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shows a close approach to constancy, except in the case of the second family, whose place seems to be taken by the dyad zinc subfamily, in which the ratio (M/p)/M2 is 13, and therefore the product n(M/p)/M2 is 26. As the subfamilies have a ratio (M/p)/M2, which is a simple multiple or submultiple of the value in the main family, we can say that in the metals n(M/p)/M2l is close to 2.8, or a simple multiple or submultiple of 2.8, except in the second family, where it is 2.0. Thus the attracting-power of a metallic atom is proportional to the square root of its volume or of a simple multiple of its volume, and also to the square root of its valency. This relation for the metals can also be stated in another interesting form, for as lp represents the potential energy due to the attractions of the molecules in unit mass, M2//(M/p) is the potential energy of mass M, which may be taken as forming the greater part of the heat of vaporization of the gramme-molecule; thus, then, the potential energy of the molecules in a gramme-equivalent (and probably the latent heat of vaporization of a gramme-equivalent) of the metals in the main families has nearly the same value for all, and in the other metals a value which is a simple multiple or submultiple of this. The methods devised in this paper for calculating M2 for inorganic compounds also lead to approximate values of their latent heats of vaporization, which are of some importance in Thermochemistry.
With this knowledge of the relation between M21 and volume in the uncombined metals, we can return to the study of the relationship of the parts carried by the metallic atoms into (M2) and MB. We have already seen that the metallic atoms in combination do not have their parts of (M2) and MB proportional to one another; and in the light of what we have just found for the free metals, we ought to compare the square of the part F carried by a metal atom into (M2) in compounds with the part B carried into M/p. Here are the values for the Li family :
The first two rows of numbers show that F2 and the volumes of the atoms in compounds run a parallel course, but are not
strictly proportional to one another as in the free state, yet the numbers in the third row show that the connexion between the two quantities in the combined state is the simple one F29B+44,
while for the free atoms the relation was
The data of this family are affected with considerable uncertainty, but the third row of numbers shows that probably in compounds
while for the free metals the relation was found to be
It is noticeable that the constant 9 for the Be family is the same as that for the Li family.
It is not possible with the existing data to follow this interesting inquiry into the other metallic families, and trace the transition from the form of relation F2=aM/p+b holding amongst the powerfully positive metals to the form F=cMB (e being a constant) characteristic of the non-metals, and apparently of some weakly electropositive metals.
We proceed now to the surface-tension method of finding M2 for the metals, chiefly to draw attention again to a discrepancy between the values of M2l for the free metals given by the Kinetic Theory of Solids and those by the method of surface-tension. This was pointed out at the end of the paper on the Kinetic Theory of Solids, but it can be brought out here in a more definite manner and with more interest, seeing that the two methods have been proved in this paper to give accordant results for binary compounds. To the gain of the Kinetic Theory of Solids, the discrepancy will be largely cleared up as due to molecular complexity.
The equation for the surface-tension of the elements has to be modified from the form in which it was used for compounds, because in the Laws of Molecular Force (Phil. Mag. March 1893) it was shown that the virial constant of compounds falls in the liquid to half of the limiting value in the gas, whereas in the elements H2, O2, and N2 it retains the same value in both states. Now the equation (9) was designed to give the value of l for compounds in the gaseous state from measurements of their surface-tensions as liquids, therefore in applying equation (2) to the elements we must provisionally divide by 2, and get
in which M will be taken as the usual atomic weight. The following are the data for the metals and a few other elements given by Quincke (Pogg. Ann. cxxxv. & cxxxviii.), with the values of M2 calculated from them by the last equation, also the values of Ml reproduced from Table XXIX. as given by the Kinetic Theory of Solids, and finally the ratio of the two values of M2l.
Where two values of a are given for one metal, they are the result of different methods of measurement; the data are all Quincke's except one of the measurements for lead made by myself. If matters were exactly as we suppose them to be, in applying equations (9) and (19) the values of the ratio in the last table ought to be 1, or a constant near to 1, such as we found in the case of the binary compounds, while the range in the actual values of the ratio is from 33 for S to 60 for K. In seeking for the reason for this discrepancy, we must remember that in using the equation (19) of the capillary method we have assumed that the characteristic equation of the metals is on the same type as that of the element gases; but there is this distinction between the metals and the element gases, that some of the metals, namely Na, K, Zn, Cd, and Hg, are known to be monatomic in the vaporous state, while the element gases are diatomic; and it is exactly for these metals and tin that the ratio in the last table has its largest values, while the smallest values belong to S, Se, and P; and S and P are well-known instances of exceptional molecular complexity, their vapour-densities at low temperatures being such as correspond to the formulas S. and P. It is therefore probable that the discrepancy has to do with an effect of molecular complexity not taken account of in equation (19). We saw that the distinction between the diatomic element gases and compounds was expressed by introducing the factor 2 into the left side of equation (2); and if we suppose that amongst the elements the effect of molecular complexity is to introduce into the right side of the equation (19), namely
a factor equal to the molecular complexity, defining the molecular complexity of an element as the number of pairs of atoms in its molecule, which is 1/2 for Na, K, Zn, Cd, and Hg, 2 for P, and 3 for S, then the values of M2 by the capillary
method in the last table would be multiplied by these factors, and also the values of the ratio. Treating the molecular complexity where unknown as 1, then the values of the ratio become :
Na. K. Cu. Ag. Ag. Au. Au. Zn. Zn. Cd. Hg. Sn. 1.3 30 .8 •7 1.2 1.4 1.9 1.2 1.1 1.2 2.5 2.7 Pb. Pb. Sb. Bi. Fe. Pd. Pt. S. 1.9 1.4 .9 1.9 .8 1.4 1.4 1-0
The numbers for K, Hg, and Sn are about double the average for the other elements; the experimental value of am is said by Quincke to be uncertain in the case of K, and also for Cu and Fe, while the pairs of values given for Ag, Au, and Pb show that in some cases a large experimental error is possible in these difficult measurements. Thus the effect which we have assumed as due to molecular complexity brings the discrepancies within the range of experimental error in all the satisfactory cases except that of Hg. But even if the explanation just suggested for the inequalities amongst the values of the ratio in the last table proves not to be correct, it has at least been demonstrated that these inequalities do not indicate any real discrepancy amongst our equations, because the effect of different molecular complexity has to be allowed for. Thus while the results of the capillary method of finding M❜l do not, in the case of the elements, confirm the values given by the Kinetic Theory of Solids, they do not invalidate them, and the complete explanation of the relation of the two sets of values may not be obtainable till we know the characteristic equations of the metals.
The transition cases from metal element to non-metal element will not be gone into at present, but it will be advantageous to gather here the values of MB and (M2)* and their ratio for the non-metal elements. We have seen reasons for believing the values in Table XXIX. to be relatively correct on the whole, although it must be expected in such an instance as that of Cl that the method there used for finding M2 must give a rather rough value in the following table the values given are reproduced from Tables VIII. and XXIX.