# 1911 Encyclopædia Britannica/Vaporization

VAPORIZATION. 1. In common language a vapour is a gaseous or elastic fluid, which emanates or evaporates from the surface of a solid or liquid at temperatures below its boiling-point. A volatile liquid or solid is one which evaporates rapidly at ordinary temperatures. It is a matter of common experience that evaporation is accelerated by currents of air, or by the use of an exhaust pump, or by any process which removes the vapour rapidly from the liquid. On the other hand, it is retarded, and finally ceases, if the vapour is allowed to accumulate in a closed space. When this equilibrium state is reached, the space is said to be saturated with the vapour; the density of the vapour is then the maximum which can exist in the presence of the liquid at the temperature of the experiment, and its pressure is called the saturation-pressure. The term vapour-pressure, when used without qualification, is also generally employed to denote the saturation or maximum pressure. Dalton showed that the saturation-pressure of a vapour depends only on the temperature, and is unaffected by the presence of any neutral gas or vapour. This relation has been more accurately verified by many subsequent observers, and the exceptions to it have been minutely studied and elucidated. The saturation-pressure invariably increases rapidly with rise of temperature, according to a regular law which has been the subject of many elaborate investigations. When the vapour-pressure of a liquid becomes equal to the external pressure, bubbles of vapour are freely formed in the interior of the liquid by the familiar process of boiling or ebullition. The temperature at which this occurs under the normal atmospheric pressure of 760 mm. of mercury (reduced to 0° C. and sea-level in latitude 45°) is termed the boiling-point (B.P.) of the liquid, and is usually determined by taking the temperature of the saturated vapour under normal pressure, to avoid error from super heating (see below, 3) of the liquid. If the external pressure remains constant, the temperature will also remain constant, provided that the liquid is pure and that its composition remains unaltered, until the whole is vaporized. If, on the other hand, the liquid is contained in a closed space, it may be made to boil at much lower temperatures by diminishing the pressure; or the temperature of the liquid may be raised considerably above the normal boiling-point, as in the boiler of a steam-engine, if the pressure is raised by preventing the free escape of the vapour. In all cases, if the temperature is given, there is a corresponding equilibrium or saturation pressure of the vapour, and vice versa, in accordance with Dalton's law. It was shown, however, by Cagniard de la Tour (Ann. Chirn. Phys., 1822, 1823) that the temperature and pressure of the liquid could not be raised indefinitely in this manner. By heating liquids in strong glass bulbs with manometers attached, he found that at a certain temperature the meniscus or curved surface separating the liquid from the vapour disappeared, and the bulb became filled with an apparently uniform substance. The temperature at which this mixing of liquid and vapour occurs is definite for each liquid, and is called the critical temperature. La Tour found the critical temperature in the case of water to be 362° C., a result which has been remarkably confirmed by later researches (Cailletet, Ann. Chirn. Phys. 25, p. 519, 1892). In many books of recent years it has been the custom, following a suggestion of Andrews, to restrict the term “gas” to temperatures above the critical temperature, and the term “vapour” to temperatures below. But this is often inconvenient in practice, as there is no sudden change in the gaseous phase at ordinary pressures on passing the critical temperature. It is more convenient to employ the terms “ vapour ” only when discussing the properties of the gaseous phase in relation to the liquid or solid, and to follow the common usage in describing substances like CO2, or even SO2 and NH3, as gases at ordinary temperatures and pressures.

2. Continuity of State.-The form of the isothermal curve, representing the compression of a vapour at constant temperature, consists, as shown in fig. 1, A, of three discontinuous branches. The relation between pressure and volume for an unsaturated vapour is represented by the branch DE, which is similar to the isothermal of a gas obeying Boyle’s law. When the saturation-pressure is reached at D the vapour begins to condense, and the volume diminishes without further increase of pressure, giving the isopiestic branch DCB. At B, when the vapour is completely liquefied, further compression produces a rapid rise of pressure, as shown by the branch BA, representing the behaviour of the liquid.

Fig. 1.
A, James Thomson Isothermal; B, Isothermals of CO2 (Andrews).

