Bible Encyclopedias
Vaporization

7. Vapour-Pressure of Solutions

The rise of boiling-point produced by a substance in solution was demonstrated by M. Faraday in 1820, but the effect had been known to exist for a long time previously. C. H. L. Babo, 1847, gave the law known by his name, that the " relative lowering" (p - po)lpo of the vapour-pressure of a solution, or the ratio of the diminution of vapour-pressure (p - po) to the vapour-pressure po of the pure solvent at the same temperature, was constant, or independent of the temperature, for any solution of constant strength. A. Wiillner (Pogg. Ann. 1858, 103, p. 529) found the lowering of the vapour-pressure to be nearly proportional to the strength of the solution for the same salt. W. Ostwald, employing Wiillner's results, found the lowering of vapourpressure produced by different salts in solution in water to be approximately the same for solutions containing the same number of gramme-molecules of salt per c.c. F. M. Raoult (Comptes Rendus, 1886-87) employed other solvents besides water, and showed that the relative lowering for different solvents and different dissolved substances was the same in many cases for solutions in which the ratio of the number of gramme-molecules n of the dissolved substance to the number of molecules N of the solvent was the same, or that it varied generally in proportion to the ratio n/N. The relative lowering of the vapour-pressure can be easily measured by Dalton's method of the barometer tube for solvents such as ether, which have a sufficient vapour-pressure at ordinary temperatures. But in many cases it is more readily determined by observing the rise of the boiling-point or the depression of the freezing-point of the solution. For the rise in the boiling-point, we have by Clapeyron's equation, dp/do = L/ov, nearly, neglecting the volume of the liquid as compared with that of the vapour v. If dp is the difference of vapourpressure of solvent and solution, and do the rise in the boiling-point, we have the approximate relation, n/N = d p/p = mLdo/Ro 2 , Raoult's law,.. (8) where is the molecular weight of the vapour, and R the gasconstant which is nearly 2 calories per degree for a gramme-molecule of gas. For the depression of the freezing-point a relation of the same form applies, but do is negative, and L is the latent heat of fusion. At the freezing-point, the solution must have the same vapour-pressure as the solid solvent, with which it is in equilibrium. The relation follows immediately from Kirchhoff's expression (below, section 14) for the difference of vapour-pressure of the liquid and solid below the freezing-point.

