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equivalents. In the case of the K and Na compounds of the dibasic acids in Table IX., we will reduce the value of (M2) # to that for equivalents not by dividing by the theoretical 2+ but by the empirical 16: for convenience the value of (M2) thus reduced to the value for equivalents will be denoted by ΣF, the relation between the two being as a rule

ΣF.=(M2)3/n3,

though in the present case we are taking 1.6 in place of 2 in order to allow for a slight difference in equation (2) as applied to the compounds of the dibasic acids; so also in Na P20, 1.6 2 will be used instead of 4.

ΣΕ

Fe

CO.

TABLE X.

Compounds of Potassium K.

SO. CrO4. Cr2O7. Cr2O10. P206. 7.6 8.2 7.9 11.1 14.1 10.9

Compounds of Sodium Na.

6

CO. SO. CrО ̧. Сг2O,. В0, WO̟. Р2О. P207. 6.4 7.1 70 9.9 10.3 7.8 99 10.3

ΣFe Thus the additive principle holds throughout the values of (M2) in Table IX. for the K and Na compounds of the monobasic acids, and in Table X. for equivalents of their compounds with dibasic acids; and if we knew the values of F for K and Na, we could at once obtain the values for the other atoms and radicals and equivalents involved. We might adopt the values already found for F for the halogens in the organic compounds, but seeing that we do not know yet how far the absolute values given by equation (2) are to be relied on, the following process is safer. From the potassium compounds we get that in (M2), Cl-F=1·2, Br-F=1.8, and I-F=2·7; while in the molecular volumes M/p, which in solids is very nearly the same as MB, we have CI-F=13.9, Br-F-20-5, and I-F=301; or, using mean values obtainable from data for the haloid compounds of Li, Na, K, and Rb given later on in Table XIII. from C. W. Clarke (Phil. Mag. ser. 5, iii.), we have Cl-F-10-3, Br-F=16·7, and I-F-26.8. It will be seen that these two series of differences in M/p are about ten times the corresponding differences in (M2), and 10 was the value that we found for the ratio of B the atomic volume to F for the halogens in organic compounds: thus then it happens that equation (2), in spite of the roughnesses in it, gives values of

F for the halogens in harmony with those already found. Accordingly we could adopt for F for the halogens in inorganic compounds the values found in the organic, but it seems to me preferable to work out the results for inorganic compounds on their own basis. To do so we need the atomic volume of fluorine F. Now from the studies made by many chemists on the molecular domains (so-called volumes) of liquids at their boiling-points there have been found values carried by the individual atoms into the molecule, and for the halogens Thorpe (Journ. Chem. Soc. 1880) gives the values F-9-2, Cl=22.7, Br=281, and I=36-6, whence for comparison with our series of differences for the solid state we get Cl-F 13.5, Br-F=18-9, and I-F=27.4, which is curiously close to identity with our series, and indicates that we may take 9 as the atomic volume of F, so that with Clarke's differences given above we have in round numbers the following atomic volumes :-F-9, Cl=19, Br=26, and I=36, to be compared with the former Cl=19, Br=24.5, and I=32 (Table V.). Accordingly the part contributed by the fluorine atom to (M27) is about 9; and with the series of differences given above we have the series of parts carried by the halogen atoms into (M2), namely:

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F=9, Cl=2.1, Br=2.7, and I=3.6

to be compared with

Cl=2.2, Br=2.5, and I=3·2 in Table V. With the above atomic volumes of the halogens and Clarke's molecular volumes of the K and Na haloid compounds given later on in Table XIII., we get the following mean atomic volumes, Na=7·4 and K=186; and with the above values of F for the halogens, we get from Table IX. that the mean value of F for Na is 3·1 and for K is 4·2. With these values of atomic volume and F for Na and K, we can derive from Tables IX. and X. the volumes of a number of negative radicals and their values of F when they are monobasic, of F/2 when dibasic, and of F/4 when tetrabasic. These are given in the next table.

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The values added in brackets in the last table are reproduced from Table VI. for purposes of comparison, and show that according to our methods of calculation, for the inorganic compounds the values of F come out about 11 times their values as found in organic compounds (see also the values for Br and I above). This is a satisfactory result so far as it goes, but on the strength of it we will not proceed to a comparison between B and F in the last table, but will wait till we have controlled these values by an independent calculation. The data for other compounds in Table IX. are too few to be worth discussing separately, but will be considered in connexion with values by another method. The only data for the surface-tension of liquid compounds at their solidifying points known to me and not included in Table IX. are those of Traube for some compounds of Na and K with the fatty acids. These will now be briefly considered.

