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Rh k of the solution and its value k0 at infinite dilution, for an electrolyte giving rise to e + i ions. The values thus found agreed in the main with Raoult's law for dilute solutions (see Solutions). For strong solutions the discrepancies from Raoult's law often become very large, even if dissociation is allowed for. Thus for calcium chloride the depression of the freezing-point, when re = 7, N = 100, is nearly 60° C. At this point n″ = 10 nearly, and the depression should be only 10.4° C. These and similar discrepancies have been very generally attributed to a loose and variable association of the mole- cules of the dissolved substance with molecules of the solvent, which, according to H. C. Jones (Amer. Chem. Jour. 1905, 33, p. 584), may vary all the way from a few molecules of water up to at least 30 molecules in the case of CaCb., or from 12 to 140 for glycerin. It has been shown, however, by Callendar (Proc. R.S.A. 1908) that, if the accurate formulae for the vapour-pressure given below are employed, the results for strong solutions are consistent with a very slight, but important, modification of Raoult's law. It is assumed that each molecule of solute combines with a molecules of solvent according to the ordinary law of chemical combination, and that the number a, representing the degree of hydration, remains con- stant within wide limits of temperature and concentration. In this case the ratio of the vapour-pressure of the solution p" to that of the solvent p′ should be equal to the ratio of the number of free molecules of solvent N -an to the whole number of molecules N -an +n in the solution. The explanation of this relation is that each of the n compound molecules counts as a single molecule, and that, if all the molecules were solvent molecules, the vapour-pressure would be p', that of the pure solvent. This assumption coincides exactly with Raoult's law for the relative lowering of vapour- pressure, if a = i, and agrees with it in the limit in all cases for very dilute solutions, but it makes a very considerable difference in strong solutions if a is greater or less than I. It appears that the relatively enormous deviations of CaCb. from Raoult's law are accounted for on the hypothesis that 0=9, but there is a slight un- certainty about the degree of ionization of the strongest solutions at-50 C. Cane-sugar appears to require 5 molecules of water of hydration both at 0Â° C. and at 100Â° C, whereas KC1 and NaCl take more water at 100 C. than at 0Â° C. The cases considered by Callendar (foe. cit.) are necessarily limited, because the requisite data for strong solutions are comparatively scarce. The vapour- pressure equations are seldom known with sufficient accuracy, and the ionization data are incomplete. But the agreement is very good so far as the data extend, and the theory is really simpler than Raoult's law, because many different degrees of hydrationare known, and the assumption = 1 (all monohydrates), which is tacitly in- volved in Raoult's law, is in reality inconsistent with other chemical relations of the substances concerned.

8. Vapour-Pressure and Osmotic Pressure.—W. F. P. Pfeffer (Osmotische. Untersuchungen, Leipzig, 1877) was the first to obtain satisfactory measurements of osmotic pressures of cane-sugar solutions up to nearly I atmosphere by means of semi-permeable membranes of copper ferrocyanide. His observations showed that the osmotic pressure was nearly proportional to the concentration and to the absolute temperature over a limited range. Van't Hoff showed that the osmotic pressure P due to a number of dissolved molecules n in a volume V was the same as would be exerted by the same number of gas-molecules at the same temperature in the same volume, or that PV = R0Â». Arrhenius, by reasoning similar to that of section 5, applied to an osmotic cell supporting a column of solution by osmotic pressure, deduced the relation between the Osmotic pressure P at the bottom of the column and the vapour-pressure p" of the solution at the top, viz. mPV/R =log,(p′−p″), which corresponds with the effect of hydrostatic pressure, and is equivalent to the assumption that the vapour-pressure of the solution at the bottom of the column under pressure P must be equal to that of the pure solvent. Poynting {Phil. Mag. 1896, 42, p. 298) has accordingly defined the osmotic pressure of a solution as being the hydrostatic pressure required to make its vapour-pressure equal to that of the pure solvent at the same temperature, and has shown that this definition agrees approximately with Raoult's law and van't Hoff's gas-pressure theory. It is probable that osmotic pressure is not really of the same nature as gas-pressure, but depends on equilibrium of vapour-pressure. The vapour-molecules of the solvent are free to pass through the semi-permeable membrane, and will continue to condense in the solution until the hydrostatic pressure is so raised as to produce equality of vapour-pressure. Lord Berkeley and E. J. G. Hartley (Phil. Trans. A. 1906, p. 481) succeeded in measuring osmotic pressures of cane-sugar, dextrose, &c, up to 135 atmospheres. The highest pressures recorded for cane-sugar are nearly three times as great as those given by van'-t Hoff's formula for the gas-pressure, but agree very well with the vapour-pressure theory, as modified by Callendar, provided that we substitute for V in Arrhenius's formula the actual specific volume of the solvent in the solution, and if we also assume that each molecule of sugar in solution combines with 5 molecules of water, as required by the observations on the depression of the freezing-point and the rise of the boiling-point. Lord Berkeley and Hartley have also verified the theory by direct measurements of the vapour-pressures of the same solutions.