The most important apparent exceptions to Raoult's law in dilute solutions are the cases, (I) in which the molecules of the dissolved substance in solution are associated to form compound molecules, or dissociated to form other combinations with the solvent, in such a way that the actual number of molecules n in the solution differs from that calculated from the molecular weight corresponding to the accepted formula of the dissolved substance; (2) the case in which the molecules of the vapour of the solvent are associated in pairs or otherwise so that the molecular weight m of the vapour is not that corresponding to its accepted formula. These cases are really included in the equation if we substitute the proper values of n or m. In the case of electrolytes, S. Arrhenius (Zeit. phys. Chem. i. p. 631) showed how to calculate the effective number of molecules n" = (1 +ek/ko)n,from the molecular conductivity k of the solution and its value ko 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 n =7, N= l oo, is nearly 60° C. At this point n" = Jo nearly, and the depression should be only 104° C. These and similar discrepancies have been very generally attributed to a loose and variable association of the molecules 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 CaC1 2, 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 constant 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 vapourpressure, if a = 1, 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 1. It appears that the relatively enormous deviations of CaC1 2 from Raoult's law are accounted for on the hypothesis that a=9, but there is a slight uncertainty 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 o° C. and at loo° C., whereas KC1 and NaCI take more water at loo° C. than at o° C. The cases considered by Callendar (loc. cit.) are necessarily limited, because the requisite data for strong solutions are comparatively scarce. The vapourpressure 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 hydration are known, and the assumption a = i (all monohydrates), which is tacitly involved 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 = ROn. 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/RB = loge(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 vapourpressure 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 vapourmolecules 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, o° 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 t° C., we have evidently the simple relation H =L+h.. .. .. (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 saturated 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 Thermodynamics). 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 1 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 +0.305t.. .. (to) 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(8'-0").. .. (II) 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 =E+p(v - b), where the intrinsic energy E is measured from the selected zero 9 0 of total heat. The external work done is p(v-b), where p is the constant pressure, v the volume of the vapour at 0, and b the volume of the liquid at Bo. 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 s (8-00), where s is the specific heat at constant volume. Taking the difference between the values of H for any two temperatures 1 " Latent Heat of Steam," Phil. Trans. A. 1895; of " Benzene," Phil. Mag. 1896. 0' and 0", we see that Rankine's result follows immediately, provided that p(v-b) is equal to (S-s)0 or Rolm, which is approximately true for gases and vapours when v is very large compared with b. We may observe that the equation (51) is accurately true for an ideal vapour, for which pv = (S-s)0, provided that the total heat is defined as equal to the change of the function (E+pv) between the given limits. Adopting this definition, without restriction to the case of an ideal vapour or to saturation-pressure, the rate of variation of the total heat with temperature (dH/dO) at constant pressure is equal to S under all conditions, whether S is constant, or varies both with p and 0. (See Thermodynamics, § 7.) I I. Specific Heat of Vapours. - The question of the measurement of the specific heat of a vapour possesses special interest on account of this simple theoretical relation between the specific heat and the variation of the latent and total heats. The first accurate calculations of the specific heats of air and gases were made by Rankine in a continuation of the paper already quoted. Employing Joule's value of the mechanical equivalent of heat, then recently published, in connexion with the value of the ratio of the specific heats of air S/s=I. 40 deduced from the velocity of sound, Rankine found for air S = 240, which was much smaller than the best previous determinations (e.g. Delaroche and Berard, S = .267), but agreed very closely with the value S=. 238, found by Regnault at a later date. Adopting for steam the same value of the ratio of the specific heats, viz. 1.40, Rankine found S = .385, a value which he used, in default of a better, in calculating some of the properties of steam, although he observed that it was much larger than the coefficient .305 in Regnault's formula for the variation of the total heat. The specific heat of steam was determined shortly afterwards by Regnault (Comptes Rendus, 36, p. 676) by condensing superheated steam at two different temperatures (about 125° and 225° C.) successively in the same calorimeter at atmospheric pressure, and taking the difference of the total heats observed. The result found in this manner, viz. S = 475, greatly increased the apparent discrepancy between Regnault's and Rankine's formulae for the total heat. The discrepancy was also noticed by G. R. Kirchhoff, who rediscovered Rankine's formula (Pogg. Ann. 103, p. 185, 1858). He suggested that the high value for S found by Regnault might be due to the presence of damp in his superheated steam, or, on the other hand, that the assumption that steam at low temperatures followed the law pv = R0 might be erroneous. These suggestions have been frequently repeated, but it is probable that neither is correct. G. A. Zeuner, at a later date (La Chaleur, p.441) employing the empirical, formula pv= BO +Cp25 for saturated steam, found the value S = 568, which further increased the discrepancy. G. A. Hirn and A. A. Cazin (Ann. Chim. Phys. iv. 10, P. 349, 1867) investigated the form of the adiabatic for steam passing through the state p= 760 mm., 0=373° Abs., by observing the pressure of superheated steam at any temperature which just failed to produce a cloud on sudden expansion to atmospheric pressure. Assuming an equation of the form log (p/760) =a log (0/373), their results give a = S/R =4.305, or S=0.474, which agrees very perfectly with Regnault's value. It must be observed, however, that the agreement is rather more perfect than the comparative roughness of the method would appear to warrant. More recently, Macfarlane Gray (Proc. Inst. Mech. Eng. 1889), who has devoted minute attention to the reduction of Regnault's observations, assuming S/s =1.400 as the theoretical ratio of specific heats of all vapours on his " aether-pressure theory," has calculated the properties of steam on the assumption S=0.384. He endeavours to support this value by reference to sixteen of Regnault's observations on the total heat of steam at atmospheric pressure with only 19° to 28° of superheat. These observations give values for S ranging from 0.30 to o46, with a mean value 0.3778. But it must be remarked that the superheat of the steam in these experiments is only I or 2%° of the total heat measured. A similar objection applies, though with less force, to Regnault's main experiments between 125° and 225° C., giving the value S =0.475, in which the superheat (on which the value of S depends) is only one-sixteenth of the total heat measured. Gray explains the higher value found by Regnault over the higher range as due to the presence of particles of moisture in the steam, which he thinks " would not be evaporated up to 524° C., but would be more likely to be evaporated in the higher range of temperature." J. Perry (Steam Engine, p. 580), assuming a characteristic equation similar to Zeuner's (which makes v a linear function of the temperature at constant pressure, and S independent of the pressure), calculates S as a function of the temperature to satisfy Regnault's formula (10) for the total heat. This method is logically consistent, and gives values ranging from 0.305 at o° to 0.345 at Ioo° C. and 0.464 at 210° C., but the difference from Regnault's S = 0.475 cannot easily be explained.