It is possible, however, to trace the branch DN for the supersaturated vapour continuously beyond D without liquefaction in the absence of nuclei. It is similarly possible to trace the liquid branch ABM beyond B to lower pressures in the absence of dissolved gases. As the temperature is raised, the length of the branch BD, representing the increase of volume in passing from the liquid to the gas, diminishes, as shown in fig. 1, B, which represents the isothermals of CO2,[1] according to Andrews (Phil. Trans. 1869). Above the critical temperature, the discontinuities at B and D disappear from the isothermal curve, and it is impossible to obtain separation of the two states, liquid and gas, however great the pressure applied. The critical pressure is the vapour-pressure of the liquid at the critical temperature. It is possible to obtain a perfectly continuous passage from the gaseous to the liquid state by keeping the vapour at a pressure greater than the critical pressure while it is cooled from a temperature above the critical point, at which it would expand indefinitely (if the pressure were reduced) without separation into two phases, to a temperature below the critical point, at which expansion would produce separation into liquid and vapour as soon as the pressure was reduced to the saturation value. It was maintained by Andrews, on the basis of these and similar observations, that the gaseous and liquid states were merely widely separated forms of the same condition of matter, since one could be converted into the other without any breach of continuity or sudden evolution of heat or change of volume; just as an amorphous solid in the process of fusion becomes gradually more and more plastic as the temperature is raised, and passes into the state of a viscous liquid with continually diminishing viscosity. The same idea was further developed by James Thomson (Proc. R.S., 1871), who suggested that the discontinuity of the isothermal at temperatures below the critical point was only apparent. He supposed that the extensions of the liquid and vapour curves BM, DN, in fig. 1, A, representing the states of superheated liquid and supersaturated vapour, might theoretically be joined by a continuous Curve MN, representing a homogeneous transformation, which, however, could not be realized in practice, as the state of the substance corresponding to this part of the curve would be unstable. Maxwell (Nature, 1875) showed that the straight line BCD representing the saturation-pressure must cut off loops BMC, CND, of equal area from this imaginary isothermal; otherwise it would be theoretically possible to obtain a balance of work without any difference of temperature by taking the substance through the isothermal cycle BCDNCMB. The theoretical isothermal of James Thomson is qualitatively represented by an equation of the type devised by Van der Waals, in which the mutual attraction of the molecules of a gas is regarded as equivalent to an internal pressure of the form a/v2 , which he supposes identical with the capillary pressure of the liquid. It has been found, however, that this simple expression is not sufficiently exact. It is probable that it is not merely a question of varying attraction between similar molecules. A vapour should rather be regarded as containing a certain proportion of compound or coaggregated molecules, which partially dissociate when the pressure is diminished or the temperature raised. A liquid similarly contains dissolved molecules of vapour, and the state of equilibrium is more nearly analogous to that between conjugate saturated solutions (e.g. water and phenol).

3. Effect of Capillary Pressure on Ebullition.—It was remarked at a very early date that water and other liquids could be raised under atmospheric pressure several degrees above their normal boiling-points in a clean glass vessel without ebullition occurring, and that, when a bubble was formed, it would expand explosively, producing the phenomenon of “bumping”; but that, if metallic filings or other bodies capable of supplying small bubbles of air were introduced, ebullition would proceed quietly at the normal temperature. L. Dufour succeeded in raising small drops of water, suspended in an oil mixture of suitable density, to a temperature of nearly 180° C. under atmospheric pressure. Similar observations lead to the conclusion that the phenomenon of ebullition, or boiling with the formation of bubbles, depends essentially on the presence of air or dissolved gas to provide nuclei for the starting-points of the bubbles. This is a natural consequence of the capillary pressure due to surface tension. The vapour-pressure p inside a small spherical bubble of radius r must exceed the pressure P in the liquid just outside the bubble by 2T/r, where T is the surface tension of the liquid. The capillary pressure 2T/r may be. very large if r is small. It is often stated on the strength of this relation that a bubble of radius r in a liquid will not expand indefinitely and rise to the surface as in ebullition, until the vapour-pressure p inside the bubble exceeds the external pressure P by 2T/r. But this neglects the effect of the air or gas contained in the bubble, which plays an essential part in the phenomenon. A bubble of vapour containing no air or gas could not exist at all in stable equilibrium in a liquid. If its radius r were such as to make 2T/r greater than p−P, it would collapse entirely. A bubble containing gas, on the contrary, is in stable equilibrium when its radius r is such that the pressure of the gas and vapour inside it balance the external pressure P together with the capillary pressure 2T/r. Any diminution of r produces an increase in the pressure of the gas which is more than sufficient to balance the increase of the capillary pressure 2T/r. Supposing that the external pressure and temperature remain constant, the partial pressure of the gas inside the bubble varies inversely as the volume of the bubble, and may be represented by a/r3 . The size of the bubble is determined by the equation p+a/r3=P+2T/r. The equilibrium is always stable if p is less than P. If p is greater than P, the equilibrium becomes unstable (and the bubble expands indefinitely), when the gas-pressure a/r3 is one-third of the capillary pressure 2T/r. This follows immediately by differentiating the above equation with respect to r, assuming the difference p−P to remain constant. Substituting 2T/3r for a/r3 we obtain the condition of stability,

