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, P0＝p0, 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, p−p0＝−gh/v. Combining these with the relation P−p＝2T/r and eliminating gh, we obtain, for the change of vapour-pressure p−p0, 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 P0＝p0), 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
| (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/p−c+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 (c−b) 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 (c−b) (p−p0) 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))
| L(θ−θ0)/θ0＝(P-P0)(V″−V′), | (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. p−p0=(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” (p−p0)/p0 of the vapour-pressure of a solution, or the ratio of the diminution of vapour-pressure (p−p0) 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
