there is no similarity between two pairs of these magnitudes. And so it should be, because equation (5) can only be applied with the observation of certain conditions, which generally speaking are not fulfilled. Nevertheless, it is possible with nearly all liquids to find a temperature To which has such properties that the magnitudes x and y are approximately the same whatever three temperatures are taken, if only they are adjacent to To and to each other. We will now turn our attention to determining this temperature. I have discovered two methods, which are here given, and for the sake of clearness applied to aqueous vapour. 6. Let us take three adjacent temperatures, T1, To, and T, differing for instance by 5°, so that T'-T=T—T1=5, and calculate the values of x and y. Then let us calculate their values for three other temperatures as near as possible to the first; and then for three more temperatures, and so on; and thus combine a more or less considerable number of observations. We thus obtain values for x and y which are different, but with the majority of substances experimented upon (i. e. for which there are tables of vapour-pressures, in a state of saturation, æ, varying uninterruptedly with the corresponding temperatures) give a maximum or minimum value. If the mean (intermediate) of the temperatures To be laid along the axis of abscissæ, and x along the axis of ordinates, then a curve is obtained with a maximum or minimum. There are, however, vapours for which the curve of a's continuously recedes from or approaches the axis of abscisse with a rise of temperature; the path of the curve indicates where the maximum or minimum lies, beyond the greatest or least of the temperatures observed. The value of c-c1 varies very inconsiderably within small limits of temperature; therefore, the magnitude x should remain constant, whatever combination of temperatures be taken, at temperatures near to that at which the vapour has the properties of a perfect gas, and when in general the above theory stands good; and this is only possible when ≈ has a value near to its maximum or minimum*, because a considerable variation in T then produces an inconsiderable variation in x. Hence the desired temperature To is that at which has a maximum or minimum value. We may here remark that at this time y varies with the temperature, as is seen from equation (4); or, more accurately, the increment of y is equal to a multiplied by the increment of temperature. *I do not consider the case when the curve has an inflexion whose direction is parallel to the axis of abscissæ, because I have not met with such an instance in my investigations upon vapours. If x has no maximum or minimum, then the vapour does not possess the properties of a perfect gas at any temperature, within the limits of observation, and the theory is not applicable to it. Then y does not vary proportionally to the temperature. To calculate the values of x and y it is sufficient to have two equations and three observed tensions corresponding to three temperatures. For the sake of simplicity let us take equidistant temperatures : T-T=To-T1=A. We shall thus have two equations of the first degree with two unknown quantities: 1 — (log-m. ToT)x+(1 − r) my=log To Po Ρί -(log-m. T)x+(-1) my = log. T Po (8) (9) To obtain to a sufficient degree of accuracy, the three temperatures for which this value is calculated must be taken as near as possible to one another, because the direction of the curve of a's can only be looked upon as parallel to the axis of abscissæ, near the maximum and minimum, for an inconsiderable distance. Moreover, this requires exceedingly careful observations of the vapour-pressures, as otherwise the series of values of a present such irregularities as to prevent the possibility of distinguishing the maximum or minimum of this quantity. These irregularities become less perceptible as the difference of temperature increases and the order of the variation of the a's becomes clearer; but in this case, according to what has been said above, the maximum or minimum value of x and the corresponding temperature may prove insufficiently accurate. Let us apply equation (8) to aqueous vapour and take, at random, temperatures 40°, 45°, and 50°. If we take the temperature of absolute zero as -273°, then these temperatures counted from the absolute zero will be 313°, 318, and 323°, and the corresponding pressures calculated by Broch according to Regnault's observations, 54.8651 millim., 71-3619 millim., and 91-9780 millim. On substituting these figures in equation (8) we find X45=3.9542. In this, and in all further calculations, a refers to the intermediate temperature, and we now mark 45 by the number 45. In a like manner we find for temperatures 45°, 50°, and 55°, X50=4.3293. Below is a table in which the upper line contains the intermediate temperatures (t) and the lower the corresponding values of x :— t 45° 50° 55° 60° 65° 70° 75° 80° 3.9542 4-3293 4.6800 5.0031 5.2109 5.2948 5.1997 3-8844 In this series of figures the greatest value of x is 5.2948, corresponding to 70°; a diminishes with a rise and fall of temperature, at 15° it is 2-0640 and at 125° 3·4263. Hence it is evident that near 70° aqueous vapour follows the laws of Boyle and Gay-Lussac; i. e. satisfies the equation If in the equation pv=DT. we replace x and AD by their values, we have 2 P 2 c-c1=x.AD=5.2948. -5.2948. =0.58831. 18 The specific heat c as determined by various experimenters shows a great diversity. According to Regnault c at 70° equals 1.0072, while according to Hendrichsen it is 1.0419. The value of c was determined by Regnault for temperatures above 100°, at 70° it is probably less. Thus, according to Regnault c-c1=0.5167, and according to Hendrichsen it is 0.5614. The latter more nearly approximates to that found * Physikalisch-chemische Tabellen von Landolt und Börnstein, § 18. above, 0.58831, and would be still nearer if a smaller figure were taken instead of 0.4805. On making the requisite substitutions in equation (9) we According to Regnault's well-known formula we find that the latent heat of vaporization at 70°, 770=557.6. The difference between the values of r, and 70 is less than 2 per cent. cent. The accordance between the results of theory and experiment must be regarded as fully satisfactory if we take into consideration the imperfect observation of the requisite conditions and especially the inexactitude of the experimental determinations of the amount of heat. If instead of 70° we take a temperature which is considerably higher or lower, then the results are less accordant; thus at 125° we find 542-8 for the latent heat according to equation (9), while, according to Regnault, 125=519-6. The value of c-c1 will then be 0.3807, which diverges still more from the results of experiment. Το 7. We will now consider the second mode of calculating and C-C1. It follows from equation (3) that This equation gives the possibility of calculating the heat of vaporization if it be assumed that the vapour follows the laws of Boyle and Gay-Lussac, and that in general the above enunciated theory contains no inexactitudes. But in order to make this calculation it is necessary to know the dp first differential coefficients of pressure the determination " dt of which, however, presents considerable difficulty. If this quantity, at T and p, be taken as equal to the difference between p and the pressure answering to the temperature T-A, divided (i. e. the difference) by A, then we find that de dt is too small, while if we take the temperature T+A it is too great. Generally the arithmetical mean of these two Variation in the Pressure of Saturated Vapours. 55 values is taken, but still it is far from being the true value, and the difference is greater the more rapidly p varies. In order to lessen the error it is necessary to make use of more or less complex interpolation formulæ. p=a+a1T+a2T2, Let where a, a, a, are arbitrary constants which may be calculated from three observations. Hence dp this equation only leads to the above method, which, as we saw, was not sufficiently accurate. In order to attain a greater degree of accuracy, let us take the more complex formula the four arbitrary constants are determined from four observations. Hence we have Let the vapour-pressures corresponding to the temperatures T1, T2, T3, and T be P1, P2, P3, and p; we will denote the differential coefficients corresponding to these pressures thus:dpi dp2 dpз dps We will assume that tables of vapour-pressures are formed for equal intervals of temperature, so that T4—·T3=T3-T2=T2—T1=A. On substituting the pressures P1, P2, P3, and P4 with their corresponding temperatures in equation (T), we obtain four equations, from which, after a very easy transmutation, we find dp1 = 11 (p1—P3) — 7 (p3—p2) +2 (pa—P1) dt = 6A (11) |