 p−P<4T/3r (1)

In other words, the temperature of a liquid containing bubbles of radius r will rise until the excess pressure given by (1) is reached, and ebullition will begin as soon as the excess pressure amounts to two-thirds of the capillary pressure, and will not be delayed until the full capillary pressure is reached, as might appear at first sight. Bubbles 1 millimetre in diameter in water at P＝760 mm. become unstable when the temperature reaches 100·05° C. approximately. To obtain a superheat of 10° C, where the excess pressure is 316 mm., the bubbles must not exceed about 1200th mm. diameter. The condensation of a vapour is also retarded by the effect of capillary pressure, but the relation in this case is somewhat different.

4. Effect of Capillary Pressure on Vapour-Pressure.—It was observed by Sir W. Thomson (Lord Kelvin) (Phil. Mag. iv. 42, p. 448, 1871) that if a capillary tube of radius r is immersed in a liquid of surface tension T, and the liquid rises to a height h above the plane surface (the whole being enclosed in a vessel of uniform temperature containing only the vapour of the liquid) the pressure of the vapour at the curved surface of the meniscus in the capillary tube will be less than that at the plane surface by the amount, gh/v, where g is the acceleration of gravity, and 1/v is the density of the vapour. But the vapour must be in equilibrium with the liquid at both surfaces. Otherwise perpetual motion would ensue in an enclosure at uniform temperature. Consequently the equili- brium value of the vapour-pressure must vary with the curvature of the surface, or with the capillary pressure due to the curvature.

If P, p are the hydrostatic pressures in the liquid and vapour close to the meniscus, the difference P−p = 2T/r. This is negative if r is negative, i.e. if the liquid rises in the tube, but is positive if the meniscus is convex and the liquid is depressed in the tube. If P0, p0 are the pressures in the liquid and vapour at the plane surface, P0p0, and if 1/V is the density of the liquid, the differences of pressure in the liquid and vapour respectively corresponding to a difference of level h, are P−P0＝−gh/V, pp0＝−gh/v. Combining these with the relation P−p＝2T/r and eliminating gh, we obtain, for the change of vapour-pressure pp0, due to change of pressure P−p0, or to curvature 1/r,

 p−p0＝(P−P0)V/v＝2TV/r−V) (2)

This increase of vapour-pressure with curvature affords a natural explanation of the fact that it is possible to cool a vapour considerably below the saturation temperature without condensation. The vapour-pressure in a fog containing small drops of radius r must exceed the normal vapour-pressure over a plane surface at the same temperature by the amount 2TV/r(v−V), which may be considerable if r is small. The same expression measures the supersaturation required to induce condensation in the presence of dust or other nuclei of radius r, and explains why it is that condensation always takes place on dust particles if any are present. This phenomenon forms the basis of J. Aitken's method of counting dust particles, or Wilson's method of counting electrical ions, which are also capable of acting as nuclei for starting condensation.

