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 ), and that L is constant. This leads immediately to the simple formula
| , . . . | (22) |
which is of the same type as , 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 = constant is not sufficiently accurate in many cases. The rate of variation of the latent heat at low pressures is equal to , where is the specific heat of the liquid. Under these conditions both and may be regarded as approximately constant, so that is a linear function of the temperature. Substituting , 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 () and 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 to formula (23)
| . . . | (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 = 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 in the formula (13). The term b, however, is only 4% of at 100° C, and the error involved in taking 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 , where so is a selected constant value of the specific heat of the liquid, and represents the difference of the actual value of at t from the ideal value Sol, and if we similarly write for the entropy of the liquid at t, where 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 . If we proceed instead by the method of integrating the equation H−h=0(v−w)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 (d−dh/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 dh＝wθdp/dθ=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 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 (c−b)/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 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＝c0 (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 (v−w). 750 mm. Hg.＝1 megadyne per sq. cm.
