Rankine (1851) was the first to show that the total heat of an ideal vapour reckoned from the state of liquid at o° C should be where S is the specific heat of the vapour at constant pressure, and L. the latent heat of vaporization at o° C. The total heat of the liquid reckoned from o° C is simply h = st, where s is specific heat of the liquid, and is taken as constant. Hence the latent heat L at any temperature t will be approximately, If the vapour also satisfies the gas law, as is generally the case at low pressures, we may substitute apV=RT in (2), and neglect v, which is generally less than a thousandth of V at atmospheric pressure. Equation (2) then reduces to the simple form, = (dp/dt)/p, which is easily integrated with the value of L from (3), and gives a theoretical equation for p, where B C = (s–S)/R, and A is the constant of integration. This equation must necessarily give a good first approximation to the saturation pressures if the values of the constants L., s, and S are correct. Thus in the case of steam, if we take L=594.3, 5= 0.9967 (the minimum specific heat of water) S = 0.4772= 13R/3, equation (4) gives values at low pres sures which are correct to o.i° C at a temperature as high as 6o° C. Beyond this point the approximation begins to fail, chiefly in the first instance because apV begins to deviate appreciably from the ideal value RT above assumed. At 1 oo° C the value of p given by (4) is already about 1.5% too small, corresponding to a defect of the same order of magnitude in the value of V. This agreement verifies Clapeyron's equation (2), and the assump tion apV=RT at low pressures, as well as equation (3) for L, but it is necessary to obtain consistent expressions for the defect of both L and V from these ideal values before equation (4) can be extended to higher pressures. Rankine, who first gave a correct deduction of (4) (Trans. R.S.E. 1865), came to the conclusion that both L and V must show an increasing defect from their ideal values as the pressure increased, but he was unable to trace the required conneetion between them. Equation (4) is generally attributed to Dupre (1869), who deduced it by assuming a linear formula for the latent heat, and used it as a purely empirical formula by calculating the coefficients from observations at high pressures. Bertrand, in his treatise on Thermodynamics, followed the same course, which is necessarily unsatisfactory, because it does not comply with the theoretical conditions, or give the true relations of the constants B and C to L and S.
Deviations of a Vapour from the Ideal State.—The manner in which a gas passes into a vapour and finally condenses, must be studied in the first instance by observing the gradual deviations from the gas law aPV=RT, as the pressure is increased. The simplest way of doing this is to find the departure from Boyle's law at various constant temperatures, which gives the whole story if the range of temperatures is sufficiently extended. Regnault (1847) was the first to make observations of this kind with sufficient accuracy in the case of Rankine (1854) suc ceeded in representing these experimental results by a fairly simple equation in a convenient form, The second term, in which A is an empirical constant, represents the deviations from the gas law, and tends to vanish when V or T is large. It is of the dimensions of a pressure, and may be inter preted as an internal pressure, varying as the square of the density, which is added to the external pressure P, and tends to reduce V below the ideal value. According to Boyle's law, the product PV of the observed values of P and V at various pressures should be constant at each temperature at which observations are made, and should give a horizontal straight line when PV is plotted against P. Equation (5) when plotted in this manner, gives a series of
parabolas, each with its axis horizontal and its vertex at a height equal to half the initial value of PV. This is easily seen by mul tiplying equation (5) by which gives it in the form of a quadratic in (PV), the two roots of which become equal when P=RT/2aV, and are imaginary at higher pressures. The equation would evidently fail before this point was reached, but was found to be quite satisfac tory for the restricted range of Regnault's experiments. It also represented his observations on the pressure-coefficient of CO2 very closely, and Joule and Thomson found that it also satisfied their results for the cooling effect in expansion through a porous plug. They found however that (5) was very inconvenient in form for their purpose, because it gave P as a quadratic function of 1/V, whereas they wanted V explicitly as a function of P. They there fore transformed equation (5) by multiplying throughout by V/P, and making the approximation PV = RT/a in the small term A/PVT, thus obtaining, The small term is now of the dimensions of a volume, and may be interpreted as a first approximation to the reduction of volume A/PVT due to the internal pressure as in equa tion (5). It also might be interpreted on the kinetic theory (which was expressly designed to explain the pressure of gases and vapours without assuming imaginary forces of attraction or re pulsion between the molecules) as being due to a reduction in the effective number of molecules per unit mass owing to the forma tion of multiple or complex molecules by coaggregation. This process is well known to occur in analogous chemical problems, where the proportions of different molecules existing in a gas mixture can be analysed and verified. On this view, the reduction of V below the ideal value RT/aP is due to a corresponding re duction in the number of effective' molecules ; and the formation of complex molecules is re garded as the first stage in the transition from the state of va pour to the state of liquid, in which nearly all the molecules are coaggregated. The reduction of volume given by the Joule-Thomson equation (6) is a function of the temperature only, and should be a very good first approximation to the coag gregation at low pressures ac cording to the kinetic theory. If the isothermals are plotted on the PV—P diagram, they will all be straight lines, as is easily seen by multiplying (6) by P, so as to give the product PV. The slope of successive isothermals is pro portional to and dimin ishes with rise of temperature.
The diminution of slope with rise of temperature is to be expected on thermodynamic grounds, if the molecules com bine with evolution of heat, which would afford the most natural explanation of the latent heat of condensation. The term A/PVT representing the reduction of volume according to Rankine's equa tion, though fitting exactly with the hypothesis of an internal pressure varying as the square of the density, might equally well represent the effect of coaggregation. We might expect that the complex molecules would begin to combine with each other as the proportion of such molecules in the mixture increased with in crease of density, and that the slope of each isothermal would increase accordingly, giving the parabolic form (5)a in place of the rectilinear-isothermals given by (6). This is certainly true at high pressures near saturation, though the isothermals approxi mate more nearly to straight lines at temperatures above the criti cal point; but neither of these equations could be expected to apply to the liquid state, or even to the vapour in extreme cases, beyond the limits of the observations on which they were founded.