5. Extension to Higher Pressures.—The approximate formula above given for the effect of hydrostatic pressure on the vapour-pressure assumes the densities of the liquid and vapour constant, and is true for small differences of pressure only. If we take P0 and p0 to represent corresponding values of the pressure in the liquid and vapour at the same level (and not necessarily at the plane surface where P0p0), and if the difference of level from P, p is small, substituting dp and dp for the small differences of pressure, we have accurately the relation vdp=VdP, where V and v are the specific volumes of the liquid and vapour under the pressures P and p respectively. In order to apply the formula to large differences of pressure, it is only necessary to integrate it at constant temperature between the required limits of P and p. We thus obtain the general equation

 ${\displaystyle \int _{p_{0}}^{p}vdp=\int _{\mbox{P}}^{\mbox{P}}{\mbox{V}}d{\mbox{P}}}$ (3)

In applying the general equation (3) to an actual case, the compressibility of the liquid is the most uncertain factor. Assuming the compressibility constant, we may write V=V0(1−aP). For the vapour we may employ equation (17) Thermodynamics, viz. v＝Rθ/pc+b, as a very close approximation over a wide range. The small quantities c and b are functions of the temperature only. Making these substitutions and integrating the equation we obtain

 Rθ loge(p−p0)＝(c−b)(p−p0)+V0(P−P0)−12aV0(P2-P20). (4)

C. T. R. Wilson (Phil. Trans. 1898) has observed that in the absence of nuclei a very fine mist is formed in a vapour on sudden expansion when its density is about eight times the saturation value. Putting p/p0＝S in equation (4), and taking for water vapour R=4-6i Xio 6 , and 8 = 300° Abs. we find P−P0 equal to 3000 atmospheres approximately as the pressure required to produce this degree of supersaturation, allowing for compressibility of V. The term (cb) may be neglected in this case, as p is small, but it would amount to about 17 % of PV at 200° C. The result obtained from the approximate formula (2) would be 9200 atmospheres, which is more than treble, and indicates the inapplicability of the simple formula in an extreme case. Taking P = 3000 atmospheres, and assuming that the formula 2T/r applies for the capillary pressure, we find the equivalent radius of a nucleus corresponding to the fine misty condensation to be 5·0×10−8. This is a quantity of molecular dimensions, and lends support to the view that a vapour contains a certain proportion of coaggregated molecules, represented by the term c in the equation, -which are capable of acting as nuclei for condensation. The analogous phenomenon of cloudy crystallization, which takes place in a supercooled liquid in the labile state, suggests that a liquid may similarly contain molecular crystals of solid, which would account, in the case of water, for its anomalous expansion and for the variation of its specific heat near the freezing-point.

For small values of the vapour-pressure p, the term (cb) (pp0) in equation (4) may generally be neglected, as in the case of water at ordinary temperatures. For moderate values of P, not exceeding say 100 atmospheres, V may be taken as nearly constant, and the equation reduces to~the simpler form PV/R0 = loge(p/p0), which is often sufficiently exact.

6. Application to a Solid.—If we imagine a vertical column of solid in a porous vessel at uniform temperature surrounded by vapour, it would appear probable by similar reasoning that it would be in equilibrium under its own hydrostatic pressure with the pressure of the vapour at different levels. This would give the same formula as (2) for the variation of vapour-pressure, with V, the specific volume of the solid, in place of V. But since the surface tension analogy does not exactly apply in the case of a solid, it is perhaps better to deduce the formula from a consideration of the effect of pressure on the fieezing-point. The freezing-point do is the point at which the solid and liquid have the same vapour-pressure p0. Otherwise they could not remain together in equilibrium. When the freezing-point is changed by pressure, the vapour-pressures p′, p″), of the solid and liquid must be the same at the new freezing-point. The rise of the freezing-point θ−θ0, for an increase of pressure P−P0, is given by the thermodynamic equation (Thermodynamics, equation (5))

where L is the latent heat of fusion, and V′, V″ are the specific volumes of the solid and liquid respectively. The difference (p′−p″) of the vapour-pressures of the solid and liquid under normal pressure P0 at a temperature θ near the normal freezing-point θ0, is deduced from the same equation (see section 24 below)

 p′−p″＝L(θ−θ0/vθ0 (6)

where v is the specific volume of the vapour. Substituting for θ in terms of P from (5), we have for the difference of the vapour-pressures at do under pressure P,

 p′−p″＝(P−P0)(V″−V′)/v. (7)

The increase of vapour-pressure of the liquid when the pressure is increased to P is given by (2), viz. pp0=(P−P0)V″/v. The increase of vapour-pressure of the solid must be less than that of the liquid by the amount given by (7), in order that their vapour-pressure may be the same at the new freezing-point . We thus obtain by subtraction

p′−p0＝(P−P0){v−(V″−V′)/v}＝(P−P0)V′/v.

