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 v−w 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,