12. Throttling Calorimeter Method

The ideal method of determining by direct experiment the relation between the total heat and the specific heat of a vapour is that of Joule and Thomson, which is more commonly known in connexion with steam as the method of the throttling calorimeter. It was first employed in the case of steam by Peabody as a means of estimating the wetness of saturated steam, which is an important factor in testing the performance of an engine. If steam or vapour is " wire-drawn " or expanded through a porous plug or throttling aperture without external loss or gain of heat, the total heat (E+pv) remains constant (Thermodynamics, § I I), provided that the experiment is arranged so that the kinetic energy of flow is the same on either side of the throttle.

Thus, starting with satu rated steam at a temperature 0' and pressure p', as represented by the point A on the p0 diagram (fig. 2), if the point B represent the state p"o" after passing the throttle, the total heat at A is the same as that at B, and exceeds that at any other point D (at the same pressure p" as at B, but at a lower temperature 0) by the amount S X (0"-0), which would be required to raise the temperature from D to B at constant pressure. We have therefore the simple relation between the total heats at A and D H A -H D =S (0"--0). (12) If the steam at A contains a fraction z of suspended moisture, the total heat H A is less than the value for dry saturated steam at A by the amount zL. If the steam at A were dry and saturated, we should have, assuming Regnault's formula (to), H A -H D = 305 (0'-O), whence, if S = .475, we have zL = .3 0 5 (0 '- 0)-. 475 (0"-O). It is evident that this is a very delicate method of determining the wetness z, but, since with dry saturated steam at low pressures this formula always gives negative values of the wetness, it is clear that Regnault's numerical coefficients must be wrong.

From a different point of view, equation (12) may be applied to determine the specific heat of steam in terms of the rate of variation of the total heat. If we assume Regnault's formula (10) for the total heat, we have evidently the simple relation S=0.305(0'-0)/(0"-o), supposing the initial steam to be dry, or at least of the same quality as that employed by Regnault. This method was applied by J. A. Ewing (B.A. Rep. 1897) to steam near Ioo° C. He found the specific heat smaller than 0.475, but no numerical results were given. A very complete investigation on the same lines was carried out by J. H. Grindley (Phil. Trans. 1900) at Owens College under the direction of Osborne Reynolds. Assuming dH/do = 0.305 for saturated steam, he found that S was nearly independent of the pressure at constant temperature, but that it varied with the temperature from o387 at 100° C. to o665 at 160° C. Writing Q for the Joule-Thomson " cooling effect," dO/dp, or the slope BC/AC of the line of constant total heat, he found that Q was nearly independent of the pressure at constant temperature, a result which agrees with that of Joule and Thomson for air and COs; but that it varied with the temperature as (1/0) 3.8 instead of (i/0) 2. These results for the variation of Q are independent of any assumption with regard to the variation of H. Employing the values of S calculated from dH/d0 = 0.305, he found that the product SQ was independent of both pressure and temperature for the range of his experiments. Assuming this result to hold generally, we should have S=0.306 at o° C., which agrees with Rankine's view; but increasing very rapidly at higher temperatures to S =1.043 at 200° C., and 1.315 at 220° C. The characteristic equation, if SQ = constant, would be of the form (v+SQ) = Roil ' , which does not agree with the well-known behaviour of other gases and vapours. Whatever may be the objections to Regnault's method of measuring the specific heat of a vapour, it seems impossible to reconcile so wide a range of variation of S with his value 5=0.475 between 125° and 225° C. It is also extremely unlikely that a vapour which is so stable a chemical compound as steam should show so wide a range of variation of specific heat. The experimental results of Grindley with regard to the mode of variation of Q have been independently confirmed by Callendar (Proc. R.S. 1900), who quotes the results of similar experiments made at McGill College in 1897, but gives an entirely different interpretation, based on a direct measurement of the - specific heat at 100° C. by an electrical method.