Which is precisely the same as relation (2) for the liquid, with V substituted for V″. Hence the effect of pressure on the vapour-pressure follows the same law for both liquid and solid (J. H. Poynting, Phil. Mag. xii. p. 40, 1881).

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” (pp0)/p0 of the vapour-pressure of a solution, or the ratio of the diminution of vapour-pressure (pp0) 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. Wullner (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 Wüllner's results, found the lowering of vapour-pressure 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, arid 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/dθ＝L/θv, nearly, neglecting the volume of the liquid as compared with that of the vapour v. If dp is the difference of vapour-pressure of solvent and solution, and dθ the rise in the boiling-point, we have the approximate relation,

 n/N＝dp/p＝mLdθ/Rθ2, Raoult's law, (8)

where m is the molecular weight of the vapour, and R the gas-constant 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 dd 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, (1) 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/k0)n, from the molecular conductivity 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/Rθ =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 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[2] 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(θ′−θ″). . . . (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 θ0 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 θ0. 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 θ′ and θ″ we see that Rankine’s result follows immediately, provided that p(vb) is equal to (S−s) or Rθ/m, which is approximately true for gases and vapours when v is very large compared with b. We may observe that the equation (11) is accurately true for an ideal vapour, for which pv＝(S−s)θ, 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/dθ) at constant, pressure is equal to S under all conditions, whether S is constant, or varies both with p and θ. (See Thermodynamics, § 7.)

11. 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＝1·40 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 dosely 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＝Rθ 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＝Bθ+Cp.25 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., θ＝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 (θ/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 0·46, with a mean value 0·3778. But it must be remarked that the superheat of the steam in these experiments is only 1 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 124° 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 0° to 0·341 at 100°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

Fig. 2.—Throttling Calorimeter Method.

of heat, the total heat (E+pv) remains constant (Thermodynamics, § 11), 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 saturated steam at a temperature θ′ and pressure p′ , as represented by the point A on the pθ diagram (fig. 2), if the point B represent the state pθ″ 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 θ) by the amount S×(θ″−θ), 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—

 HA−HD＝S(θ″−θ). (12)

If the steam at A contains a fraction z of suspended moisture, the total heat HA 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 (10), HA−HD＝·305 (θ′−θ), whence, if S＝·475, we have zL＝·305 (θ′−θ)−·475 (θ′−θ). 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(θ′−θ)/(θ″−θ), 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 100° 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/dθ＝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 0·387 at 100° C. to 0·665 at 160° C. Writing Q for the Joule-Thomson “cooling effect,” dθ/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 CO2; but that it varied with the temperature as (1/θ)3·8 instead of (1/θ)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/dθ＝ 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 0° 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)＝Rθ/p, 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 S＝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 (12). 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 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 (10) must be wrong, although the numerical value of the coefficient 0·305 may approximately represent the average rate of variation over the range (100° 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 hear 10 to 6 calories too large between 0° 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)＝Rθ/p−c0(θ0/θ)n＝V−c. (13)

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 + 1)c−b. (14)

The corresponding formula for the total heat is

 H−H0＝S0(θ-θ0)-(n+1) (cp−c0p0)+b(p−p0), (15)

and for the variation of the specific heat with pressure

 S＝S0+n(n+ 1)pc/θ, (16)

where S is the value of S when p＝0, and is assumed to be independent of θ, as in the case of an ideal gas.

Calendar's experiments on the cooling effect for steam by the throttling calorimeter method gave 71 = 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 S0 =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+1. The assumption n=s/R simplifies the adiabatic equation, but the value n = 3·5 gives S = 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 0° 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/dθ＝S,(1−Qdp/dθ) (17)

and diminishes from 0·478 at 0° C. to about 0·40 at 100° and 0·20 at 200° C, decreasing more rapidly at higher temperatures. The mean value, 0·313 of dH/dθ, between 100° 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/dO 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

