The theory of the continuity of state, which is now so familiar as an explanation of the relations between liquid and vapour, appears to have originated from the experiments of Andrews (Phil. Trans. 1869) on the properties of carbonic acid (CO2) between the temperatures of 13° and 43° C, including the critical region. The vapour was confined in a capillary tube by a pellet of mercury, and the apparatus was ar ranged so that the volume could be varied by means of a screw plunger, while the tube was maintained at a steady temperature in a water bath. A similar capillary containing air subject to the same pressure was employed as a manometer (for details of apparatus see article LIQUEFACTION OF GASES). The isothermal curves obtained by plotting simultaneous values of P and V ob served at the temperatures of 13.1°, 21.5°, 31.1°, 35.5° and 481° C, are shown in fig. 2B. They showed the well known dis continuities, required by Dalton's law, at the commencement and conclusion of condensation, but the sharp corners were rounded off at these high pressures, and the pressure did not remain quite constant during condensation, but showed a rise of about 2 per cent attributed by Andrews to an impurity of about 2 parts of air in 2,000 of
The dotted line shown in the figure has been added to Andrews' original diagram to indicate the satu ration lines for the liquid and vapour, meeting at the critical point 31.1°, with the critical isothermal as a common tangent. At any point within the dotted boundary, e.g., at 21.5°, the fluid exists partly in the state of vapour and partly in that of liquid, in stable equilibrium with the vapour, as is usually observed during the condensation of a vapour by compression at constant tempera ture. But Andrews showed very conclusively that a vapour could be transformed into a liquid by a continuous process without any breach of homogeneity, or separation into the two states of liquid and vapour, provided that the change was effected along any path on the diagram which did not intersect the dotted boundary curve. Thus if the vapour were taken at a temperature of 48°, and at a pressure of 8o atmospheres, and were then cooled at constant pressure to a temperature of 13°, at which it was certainly liquid, the fluid remained homogeneous throughout the process, and showed no sign of separation into two states at any point. He concluded from these experiments that "The gaseous and liquid states are only widely separated forms of the same condition of matter, and may be made to pass into one another by a series of gradations so gentle that the passage shall nowhere present any interruption or breach of continuity." James Thomson (Proc. R.S. 1871) extended this view to the re gion included by the boundary curve. He maintained that the dis continuities ordinarily shown at the beginning and end of con densation, were apparent rather than real. He quoted Donny's experiments as showing that, in the absence of dissolved air, the liquid curve AB in fig. 2 A, could be traced along the extension
BM to pressures far below the saturation line BCD, without any formation of vapour. It was possible to imagine the vapour curve ED similarly extended in the direction DN to pressures above saturation. This had been suggested by Kelvin (Phil. Meg. 187o) on theoretical grounds, though it had not then been proved by experiment. The two curves thus extended might be joined into one continuous isothermal by the intermediate branch MCN. This branch would involve unstable conditions (V increasing with P) and could not be realized in practice, though it might be repre sented by some type of equation such as a cubic. If this could be done, it would have the great advantage of representing both liquid and vapour by a single equation of state, which would take account of all possible modes of transformation.
The continuous iso thermal of James Thomson was first realized in the form of an equation by van der Waals in his famous essay On the Continuity of the Liquid and Gaseous States (Leyden, 1873), which put the whole theory in a more definite form. He assumed that the co hesion of the liquid, shown by . Donny's experiments, could be attributed to an internal pressure, such as that invoked by Laplace to account for the surface tension. He considered that this internal pressure should depend on the attraction of contiguous parts of the fluid for each other, and should vary as the square of the density, or as
It would then also explain the deviations of the vapour from the ideal state, as in Rankine's equation for CO2. Thus both liquid and vapour would be represented by the same equation, which implied that the molecules were identical, and that the two states differed only in density. The only modification necessary in Rankine's equation was the inclusion of the co-volume b in the last term, to represent the irreducible volume of the mole cules themselves at high pressures, as demonstrated by Natterer (1854). This also had the effect of transforming the equation into a cubic in V as required by the theory of the James Thomson isothermal. We thus obtain the equation, The factor T in Rankine's expression for the internal pressure was omitted by van der Waals on the ground that an internal pres sure, representing a force of attraction between the molecules, should be independent of the temperature. But this does not make any material difference to the critical relations. It will be evident that, for given values of P and T, equation (7) gives a cubic in V, and may have three real roots within certain limits represent ing the points of intersection, B,C,D, in fig. IA, of the isothermal of T with the line of constant pressure P. The smallest of these roots, given by the point B, may be identified with the volume of the liquid, v, and the largest, given by the point D, with that of