The method of deducing the specific heat from Regnault's formula for the variation of the total heat is evidently liable in a greater degree to the objections which have been urged against his method of determining the specific heat, since it makes the value of the specific heat depend on small differences of total heat observed under conditions of greater difficulty at various pressures. The more logical method of procedure is to determine the specific heat independently of the total heat, and then to deduce the variations of total heat by equation (52). The simplest method of measuring the specific heat appears to be that of supplying heat electrically to a steady current of vapour in a vacuum-jacket calorimeter, and observing the rise of temperature produced. Employing this method, Callendar finds S = 0.497 for steam at one atmosphere Temperature Centigrade FIG. 2. - Throttling Calorimeter Method.

between 103° C. and 113 ° C. This is about 4% larger than Regnault's value, but is not really inconsistent with it, if we suppose that the specific heat at any given pressure diminishes slightly with rise of temperature, as indicated in formula (16) below.

13. Corrected Equation of Total Heat

Admitting the value S =0.497 for the specific heat at 108° C., it is clear that the form of Regnault's equation (io) must be wrong, although the numerical value of the coefficient 0.305 may approximately represent the average rate of variation over the range (loo° to 190° C.) of the experiments on which it chiefly depends. Regnault's experiments at lower temperatures were extremely discordant, and have been shown by the work of E. H. Griffiths (Proc. R.S. 1894) and C. H. Dieterici (Wied. Ann. 37, p. 504, 1889) to give values of the total heat to to 6 calories too large between o° and 40° C. At low pressures and temperatures it is probable that saturated steam behaves very nearly as an ideal gas, and that the variation of the total heat is closely represented by Rankine's equation with the ideal value of S. In order to correct this equation for the deviations of the vapour from the ideal state at higher temperatures and pressures, the simplest method is to assume a modified equation of the Joule-Thomson type (Thermodynamics, equation (17)), which has been shown to represent satisfactorily the behaviour of other gases and vapours at moderate pressures. Employing this type of equation, all the thermodynamical properties of the substance may conveniently be expressed in terms of the diminution of volume c due to the formation of compound or coaggregated molecules, (v - b) =RO/p - co(Oo/O) n =V - c. . . (is) The index n in the above formula, representing the rate of variation of c with temperature, is approximately the same as that expressing the rate of variation of the cooling effect Q, which is nearly proportional to c, and is given by the formula SQ= (n+i)c - b.. .. . (14) The corresponding formula for the total heat is H - Ho=So(O-00) - (n 1) (cp - copo)+ b(p - po), 0E5) and for the variation of the specific heat with pressure S = So+n(n+ i) pc/O,. .. (16) where So is the value of S when p=o, and is assumed to be independent of 0, as in the case of an ideal gas.

Callendar's experiments on the cooling effect for steam by the throttling calorimeter method gave n =3-33 and c =26.3 c.c. at 100° C. Grindley's experiments gave nearly the same average value of Q over his experimental range, but a rather larger value for n, namely, 3.8. For purposes of calculation, Callendar (Proc. R.S. 1900) adopted the mean value n=3.5, and also assumed the specific heat at constant volume s =3.5 R (which gives So=4.5 R) on the basis of an hypothesis, doubtfully attributed to Maxwell, that the number of degrees of freedom of a molecule with m atoms is 2m +I. The assumption n=s/R simplifies the adiabatic equation, but the value n=3.5 gives So =0.497 at zero pressure, which was the value found by Callendar experimentally at 108° C. and 1 atmosphere pressure. Later and more accurate experiments have confirmed the experimental value, and have shown that the limiting value of the specific heat should consequently be somewhat smaller than that given by Maxwell's hypothesis. The introduction of this correction into the calculations would slightly improve the agreement with Regnault's values of the specific heat and total heat between 100° and 200° C., where they are most trustworthy, but would not materially affect the general nature of the results.

Values calculated from these formulae are given in the table below. The values of H at o° and 40° agree fairly with those found by Dieterici (596.7) and Griffiths (613.2) respectively, but differ considerably from Regnault's values 606.5 and 618.7. The rate of increase of the total heat, instead of being constant for saturated steam as in Regnault's formula, is given by the equation dH/d0 =S(1 - Qdp/d0). . (17) and diminishes from 0.478 at o° C. to about o40 at loo° and 0.20 at 200° C., decreasing more rapidly at higher temperatures. The mean value, 0.313 of dH/d0, between loo° and 200° agrees fairly well with Regnault's coefficient 0.305, but it is clear that considerable errors in calculating the wetness of steam or the amount of cylinder condensation would result from assuming this important coefficient to be constant. The rate of change of the latent heat is easily deduced from that of the total heat by subtracting the specific heat of the liquid. Since the specific heat of the liquid increases rapidly at high temperatures, while dH/d0 diminishes, it is clear that the latent heat must diminish more and more rapidly as the critical point is approached. Regnault's formula for the total heat is here again seen to be inadmissible, as it would make the latent heat of steam vanish at about 870° C. instead of at 365° C. It should be observed, however, that the assumptions made in deducing the above formulae apply only for moderate pressures, and that the formulae cannot be employed up to the critical point owing to the uncertainty of the variation of the specific heats and the cooling effect Q at high pressures beyond the experimental range. Many attempts have been made to construct formulae representing the deviations of vapours from the ideal state up to the critical point. One of the most complete is that proposed by R. J. E. Clausius, which may be written RO i p - v = RO (v - b) (A - B0 n) /p(v+a)'0,ti;. (18) but such fomulae are much too complicated to be of any practical use, and are too empirical in their nature to permit of the direct physical interpretation of the constants they contain.

