Note 34 — Chapter II · THE COMPOSITION OF WATER, HYDROGEN

After the idea of the absolute temperature of ebullition (tc, Note 29) had been worked out (about 1870), and its connection with the deviations from Mariotte's law had become evident, and especially after the liquefaction of permanent gases, general attention was turned to the development of the fundamental conceptions of the gaseous and liquid states of matter. Some investigators directed their energies to the further study of vapours (for instance, Ramsay and Young), gases (Amagat), and liquids (Zaencheffsky, Nadeschdin, and others), especially to liquids near tc and pc; others (Konovaloff and De Heen) endeavoured to discover the relation between liquids under ordinary conditions (removed from tc and pc) and gases, whilst a third class of investigators (van der Waals, Clausius, and others), starting from the generally-accepted principles of the mechanical theory of heat and the kinetic theory of gases, and assuming in gases the existence of those forces which certainly act in liquids, deduced the connection between the properties of one and the other. It would be out of place in an elementary handbook like the present to enunciate the whole mass of conclusions arrived at by this method, but it is well to give an idea of the results of van der Waals' considerations, for they explain the gradual uninterrupted passage from a liquid into a gaseous state in the simplest manner, and, although the deduction cannot be considered as complete and decisive (see Note 25), nevertheless it penetrates so deeply into the essence of the matter that its signification is not only reflected in a great number of physical investigations, but also in the province of chemistry, where instances of the passage of substances from a gaseous to a liquid state are so common, and where the very processes of dissociation, decomposition, and combination must be identified with a change of physical state of the participating substances, which has been elaborated by Gibbs, Lavenig, and others.

For a given quantity (weight, mass) of a definite substance, its state is expressed by three variables—volume v, pressure (elasticity, tension) p, and temperature t. Although the compressibility—[i.e., d(v)/d(p)]—of liquids is small, still it is clearly expressed, and varies not only with the nature of liquids but also with their pressure and temperature (at tc the compressibility of liquids is very considerable). Although gases, according to Mariotte's law, with small variations of pressure, are uniformly compressed, nevertheless the dependence of their volume v on t and p is very complex. This also applies to the coefficient of expansion [= d(v)/d(t), or d(p)/d(t)], which also varies with t and p, both for gases (see Note 26), and for liquids (at tc it is very considerable, and often exceeds that of gases, 0·00367). Hence, the equation of condition must include three variables, v, p, and t. For a so-called perfect (ideal) gas, or for inconsiderable variations of density, the elementary expression pv = Ra(1 + at), or pv = R(273 + t) should be accepted, where R is a constant varying with the mass and nature of a gas, as expressing this dependence, because it includes in itself the laws of Gay-Lussac and Mariotte, for at a constant pressure the volume varies proportionally to 1 + at, and when t is constant the product of tv is constant. In its simplest form the equation may be expressed thus:

pv = RT;

where T denotes what is termed the absolute temperature, or the ordinary temperature + 273—that is, T = t + 273.

Starting from the supposition of the existence of an attraction or internal pressure (expressed by a) proportional to the square of the density (or inversely proportional to the square of the volume), and of the existence of a real volume or diminished length of path (expressed by b) for each gaseous molecule, van der Waals gives for gases the following more complex equation of condition:—

(p + a / v² )(v - b) = 1 + 0·00367t;

