Further 79.76 parts of bromine, 126.53 parts of iodine, and probably 19.06 of fluorine are isomorphous with 35-37 parts by weight of chlorine.

There are several other large groups of isomorphous elements. The phosphates, vanadates, and arsenates are isomorphous. In the free state arsenic is isomorphous with antimony, bismuth, and tellurium. But it is obvious that the crystallographic equivalent can only be deduced from the isomorphism of the compounds, and not from the isomorphism of the free elements, for in the latter case there are no means of ascertaining what weight of the one element can replace a given weight of the other. It is assumed in certain minerals that sulphur is isomorphously replaced by arsenic and antimony; if this is really the case then we have the following crystallographic equivalents:30.96 parts by weight of phosphorus

Silicon, titanium, zirconium, thorium, and tin form another isomorphous group, and tin is related to the isomorphous group of the platinum metals containing platinum, iridium, osmium, palladium, rhodium, and ruthenium; these two groups are thus brought into relation with each other, and the following equivalents are obtained ::

If the somewhat doubtful isomorphism of titanium and iron be admitted, this group may thus be brought into relationship with the iron group. Some of the crystallographic equivalent weights arrived at in this way may not be correct, for doubtless some of the cases of isomorphism may prove not to be genuine. But the great advantage of this method is that it can only give one equivalent weight for each element, whereas the other methods would yield two or more equivalent weights.

After the discovery of the law of isomorphism Berzelius regarded the crystallographic equivalent weights as identical with the atomic weights, except in the case of K, Na, Li, Ag, of which he determined the atomic weights by the use of their electrolytic equivalents. The identification of the crystallographic equivalents with the atomic weights offered a lucid explanation of the phenomena of isomorphism. Imagine that a crystal is a regular structure composed of small particles of matter, called molecules, the molecules being themselves systematically built up of a definite number of atoms. If in each of these particles one atom is replaced by another of similar shape and size it is clear that the whole structure will remain unaltered in shape and arrangement. This is obvious, for experience shows us that, although the crystallographic equivalents vary considerably in weight, they all occupy approximately the same space.

We are, however, acquainted with a series of cases in which it is apparently not permissible to assume that replacement takes place atom for atom. In innumerable compounds the equivalent of potassium (39.03) is isomorphously replaced by 14.01 parts of nitrogen and 4 parts of hydrogen, and scarcely the slightest difference between the two classes of compounds is to be found. Consequently since Mitscherlich's first discoveries it has been assumed that an atom of potassium can be isomorphously replaced by the compound radical' ammonium (NH=18.01), composed of one atom of nitrogen (N=14·01)

THERMIC EQUIVALENTS

21

and four atoms of hydrogen. But if this is possible in one case, it may frequently happen that several atoms replace one single atom. If we admit this, then the whole foundation of these considerations is weakened: 107.66 parts by weight of silver are isomorphously replaced by 63.18 parts by weight of copper: this may mean that one atom of silver is replaced by one of copper. But if we assume that each atom of silver is replaced by two atoms of copper, then the atomic weight of copper will be 31.59, which is identical with the chemical equivalent weight given in § 11.

Another weak point in determining atomic weights by isomorphism is that many elements must be omitted, and hydrogen amongst the number. As a natural consequence the atomic weights can only be referred to the unit by making certain arbitrary assumptions. In fact we have already started with the arbitrary assumption that the equivalent weight of potassium as compared with hydrogen is 39.03. If we had taken it as half, or, like Berzelius, as double this value, we should have obtained for all the other elements values half or double their present atomic weights, and it would not have been possible to prove that these values were incorrect.

§ 14. Thermic Equivalents. In 1819, at the time when Mitscherlich discovered the law of isomorphism, two French chemists, Dulong and Petit, observed the existence of another simple relation between the chemical combining weights of the elements and a physical property, viz. their specific heat or capacity for heat in the solid state. The atomic weights are approximately inversely proportional to the specific heats, and consequently the product of these two values is nearly the same for all elements. In order to make this law valid Dulong and Petit found it necessary to alter the combining weights of some of the elements. Although these changes were not at the time generally welcomed, they are now universally adopted (except in the case of a few small errors), and all the more recent specific heat determinations have confirmed the accuracy of the law ✔of Dulong and Petit. This important law is of general application. It gives the same values as the law of isomorphism and meets with the same difficulty, for here again the results cannot be directly referred to hydrogen as the standard, for solid

hydrogen has not yet been investigated. If we take any of the crystallographic equivalents mentioned in the preceding paragraph, and multiply each by the specific heat of the element, we obtain approximately the same product.

The explanation of this fact is very simple. As the specific heat is the amount of heat required to raise the unit weight of a substance from 0° to 1° C. this product represents the amount of heat required to raise the equivalent weight by 1°C. The weight of the given element, which is heated 1° C., is termed the thermic equivalent weight. If we regard this as the atomic weight, then the product of the atomic weight into the specific heat is the atomic heat, i.e. the amount of heat taken up by one atom. It is clear that the atoms of the different elements have the same capacity for heat. The law may be simply expressed by saying that the atomic heats of all the elements are approximately equal.

This law applies without exception to all the malleable metals, to almost all the brittle metals, and to the majority of the non-metals. The following table contains in the first column the names of the elements, in the second column under c the specific heats, in the third the thermic equivalent or thermic atomic weight A, and in the fourth the product A. c, the atomic heat. The specific heat of most of the elements has been determined between the boiling point of water and the mean temperature of the atmosphere, but in the case of easily fusible elements the determinations are made at temperatures below their melting points, as most bodies exhibit abnormal specific heats at a temperature near their melting point.

The atomic heats in this table do not exhibit absolute uniformity—the values vary between 54 and 6.8. The thermic equivalent or atom may now be defined as that stœchiometric quantity which on multiplication by the specific heat yields a constant which is approximately 6. If the specific heat of ice is taken as the unit, or the equivalent weight of some other element instead of hydrogen is taken as the standard, then different values would be obtained. If the atomic weight of oxygen is taken as 100, then the atomic heats would vary between 38 and 40.

As the atomic heat is almost constant, and as

A. c = const. = 6.3 approximately,