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These values of (M27) are exceptional, for they do not increase by 9 for each addition of CH2 to the acid radical: with 4.2 for K and 3.1 for Na, also 2.9 for HCOO and 9 for CH2, we can calculate the following values of (M2) for comparison with those just obtained from experiment—

8.0 6.0 6.9 7.8 9.6

21.3.

7.1 The reason for the discrepancies in these two sets of values would have to be sought for by a special inquiry.

2 (b). Second Method of determining (M21) for Inorganic Compounds, namely by the Kinetic Theory of Solids.

In "A Kinetic Theory of Solids" (Phil. Mag. ser. 5, xxxii.), it was shown that, for a homogeneous isotropic solid composed of molecules which can be regarded as spheres of diameter E whose centres are at an average distance e from their nearest neighbours, D being the average kinetic energy of each,

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E is also the distance apart of the centres of two when they

are in contact, and (r) the attraction between two molecules at distance r apart, the summation to extend to all molecules within appreciable action of any one. It may be more convenient to sum both members of the above equation for all the molecules in unit mass, when we get

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If the law of force is given by (r) = 3Am2/r1, then ΣΣro(r)/6 reduces to what is denoted by lp in the notation of the "Laws of Molecular Force," and e3=m/p, so that

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This equation was established on the assumption that the molecules collide as though they were perfectly restitutional spheres.

Now Σ2D is twice the kinetic energy of the motion of the centres of mass of the molecules in unit mass, and, if the energy of other motions is negligible, equals 2Jc0, where 0 is the temperature and c the specific heat. e2(e-E)=E3 (e/E−1) approximately; if the molecules are invariable with temperature, e/E-1=60, where b is the coefficient of linear expansion of the solid, and it was shown that the metals behave as if E diminished with rising temperature in such a way as to make e/E-1=760 approximately; and as E=m/p nearly, we have

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m being the actual mass of a molecule, and M the ordinary molecular mass referred to that of the hydrogen atom. Now cM, by Dulong and Petit's law, is nearly 64 for the metals, so that M2/=61J (M/p)/b.

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(7)

The values of b have not been found experimentally for a number of the most interesting metals, but can be got by means of an empirical relation given in "A New Periodic Property of the Elements" (Phil. Mag. ser. 5, xxx., also xxxii. p. 540), namely, if T is the absolute melting-point bTM044, and then

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Taking J as 4.2 x 10' ergs and using 1012 dynes as the unit of force as hitherto, then finally

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This equation applies, so far as we know at present, only to the metals; it has been deduced from (3), which applies only Phil. Mag. S. 5. Vol. 39. No. 236. Jan. 1895. C

to solids in which the molecules are monatomic or composed of equal atoms, as in the case of the elements, all the atoms being treated as separate spheres. The corresponding equation for compound solids is sketched in section 9 of the "Kinetic Theory of Solids" (Phil. Mag. ser. 5, xxxii. p. 550), but in a form which is not correct unless a strained interpretation is put on certain symbols; but we can easily establish the correct form now. There would be no need to establish a separate form of equation for solid compounds if we knew that the molecules move as wholes; but, on the contrary, we have evidence that the atoms in the molecule move almost independently of one another, for according to Joule and Kopp's law the molecular specific heat of solid compounds is the sum of the atomic specific heats of the atoms in the molecule. In a certain sense a solid compound is like a mechanical mixture of its atoms: in the act of combining the atoms have produced mutual changes in their sizes and attracting-power and other properties, and the solid is like a mechanical mixture of these changed atoms. Now in the establishment of equation (3) for elements, the first term 2D/3e2(e-E) is calculated as the collisional pressure per unit surface, that is the force transmitted across that surface by the collisions of molecules, or rather atoms, of diameter E, average distance e from next neighbour, and kinetic enegy D, while the term Erp(r)/6e3 is the resultant attraction which equilibrates this collisional pressure. Let N be the number of atoms in unit volume, then e31/N, and the collisional pressure can be written 2DN/3(1-E/e). Now if we have N molecules of a compound solid in unit volume, and if each molecule contains n1 atoms of an element A1, ng atoms of an element A2, and so on, then the collisional pressure due to the n atoms of A, in unit volume is m1v2Nn1/3 (1— E1/e1), and similarly for the other atoms, the total collisional pressure being their sum, so that we have the equation

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denoting the mean distance of two neighbour molecules from one another by e: thus we get for a compound solid free from external force the equation

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+

+

1

3e31-E1/e 1-Es/ea .....) = 63Σrp(r). (11)

We cannot assign the values of 1-E1e, for atoms in compounds in the present state of our knowledge; but as it was shown that for the metals 1-E/e=760, where b is the coefficient of linear expansion and the absolute temperature,

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