 Rθ/p−v＝Rθ(v−b)(A−Bθ)/p(v+a)2θn; (18)

but such formulae 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 very 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 θ. 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+Bθ (Dalton, 1800) . . . . . . . . . . . . (19) log p＝C log (A+Bθ) (Young, 1820). log p＝Aθ/(B+Cθ) (Roche, 1830). log p＝A+Bbθ+Ccθ (Biot, 1844; Regnault). log p＝A+B/θ+C/θ2 (Rankine, 1849). log p＝A+B/θ)+C log θ (Kirchhoff,1858; Rankine, 1866). log p＝A+B/θb (Unwin, 1887). log p＝A+B log θ+ C log (θ+c) (Bertrand, 1887). log p＝A+B/(θ+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/dθ) divided by p should be constant and equal to B ; but observation shows that this ratio decreases, e.g. from 0·0722 at 0° 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/θ represents a much closer approximation to experiment. As a matter of fact, the two terms A+B/θ 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/θ for two different substances which have the same vapour-pressure p at the absolute temperatures θ′ and θ″ respectively, we may write

 log p＝A'+B/θ′＝A″+B″/7θ″, (20)

from which we deduce that the ratio θ′/θ″ of the temperatures at which the vapour-pressures are the same is a linear function of the temperature θ′ 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 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/dθ), which is given by the second law of thermodynamics, viz.

 θ(dp/dθ)＝L/(v−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 vw are known as functions of θ. 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 (10) 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, 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 non-condensible 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 ${\displaystyle p(v-w)=\mathrm {R} \theta }$), and that L is constant. This leads immediately to the simple formula

 ${\displaystyle \log _{e}\left(p/p_{e}\right)=\left(1/\theta _{0}-1/\theta \right)\mathrm {L} /\mathrm {R} }$, . . . (22)

which is of the same type as ${\displaystyle \log p=\mathrm {A} +\mathrm {B} /\theta }$, 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 ${\displaystyle L}$ = constant is not sufficiently accurate in many cases. The rate of variation of the latent heat at low pressures is equal to ${\displaystyle S-s}$, where ${\displaystyle s}$ is the specific heat of the liquid. Under these conditions both ${\displaystyle S}$ and ${\displaystyle s}$ may be regarded as approximately constant, so that ${\displaystyle L}$ is a linear function of the temperature. Substituting ${\displaystyle L=L_{o}+\left(S-s\right)\left(\theta -\theta _{0}\right)}$, and integrating between limits, we obtain the result

 logep＝A + B/θ+C logeθ, (23)

where

C＝(S−s)/R, B＝−[L0 + (s−S)θ0 /R,

and

A＝logep0−B/θ0 − C logeθ0,

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 H2SO4 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/dθ)s−(dp/dθ)l of the vapour-pressure curves of a solid and liquid at the triple point, which is immediately deducible from (21), viz.

 dp/dθ)s−(dp/dθ)l＝(Ls-Ll)/(v−w)＝Lf/(v−w), (24)

in which Ls and Ll are the latent heats of vaporization of the solid and liquid respectively, the difference of which is equal to the latent heat of fusion Lf. 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, Ls＝674·7, Ll＝595·2, Lf＝79·5, R＝0·1103 cal./deg., p＝4·61 mm., s-S＝·5l9 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 log10p = 0·6640+9·73t/θ, Water⁠ log10p = 0·6640+8·585t/θ−4·70(log10θ/θ0−Mt/θ),

where t = −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:—

 Temperature, C. −20° −10° 0° +10 +20° V.P. of ice, mms. 0·79 1·97 4·61 10·20 21·27 V.P. of water, mms. 0·96 2·17 4·61 9·27 17·58

The error of the formula for water is less than 1 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 (11) for the total heat of gasification with (21), and assuming an ideal vapour. A formula of the same type was given by Athenase Dupré (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 (10) for steam. It is generally called Dupré’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 0° C, which are impossible, and the values of p given by his formula (e.g. 763 mm. at 100° 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. 177, 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 1 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 errors 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 (${\displaystyle v-w}$) and ${\displaystyle L=H-h}$ from equations (13) and (15) 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 ${\displaystyle \left(c-b\right)/V}$ to formula (23)