if at 0° under a pressure p = 1 (for example, under the atmospheric pressure), the volume (for instance, a litre) of a gas or vapour he taken as 1, and therefore v and b be expressed by the same units as p and a. The deviations from both the laws of Mariotte and Gay-Lussac are expressed by the above equation. Thus, for hydrogen a must be taken as infinitely small, and b = 0·0009, judging by the data for 1,000 and 2,500 metres pressure (Note 28). For other permanent gases, for which (Note 28) I showed (about 1870) from Regnault's and Natterer's data, a decrement of pv, followed by an increment, which was confirmed (about 1880) by fresh determinations made by Amagat, this phenomena may be expressed in definite magnitudes of a and b (although van der Waals' formula is not applicable in the case of very small pressures) with sufficient accuracy for contemporary requirements. It is evident that van der Waals' formula can also express the difference of the coefficients of expansion of gases with a change of pressure, and according to the methods of determination (Note 26). Besides this, van der Waals' formula shows that at temperatures above 273( 8a / 27b -1) only one actual volume (gaseous) is possible, whilst at lower temperatures, by varying the pressure, three different volumes—liquid, gaseous, and partly liquid, partly saturated-vaporous—are possible. It is evident that the above temperature is the absolute boiling point—that is (tc) = 273( 8a / 27b - 1). It is found under the condition that all three possible volumes (the three roots of van der Waals' cubic equation) are then similar and equal (vc = 3b). The pressure in this case (pc) = a / 27b² . These ratios between the constants a and b and the conditions of critical state—i.e. (tc) and (pc)—give the possibility of determining the one magnitude from the other. Thus for ether (Note 29), (tc) = 193°, (tp) = 40, hence a = 0·0307, b = 0·00533, and (vc) = 0·016. That mass of ether which at a pressure of one atmosphere at 0° occupies one volume—for instance, a litre—occupies, according to the above-mentioned condition, this critical volume. And as the density of the vapour of ether compared with hydrogen = 37, and a litre of hydrogen at 0° and under the atmospheric pressure weighs 0·0896 gram, then a litre of ether vapour weighs 3·32 grams; therefore, in a critical state (at 193° and 40 atmospheres) 3·32 grams occupy 0·016 litre, or 16 c.c.; therefore 1 gram occupies a volume of about 5 c.c., and the weight of 1 c.c. of ether will then be 0·21. According to the investigations of Ramsay and Young (1887), the critical volume of ether was approximately such at about the absolute boiling point, but the compressibility of the liquid is so great that the slightest change of pressure or temperature has a considerable effect on the volume. But the investigations of the above savants gave another indirect demonstration of the truth of van der Waals' equation. They also found for ether that the isochords, or the lines of equal volumes (if both t and p vary), are generally straight lines. Thus the volume of 10 c.c. for 1 gram of ether corresponds with pressures (expressed in metres of mercury) equal to 0·135t - 3·3 (for example, at 180° the pressure = 21 metres, and at 280° it = 34·5 metres). The rectilinear form of the isochord (when v = a constant quantity) is a direct result of van der Waals' formula.

When, in 1883, I demonstrated that the specific gravity of liquids decreases in proportion to the rise of temperature [S_t = S₀ - Kt or S_t = S₀(1 - Kt)], or that the volumes increase in inverse proportion to the binomial 1 - Kt, that is, V_t = V₀(1-Kt)^-1, where K is the modulus of expansion, which varies with the nature of the liquid, then, in general, not only does a connection arise between gases and liquids with respect to a change of volume, but also it would appear possible, by applying van der Waals' formula, to judge, from the phenomena of the expansion of liquids, as to their transition into vapour, and to connect together all the principal properties of liquids, which up to this time had not been considered to be in direct dependence. Thus Thorpe and Rücker found that 2(tc) + 273 = 1/K, where K is the modulus of expansion in the above-mentioned formula. For example, the expansion of ether is expressed with sufficient accuracy from 0° to 100° by the equation S_t = 0·736/(1 - 0·00154t), or V_t = 1/(1 - 0·00154t), where 0·00154 is the modulus of expansion, and therefore (tc) = 188°, or by direct observation 193°. For silicon tetrachloride, SiCl₄, the modulus equals 0·00136, from whence (tc) = 231°, and by experiment 230°. On the other hand, D. P. Konovaloff, admitting that the external pressure p in liquids is insignificant when compared with the internal (a in van der Waals' formula), and that the work in the expansion of liquids is proportional to their temperature (as in gases), directly deduced, from van der Waals' formula, the above-mentioned formula for the expansion of liquids, V_t = 1/(1 - Kt), and also the magnitude of the latent heat of evaporation, cohesion, and compressibility under pressure. In this way van der Waals' formula embraces the gaseous, critical, and liquid states of substances, and shows the connection between them. On this account, although van der Waals' formula cannot be considered as perfectly general and accurate, yet it is not only very much more exact than pv = RT, but it is also more comprehensive, because it applies both to gases and liquids. Further research will naturally give a closer proximity to truth, and will show the connection between composition and the constants (a and b); but a great scientific progress is seen in this form of the equation of state.

Clausius (in 1880), taking into consideration the variability of a, in van der Waals' formula, with the temperature, gave the following equation of condition:—

(p + a / T(v + c)² )(v - b) = RT.

Sarrau applied this formula to Amagat's data for hydrogen, and found a = 0·0551, c = -0·00043, b = 0·00089, and therefore calculated its absolute boiling point as -174°, and (pc) = 99 atmospheres. But as similar calculations for oxygen (-105°), nitrogen (-124°), and marsh gas (-76°) gave tc higher than it really is, the absolute boiling point of hydrogen must lie below -174°.