14. Empirical Formulae for the Saturation-Pressure

The values of the saturation-pressure have been ver y accurately determined for the majority of stable substances, and a large number of empirical formulae have been proposed to represent the relation between pressure and temperature. These formulae are important on account of the labour and ingenuity expended in devising the most suitable types, and also as a convenient means of recording the experimental data. In the following list, which contains a few typical examples, the different formulae are arranged to give the logarithm of the saturation-pressure p in terms of the absolute temperature 0. As originally proposed, many of these formulae were cast in exponential form, but the adoption of the logarithmic method of expression throughout the list serves to show more clearly the relationship between the various types. log p= A+BO. .. (Dalton, 1800).

log p=C log (A+BO). (Young, 1820).

log p=A0/(B +CO). .. (Roche, 1830).

log p=A+Bbe+Cce.. . (Biot, 1844; Regnault).

log p= A+B/0+C/0 2. (Rankine, 1849).

log p=A+B/0+C log 0.. (Kirchhoff,1858;Rankine,1866).

log p = A+B/0 b (Unwin, 1887).

log p=A+B log 0+C log (0+c). (Bertrand, 1887).

log p=A+B/(0+C).. . (Antoine, 1888).

The formula of Dalton would make the pressure increase in geometrical progression for equal increments of temperature. In other words, the increase of pressure per degree (dp/d0) divided by p should be constant and equal to B; but observation shows that this ratio decreases, e.g. from 0.0722 at o° C. to 0.0357 at 100° C. in the case of steam. Observing that this rate of diminution is approximately as the square of the reciprocal of the absolute temperature, we see that the almost equally simple formula log p=A+B/0 represents a much closer approximation to experiment. As a matter of fact, the two terms A+B/0 are the most important in the theoretical expression for the vapour-pressure given below. They are not sufficient alone, but give good results when modified, as in the simple and accurate formulae of Rankine, Kirchhoff, L. C. Antoine and Unwin. If we assume formulae of the simple type A+B/0 for two different substances which have the same vapour-pressure p at the absolute temperatures 0' and 0" respectively, we may write log p=A'+B'/0'= A"+B"/0", . . (20) from which we deduce that the ratio 0'/0" of the temperatures at which the vapour-pressures are the same is a linear function of the temperature 0' of one of the substances. This approximate relation has been employed by Ramsay and Young (Phil. Mag. 1887) to deduce the vapour-pressures of any substance from those of a standard substance by means of two observations. More recently the same method has been applied by A. Findlay (Proc. R.S. 1902), under Ramsay's direction, for comparing solubilities which are in many respects analogous to vapour-pressures. The formulae of Young and Roche are purely empirical, but give very fair results over a wide range. That of Biot is far more complicated and troublesome, but admits greater accuracy of adaptation, as it contains five constants (or six, if 0 is measured from an arbitrary zero). It is important as having been adopted by Regnault (and also by many subsequent calculators) for the expression of his observations on the vapour-pressures of steam and various other substances. The formulae of Rankine and Unwin, though probably less accurate over the whole range, are much simpler and more convenient in practice than that of Biot, and give results which suffice in accuracy for the majority of purposes.

15. Theoretical Equation for the Saturation-Pressure

The empirical formulae above quoted must be compared and tested in the light of the theoretical relation between the latent heat and the rate of increase of the vapour-pressure (dp/d0), which is given by the second law of thermodynamics, viz.

