vapour to be calculated over a series of temperatures. This method is simple but time-consuming, and requires considerable experimental care. 

A superior method, in terms of speed, is the paired-tube method used by Ambrose and co-workers. The method is suitable only for pure substances and cannot be used for mixtures. The method involves sealing varying amounts of the substance, as a liquid, in a series of thick-walled tubes, so that between 15 and 65 per cent of the internal volume of the tubes is occupied by liquid. The volumes of the tubes are calculated from a knowledge of the bore length and the dimensions of the sealed end. The fraction of this volume which is occupied by liquid is measured at room temperature and from about 120 K below the critical temperature to between 10 and 20 K below the critical temperature. 

Rowlinson, Liquid and Liquid Mixtures , 2nd edn., Butterworths, London, 1969, 

The theory of the method is as follows. Consider the effect of temperature upon the proportion of the tube occupied by liquid. Let F be the fraction occupied by liquid, V the internal volume of the tube, m the mass of substance present, py and p are the densities of vapour and liquid respectively, and a the cubical coefficient of expansion of glass, and let the subscripts 0 and T refer to room temperature and temperature T respectively. 

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Xi j = mole fraction of component or / in the liquid mixture V I = liquid molar volume of pure component or / at temperature T, mVkmol... 

If Vi1 is the liquid molar volume of pure component i and Vi1 is the partial molar volume of component i in the liquid mixture, equilibrium between the liquid and vapor phases is given as... 

Component Molar volume of pure component (cm ) Partial molar volume between 11 and 33 mole % R2O... 

On the molar volumes of pure components, we used the following values in Equation (13) by assuming that the temperature dependence of the molar volume and the volume change due to the melting are neglected because the effect of the excess Gibbs energy and the surface tension on the phase equilibria is focused. 

Vi are the molar volumes of pure components, and the average surface area per molecule, A, is given by the equation... 

If the reference point for the integration is a solution consisting of pure component A (xg = 0), then the integration constant required is the molar volume of pure component A, and one may write... 

Here 5°, ]J°, and V° are the molar entropy, internal energy, and volume of pure component i in some reference state, and Cv.i is its constant-volume heat capac-ity. 

The quantity v p ) is the molar volume of pure component / at temperature T and pressure p. ... 

In solutions, particularly electrolyte solutions, the standard state for the solvent is always the pure phase (pure water), so that, for example, refers to the molar volume of pure component 1, that is, pure water. For the solute, the standard state for most properties is, as just mentioned, the state of infinite dilution, so we could use for tho partial molar volume of the solute in the standard state. However, this proves a bit confusing, so for clarity we introduce superscript °° to indicate the infinite dilution state (1 ), and we understand that this is also the standard state for most properties. This raises the question of what symbol to use for the solute in its pure state. The lUPAC recommends the use of for pure substances, but our examples involve only minerals so we will just use the mineral name. Thus we use for the molar volume of pure NaCl. 

The density of the melts of the system KCl-KBF4-K2TiF6 has been measured in [10]. Considering the linear temperature dependence of the molar volumes of pure components as well as of the binary and ternary interactions the following equation was obtained for the molar volume of the ternary system... 

These equations are based on the integration of the partial molar Gibbs energy equation dG = -SdT + V dP for component k at constant T and volume of pure component. Substituting Eq. (10.2) into Eq. (10.1), we have the equilibrium condition... 

With the aid of the principle of compensation, the end state represented by Fig. 24b may be reached if only two components are present. That is not the only condition that must be fulfilled, however. To isolate such large volumes of pure components, quite appreciable amounts of electricity must be passed through the apparatus. It is also required, therefore, that the apparatus itself be built for such large amounts of electricity. This relationship between the dimensions of an electrophoresis instrument and the amount of electricity which it is possible, to send through it was pointed out by Tiselius (1937a). When reversible... 

Y volume of equilibrium stage molar volume of pure component i... 

This is related to the function v (T) which is the reduced volume of pure components. We have... 

The molecular parameters d and p have been deduced from the critical temperatures, pressures and volumes of pure components (cf. Ch. IX, 2). The signs of t and may then be read at once from Fig. 12.6.1 and 12.6.2. The results are summarized in Table 12.7.1. 

Formally, the equation of state for a mixture is identical with equation (171), with reduction parameters obtained by mixing rules consistent with the character of the model. The close-packed volume V of the mixture is related to the averaged close-packed volume r per monomer unit and the close-packed molecular volumes of pure components... 

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