9. Total Heat and Latent Heat.—To effect the conversion of a solid or liquid into a vapour without change of temperature, it is necessary to supply a certain quantity of heat. The quantity required per unit mass of the substance is termed the latent heat of vaporization. The total heat of the saturated vapour at any temperature is usually defined as the quantity of heat required to raise unit mass of the liquid from any convenient zero up to the temperature considered, and then to evaporate it at that temperature under the constant pressure of saturation. The total heat of steam, for instance, is generally reckoned from the state of water at the freezing-point, 0° C. If h denote the heat required to raise the temperature of the liquid from the selected zero to the temperature t° C, and if H denote the total heat and L the latent heat of the vapour, also at tÂ° C, we have evidently the simple relation

H=L+ft. ..... (9)

The pressure under which the liquid is heated makes very little difference to the quantity h, but, in order to make the statement definite, it is desirable to add that the liquid should be heated under a constant pressure equal to the final saturation-pressure of the vapour. The usual definition of total heat applies only to a satu- rated vapour. For greater simplicity and generality it is desirable to define the total heat of a substance as the function (E +pv), where E is the intrinsic energy and v the volume of unit mass (see ). This agrees with the usual definition in the special case of a saturated vapour, if the liquid is heated under the final pressure p, as is generally the case in heat engines and in experimental measurements of H.

The method commonly adopted in measuring the latent heat of a vapour is to condense the vapour at saturation-pressure in a calorimeter. The quantity of heat so measured is the total heat of the vapour reckoned from the final temperature of the calorimeter, and the heat of the liquid h must be subtracted from the total heat measured to find the latent heat of the vapour at the given temperature. It is necessary to take special precautions to ensure that the vapour is dry or free from drops of liquid. Another method, which is suitable for volatile liquids or low temperatures, is to allow the liquid to evaporate in a calorimeter, and to measure the quantity of heat required for the evaporation of the liquid at the temperature of the calorimeter and at saturation-pressure. The first method may be called the method of condensation. It was applied in the most perfect manner by Regnault to determine the latent heats of steam and several other vapours at high pressures. The second method may be called the method of evaporation. It is more difficult of application than the first, but has given some good results in the hands of Griffiths and Dieterici, although the experiments of Regnault by this method were not very successful.

It was believed for many years, in consequence of some rough experiments made by J. Watt, that the total heat of steam was constant. This was known as Watt's law, and was sometines extended to other vapours. An alternative supposition, due to J. Southern, was that the latent heat was constant. The very careful experiments of Regnault, published in 1847, showed that the truth lay somewhere between the two. The formula which he gave for the total heat H of steam at any temperature t" C, which has since been universally accepted and has formed the basis of all tables of the properties of steam, was as follows:—

H =606·5+0·305/. . . . (10)

He obtained similar formulae for other vapours, but the experiments were not so complete or satisfactory as in the case of steam, which may conveniently be taken as a typical vapour in comparing theory and experiment.

10. Total Heat of Ideal Vapour.—It was proved theoretically by W. J. M. Rankine (Proc. R.S.E. vol. xx. p. 173) that the increase of the total heat of a saturated vapour between any two temperatures should be equal to the specific heat S of the vapour at constant pressure multiplied by the difference of temperature, provided that the saturated vapour behaved as an ideal gas, and that its specific heat was independent of the pressure and temperature. Expressed in symbols, the relation may be written

H′−H″ = S(′−″). . . . (11)

This relation gives a linear formula for the variation of the total heat, a result which agrees in form with that found by Regnault for steam, and implies that the coefficient of t in his formula should be equal to the specific heat S of steam. Rankine's equation follows directly from the first law of thermodynamics, and may be proved as follows: The heat absorbed in any transformation is the change of intrinsic energy plus the external work done. To find the total heat H of a vapour, we have H=K-{-p(v-b), where the intrinsic energy E is measured from the selected zero 0 of total heat. The external work done is p(v-b), where p is the constant pressure, the volume of the vapour at 6, and b the volume of the liquid at 0. If the saturated vapour behaves as a perfect gas, the change of intrinsic energy E depends only on the temperature limits, and is equal to j> (0-0<j), where s is the specific heat at constant volume. Taking the difference between the values of H for any two temperatures