0(dp/d0) =L/(

w),.. . (21) in which v and w are the volumes of unit mass of the vapour and liquid respectively at the saturation-point (Thermodynamics, § 4). This relation cannot be directly integrated, so as to obtain the equation for the saturation-pressure, unless L and v - w are known as functions of 0. Since it is much easier to measure p than either L or v, the relation has generally been employed for deducing either L or v from observations of p. For instance, it is usual to calculate the specific volumes of saturated steam by assuming Regnault's formulae for p and L. The values so found are necessarily erroneous if formula (io) for the total heat is wrong. The reason for adopting this method is that the specific volume of a saturated vapour cannot be directly measured with sufficient accuracy on account of the readiness with which it condenses on the surface of the containing vessel. The specific volumes of superheated vapours may, however, (19) be measured with a satisfactory degree of approximation. The deviations from the ideal volume may also be deduced by the method of Joule and Thomson. It is found by these methods that the behaviour of superheated vapours closely resembles that of noncondensible gases, and it is a fair inference that similar behaviour would be observed up to the saturation-point if surface condensation could be avoided. By assuming suitable forms of the characteristic equation to represent the variations of the specific volume within certain limits of pressure and temperature, we may therefore with propriety deduce equations to represent the saturation-pressure, which will certainly be thermodynamically consistent, and will probably give correct numerical results within the assigned limits.

The simplest assumptions to make are that the vapour behaves as a perfect gas (or that p(v-w) = Re), and that L is constant. This leads immediately to the simple formula loge (p/po) = (t/90t/6)L/R,. .. (22) which is of the same type as log p =A-}-B/6, and shows that the coefficient B should be equal to L/R. A formula of this type has been widely employed by van't Hoff and others to calculate heats of reaction and solution from observations of solubility and vice versa. It is obvious, however, that the assumption L =constant is not sufficiently accurate in many cases. The rate of variation of the latent heat at low pressures is equal to S-s, where s is the specific heat of the liquid. Under these conditions both S and s may be regarded as approximately constant, so that L is a linear function of the temperature. Substituting L= Lo+ (S-s)(6-Bo), and integrating between limits, we obtain the result log e p=A+B/o+C log e 6,.. (23) C= (S-s)/R, B = -[Lo+(s-S)90]/R, A = log e po - B /9 0 - C logeeo.

A formula of this type was first obtained by Kirchhoff (Pogg. Ann. 103, p. 185, 1858) to represent the vapour-pressure of a solution, and was verified by Regnault's experiments on solutions of H 2 SO 4 in water, in which case a constant, the heat of dilution, is added to the latent heat. The formula evidently applies to the vapour-pressure of the pure solvent as a special case, but Kirchhoff himself does not appear to have made this particular application of the formula. In the paper which immediately follows, he gives the oft-quoted expression for the difference of slope (dp/d9) 8 -(dp/de) 1 of the vapour-pressure curves of a solid and liquid at the triple point, which is immediately deducible from (21), viz.

e(dp/de)8-o(dp/de)1= (L8-Li)/(v-w) = Lsl (v-w), (24) in which L. and L 1 are the latent heats of vaporization of the solid and liquid respectively, the difference of which is equal to the latent heat of fusion L1. He proceeds to calculate from this expression the difference of vapour-pressures of ice and water in the immediate neighbourhood of the melting-point, but does not observe that the vapour-pressures themselves may be more accurately calculated for a considerable interval of temperature by means of formula (23), by substituting the appropriate values of the latent heats and specific heats. Taking for ice and water the following numerical data, L = 674.7, 6 74.7, L 1 =595.2, L r = 79.5, R = o 03 cal./deg., po = 4.61 mm., s-S = 519 cal./deg., and assuming the specific heat of ice to be equal to that of steam at constant pressure (which is sufficiently approximate, since the term involving the difference of the specific heats is very small), we obtain the following numerical formulae, by substitution in (23), Ice.. lo g l op = 0 . 6640 +9.731/x, Water. lo g i op =o6640+8.585t/e-4.70(log109/Bo-Mt/6), where t=9 -273, and M =0.4343, the modulus of common logarithms. These formulae are practically accurate for a range of 20° or 30° C. on either side of the melting-point, as the pressure is so small that the vapour may be treated as an ideal gas. They give the following numerical values The error of the formula for water is less than t mm. (or a tenth of a degree C.), at a temperature so high as 60° C.

