2. Before applying equation (5) to various liquids we will investigate its properties. The equation may be given in another simpler form, more convenient for calculation. To T-To do this let us resolve log into a series in powers of T T and cast aside the members higher than the second power. In this manner we shall have, instead of equation (5), the following: log P Ро = c-c1 T-T2 m 2AD T 2 + mro/T AD 0 This equation serves only in the case when the fraction T-To is small, and when, consequently, the temperature T is T not too far removed from To; the results given by it will be particularly near to those of experiment if c-c1 be small. I have applied this equation to bisulphide of carbon [12]* in order to form an idea of its accuracy, and found that the simplified equation (5′) and the complete equation (5) give similar, or very nearly similar, results. Experiments have shown that the difference c-c1 is exceedingly small, compared to the heat of vaporization, for all investigated substances. Therefore the first member of the equation may be neglected, and we shall have Whence a formula analogous to that of Roche † can be easily obtained by integration, t p=aal+mt. c-c1 approaches to zero a, a, and m are arbitrary constants. By choosing them in the most advantageous manner Regnault obtained a formula which amply satisfied the results of observation, and which was hardly less satisfactory than Biot's formula with five arbitrary coefficients. The nearer the better does formula (r) agree with the results of experiment. Among the substances for which I made calculations there was one, namely ethyl bromide [14], for which c-c1 closely equalled zero. I calculated its vapour-pressure according to formula (r). Thus Roche's formula has a *The figures placed in brackets refer to the numbers of the sections in this paper. + Mémoires de l'Acad. des Sci. t. xxi, p. 585. theoretical basis; this explains its accordance with the results of experiments. The same formula may be deduced upon other bases. According to Southern and Crighton's law, the inaccuracy of which was, however, proved by Regnault's experiments, the heat of vaporization does not depend upon the pressure on the surface of a liquid remaining constant at all temperatures. If this law were true, then instead of equation (4) we should have the less accurate one r=ro. Then equation (4') would be converted into the following: whence by integration we obtain equation (r). Thus Southern's inaccurate law and equation (r) proceed from one and the same supposition, that c-c, is negligible compared with r. This is sufficiently near the truth, and may be regarded as the very first approximation. From the preceding we have the right to conclude that Roche's formula, although roughly, still presents a law of nature. All the other empirical formula-Young's, Arago's, Dulong's, Tredgold's, Cariolis', and others, and among them Biot's have no theoretical basis, and with the exception of the latter do not satisfy the results of experiments. It is true that Biot's formula gives vapour-pressures very nearly approaching those found by experiment; but this is not due to its representing in itself the nature of a vapour, but because of the large number (five) of arbitrary coefficients, or, in geometrical language, because the curve expressed by Biot's formula has a large number (five) of points common to the curve of the actual vapour-pressures in a state of saturation. 3. Let us investigate the curve expressed by the equation (5), taking as abscissæ the absolute temperatures T, and as ordinates the pressures p. From this formula it is easy to obtain It is evident from equation (p) that the curve passes through the origin of the coordinates, and from equation (p1) that the curve touches the axis of abscissæ at this point. From d'p equation (p2) it follows that becomes zero twice as T 1 when T1 = (1 When T> T2, consequently p then attains its maximum. Thus, starting from the origin of the coordinates, the curve diverges from the axis of abscissæ and turns its convexity towards it. When T R (1) an inflexion takes place; the curve turns R x its concavity towards the axis of abscissæ ; p continuing to augment attains a maximum when T=. Then p decreases; a second inflexion occurs, and then the curve asymptotically approaches the axis of abscissæ. For the subject under consideration the entire curve is not of importance, but only that inconsiderable portion of it between the origin of coordinates and the first inflexion, beyond which observations are never carried. Moreover, it must be remembered that the formula under consideration refers only to certain special conditions, one of which is never actually fulfilled, namely—that the volume of a liquid is equal to zero compared to the volume of the vapour formed from it, so that the curve represented by equation (p) serves to express the actual variation of the pressure of a vapour, and would only fully express these variations if the above conditions were strictly observed*. The temperature of the first point of inflexion is very high. For water, for instance†, R5106.2+5.2948 x 343=6922.3; The highest temperature of steam attained by Regnault was only 230°. So also it is easy to find that for sulphuric ether the first inflexion of the curve corresponds to 74302, for benzene 886° 7, sulphur 1279°. With the majority of the substances I calculated for, ≈ is a positive quantity, but it is sometimes negative and may be equal to zero; R was always positive. If a <0, when its absolute magnitude >1, then it is easy to prove, from the abovementioned equations (p), (pi), and (p2), that as T varies from zero to infinity, the curve diverges continuously from the axis of abscissæ, and always turns its convexity towards this axis, without giving any special points. We may add that it would be absurd to deduce any properties of a vapour from the properties of the curves and *My paper "On Van der Waals' Formula" (Russian Physico-Chemical Society, vol. xix.) proves that for all substances the second differential coefficient dp, where p is a function of and t, is greater than zero, and that it is only for perfect gases that =0. This may seem a contradiction to the above, because the value of for the curve (5) is either dt2 dp dt2 d2p less or equal to zero. In reality there is no contradiction at all. To be concise, I will only observe that the curve expressed by equation (p) presents the variation of the vapour-pressure under certain conditions and within certain limits; in the given instance it only applies for that portion of the curve which lies between the origin of coordinates and the first point of inflexion, and then the condition that > 0 is fulfilled. d2p dt2 † In order to calculate R and T, it is necessary to know a aid T1; the mode of determining these quantities is given below. For water x= = 5.2948, T=343° [6]. Phil. Mag. S. 5. Vol. 37. No. 224. Jan. 1894. E equation (5) we have just considered, because they do not refer to actual vapours but only to imaginary ones such as do not really exist. 4. I will now pass to the applications of equation (5) and to proving that its theoretical bases are confirmed by experiment. The fundamental proposition upon which equation (5) is based is that at a certain temperature a vapour in a state of saturation has the properties of a perfect gas. Above and below this temperature it diverges from these properties, but for the present investigation it does not matter in which direction. It follows from the calculations given below that such a temperature does actually exist for the majority of substances yet experimented upon, and which are liquids at the ordinary temperature and pressure. The vapours of other liquids are not subject to the laws of Boyle and Gay-Lussac within the limits of the observations made by us. Without fresh observations it is impossible to say if these vapours are able to exist in the state of a perfect gas outside the temperatures observed. Equation (5) is naturally as yet inapplicable to such vapours. Our theory should not be applied to gases which are able to take the form of liquid only under very high pressures, because the volume of the resultant liquid would then be somewhat considerable compared with the volume of the gas and could not, therefore, be taken as zero, as our theory demands. 5. Suppose that in equation (5) T To T-T log2 = [log-m] + ( − 1)my. (7) Po T Pi were If x and y be regarded as arbitrary constants, then they might be determined, if any vapour-pressures p, Po, and known corresponding to temperatures T1, To, and T. If, moreover, vapours in a state of saturation followed Boyle's and Gay-Lussac's laws, and if in general the propositions which served for the deduction of equations (5) or (7) were unimpeachable and perfectly accurate, then the pair of unknown quantities x and y would be the same whatever were the temperatures T1, To, and T', and their corresponding tensions P1, Po, and p'. In reality just the contrary occurs. Every three observations, taken at random, give in general different magnitudes for x and y, so that not unfrequently |