 ${\displaystyle log\ p=A+B/\theta +C\log \theta +\left(c-b\right)/V}$ . . . (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 saturation- pressure itself. The value of A for steam is determined by the consideration that ${\displaystyle p}$= 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 ${\displaystyle b}$ in the formula (13). The term b, however, is only 4% of ${\displaystyle c}$ at 100° C, and the error involved in taking ${\displaystyle b}$ 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 ${\displaystyle h=S_{o}t+dh}$, where so is a selected constant value of the specific heat of the liquid, and ${\displaystyle dh}$ represents the difference of the actual value of ${\displaystyle h}$ at t from the ideal value Sol, and if we similarly write ${\displaystyle \theta =s_{0}\log _{e}\left(\theta /\theta _{0}\right)+d\phi }$ for the entropy of the liquid at t, where ${\displaystyle d\theta }$ 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 ${\displaystyle -\left(d_{\theta }=dh/\theta \right)/R}$. If we proceed instead by the method of integrating the equation H−h=0(vw)dp/d8, we observe that the expression above given results from the integration of the terms −dh/Rθ2+w(dp/dθ)/Rθ, 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 100° and 200° C, we find the values of the difference (ddh/θ) to be less than one-tenth of dφ 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 dhwθdp/dθ=wL/ (vw). 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/(vw) 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 Li 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 (cb)/V to the expression for log p) is independent of the assumption that c varies inversely as the nth power of θ, and is true generally provided that cb 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 cc0 (θ0/θ)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＝100° C. or 373·0° Abs. pV/d＝0·11030 calories per degree for ideal steam. S0=0·478 calories per degree at zero pressure, L＝540·2 calories at 100° C. (Joly- Callendar), n＝3·33, c100＝26·30 c.c, b＝1 c.c, h=0·9970t+wL (vw). 750 mm. Hg.＝1 megadyne per sq. cm.

Table of Properties of Saturated Steam[3]

 Total Cent. Coaggre-gation, c, cub. cms. Total Heat, H, calories. Latent Heat, L, calories. Specific Heat, S, cals./deg. Saturation-Pressure, p, mm. of Hg. ⁠0° 74·43 595·2 595·2 ·4786 ⁠4·6 20° 58·81 604·7 584·7 ·4796 17·6 40° 47·49 614·0 574·0 ·4318 55·4 60° 38·68 623·1 536·1 ·4860 149·4 80° 31·60 631·9 551·9 ·4926 355·0 100° 26·30 640·3 540·2 ·5027 760·0 120° 21·93 648·1 527·8 ·5163 1490·4 140° 18·73 655·1 514·5 ·5347 2715·8 160° 16·00 661·4 500·3 ·5571 4647 l80° 15·76 666·9 485·3 ·5834 7534 200° 11·92 671·6 469·3 ·6134 11660⁠

The values of the co aggregation-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 isothermal 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 colorimetric 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:—

 Hydrogen −252°·6 Benzophenone⁠ +305°·8 Oxygen −182°·8 Mercury +356°·7 Carbon dioxide − 78°·3 Sulphur . . +444°·5 Sulphur dioxide⁠ − 10°·0 Cadmium . +756° Aniline +184°·1 Zinc. . +916° Naphthalene +218°·0
 Alphabetical Index of Symbols A, B, C, Empirical constants in formulae; section 14. b, Minimum volume or co-volume of vapour, equation (13). C, Concentration of solution, gm. mols. per c.c. c, Coaggregation-volume of vapour, equation (13). D, d, Density of liquid and vapour. E, Intrinsic energy of V3.p0llI'. g, Acceleration of gravity. H, Total heat of vapour. h, Heat of the liquid; height of capillary ascent. L, Latent heat of vaporization. M, Modulus of logarithms. m, Molecular weight. n, Index of θ in expression for e, equation (13). P, Osmotic or capillary pressure. p, Pressure of vapour. Q, Cooling effects in adiathermal expansion. R, Constant in gas equation, pv=Rθ. 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. t, 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. θ, Temperature on thermodynamic scale. φ, Entropy of vapour or liquid.

1. The slight increase of pressure observed during condensation was attributed by Andrews to the presence of a trace of air in the CO2.
2. "Latent Heat of Steam," Phil. Trans. A. 1895; of "Benzene," Phil. Mag. 1896.
3. 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 und Diagramme fur Wasserdampf, published by J. Springer (Berlin, 1906).