Formula (23) for the vapour-pressure was subsequently deduced by Rankine (Phil. Mag. 1866) by combining his equation (II) for the total heat of gasification with (21), and assuming an ideal vapour. A formula of the same type was given by Athenase Dupre (Theorie de chaleur, p. 96, Paris, 1869), on the assumption that the latent heat was a linear function of the temperature, taking the instance of Regnault's formula (io) for steam. It is generally called Dupre's formula in continental text-books, but he did not give the values of the coefficients in terms of the difference of specific heats of the liquid and vapour. It was employed as a purely empirical formula by Bertrand and Barus, who calculated the values of the coefficients for several substances, so as to obtain the best general agreement with the results of observation over a wide range, at high as well as low pressures. Applied in this manner, the formula is not appropriate or satisfactory. The values of the coefficients given by Bertrand, for instance, in the formula for steam, correspond to the values S = 576 and L =573 at o° C., which are impossible, and the values of p given by his formula (e.g. 763 mm. at loo° C.) do not agree sufficiently with experiment to be of much practical value. The true application of the formula is to low pressures, at which it is very accurate. The close agreement found under these conditions is a very strong confirmation of the correctness of the assumption that a vapour at low pressures does really behave as an ideal gas of constant specific heat. The formula was independently rediscovered by H. R. Hertz (Wied. Ann. 17, p. 1 77, 1882) in a slightly different form, and appropriately applied to the calculation of the vapour-pressures of mercury at ordinary temperatures, where they are much too small to be accurately measured.

16. Corrected Equation of Saturation-Pressure

The approximate equation of Rankine (23) begins to be I or 2% in error at the boiling-point under atmospheric pressure, owing to the coaggregation of the molecules of the vapour and the variation of the specific heat of the liquid. The errcrs from both causes increase more rapidly at higher temperatures. It is easy, however, to correct the formula for these deviations, and to make it thermodynamically consistent with the characteristic equation (13) by substituting the appropriate values of (v-w) and L =H -h from equations (13) and (is) in formula (21) before integrating. Omitting w and neglecting the small variation of the specific heat of the liquid, the result is simply the addition of the term (c-b)/V to formula (23) log p=A+B/B - I - C log B-f-(c-b)IV.. . (25) The values of the coefficients B and C remain practically as before. The value of c is determined by the throttling experiments, so that all the coefficients in the formula with the exception of A are determined independently of any observations of the saturationpressure itself. The value of A for steam is determined by the consideration that p = 760 mm. by definition at 100° C. or 373° Abs. The most uncertain data are the variation of the specific heat of the liquid and the value of the small quantity b in the formula (13). The term b, however, is only 4% of c at 100° C., and the error involved in taking b equal to the volume of the liquid is probably small. The effect of variation of the specific heat is more important, but is nearly eliminated by the form of the equation. If we write h=sot+dh, where so is a selected constant value of the specific heat of the liquid, and dh represents the difference of the actual value of h at t from the ideal value sot, and if we similarly write q5 = sologe(6/90)+dcp for the entropy of the liquid at t, where do represents the corresponding difference in the entropy (which is easily calculated from a table of values of h), it is shown by Callendar (Proc. R.S. 1900, loc. cit.) that the effect of the variation of the specific heat of the liquid is represented in the equation for the vapour-pressure by adding to the right-hand side of (23) the term - (d4-dh/9)/R. If we proceed instead by the method of integrating the equation H -h =6(v-w)dp/d6, we observe that the expression above given results from the integration of the terms -dh/R0 2 +w(dp/d9)/R9, which were omitted in (25). Adopting the formula of Regnault as corrected by Callendar (Phil. Trans. R.S. 1902) for the specific heat of water between ioo° and 200° C., we find the values of the difference (d4-dh/9) to be less than one-tenth of do at 200° C. The whole correction is therefore probably of the same order as the uncertainty of the variation of the specific heat itself at these temperatures. It may be observed that the correction would vanish if we could write dh=wodp/do=wL/(v-w). This assumption is made by Gray (Proc. Inst. C.E. 1902). It is equivalent, as Callendar (loc. cit.) points out, to supposing that the variation of the specific heat is due to the formation and solution of a mass w/(v-w) of vapour molecules per unit mass of the liquid. But this neglects the latent heat of solution, unless we may suppose it included by writing the internal latent heat L i in place of L in Callendar's formula. In any case the correction may probably be neglected for practical purposes below 200° C.

It is interesting to remark that the simple result found in equation (25) (according to which the effect of the deviation of the vapour from the ideal state is represented by the addition of the term (c-b)/V to the expression for log p) is independent of the assumption that c varies inversely as the n th power of 9, and is true generally provided that c-b is a function of the temperature only and is independent of the pressure. But in order to deduce the values of c by the Joule-Thomson method, it is necessary to assume an empirical formula, and the type c=co(6019) n is chosen as being the simplest. The justification of this assumption lies in the fact that the values of c found in this manner, when substituted in equation (25) for the saturation-pressure, give correct results for p within the probable limits of error of Regnault's experiments.

17. Numerical Application to Steam

As an instance of the application of the method above described, the results in the table below are calculated for steam, starting from the following fundamental data: p = 760 mm. at t = too° C. or '373.0° Abs. pV/6 '=0 .' 11030 calories per degree for ideal steam. So =0.478 calories per degree at zero pressure, L=5402 calories at loo° C. (JolyCallendar), n= 3'33, cioo = 26.30 c.c., b=t c.c., h=o9970t+wL (v -w). 750 mm. Hg. =1 megadyne per sq. cm.

where and Table Of Properties Of Saturated Steam The values of the coaggregation-volume c, which form the starting-point of the calculation, are found by taking n =10/3 for convenience of division in formula (13). The unit of heat assumed in the table is the calorie at 20° C., which is taken as equal to 4.180 joules, as explained in the article Calorimetry. The latent heat L (formula 9) is found by subtracting from H (equation 15) the values of the heat of the liquid h given in the same article. The values of the specific heat in the next column are calculated for a constant pressure equal to that of saturation by formula (16) to illustrate the increase of the specific heat with rise of pressure. The specific heat at any given pressure diminishes with rise of temperature. The values of the saturation-pressure given in the last column are calculated by formula (25), which agrees with Regnault's observations better than his own empirical formulae. The agreement of the values of H with those of Griffiths and Dieterici at low temperatures, and of the values of p with those of Regnault over the whole range, are a confirmation of the accuracy of the foregoing theory, and show that the behaviour of a vapour like steam may be represented by a series of thermodynamically consistent formulae, on the assumption that the limiting value of the specific heat is constant, and that the isothermals are generally similar in form to those of other gases and vapours at moderate pressures. Although it is not possible to represent the properties of steam in this manner up to the critical temperature, the above method appears more satisfactory than the adoption of the inconsistent and purely empirical formulae which form the basis of most tables at the present time.

A similar method of calculation might be applied to deduce the thermodynamical properties of other vapours, but the required experimental data are in most cases very imperfect or even entirely wanting. The calorimetric data are generally the most deficient and difficult to secure. An immense mass of material has been collected on the subject of vapour-pressures and densities, the greater part of which will be found in Winkelmann's Handbook, in Landolt's and Bornstein's Tables, and in similar compendiums. The results vary greatly in accuracy, and are frequently vitiated by errors of temperature measurement, by chemical impurities and surface condensation, or by peculiarities of the empirical formulae employed in smoothing the observations; but it would not be within the scope of the present article to discuss these details. Even at the boiling-points the discrepancies between different observers are frequently considerable. The following table contains the most probable values for a few of these points which have been determined with the greatest care or frequency: Table of Boiling-Points at Atmospheric Pressure on Centigrade Scale Alphabetical Index of Symbols Empirical constants in formulae; section 14. Minimum volume or co-volume of vapour, equation (I Concentration of solution, gm. mols. per c.c. Coaggregation-volume of vapour, equation (13). Density of liquid and vapour.

Intrinsic energy of vapour.

Acceleration of gravity.

Total heat of vapour.

Heat of the liquid; height of capillary ascent. Latent heat of vaporization.

Modulus of logarithms.

Molecular weight.

Index of 0 in expression for c, equation (13).

Complete tables of the properties of steam have been worked out on the basis of Callendar's formulae by Professor Dr R. Mollier of Dresden, Neue Tabellen and Diagramme far Wasserdampf, published by J. Springer (Berlin, 1906).

P, Osmotic or capillary pressure. p, Pressure of vapour.

Q, Cooling effects in adiathermal expansion.

R, Constant in gas equation, pv = RU. r, Radius of curvature, formula (1).

S, Specific heat of vapour at constant pressure.

s, Specific heat of liquid, equation (23).

Specific heat of vapour at constant volume; section 8.

T, Surface tension of liquid.

1, Temperature Centigrade.

V, Ideal volume of vapour, equation (13). Specific volume of solid or liquid, equation (5).

v, Specific volume of vapour or steam.

w, Specific volume of water or liquid. 0, Temperature on thermodynamic scale. 0, Entropy of vapour or liquid. (H. L. C.)