completed. Suppose that 10 per cent. of B has passed into A before the brake of passive resistance is sufficiently powerful to prevent any further change, then, if the temperature be raised to 230°, the brake is relaxed, and, say, 10 per cent. more of B passes into A. The remaining 80 per cent. is prevented from passing into A by the passive resistance. Again, by raising the temperature to 300°, the brake will be sufficiently relaxed to allow a little more of B to pass into A; and generally, the higher the temperature, the less the passive resistance. At Ac1 the brake appears to be sufficiently relaxed to allow the whole of B to pass into A.

This analogy must not be pushed too far. For this reason. If the passage from B into A below the Ar1 obeys the laws of all chemical changes, then at any fixed temperature the velocity of the transformation of B into A, at any instant, will be proportional to the amount of B remaining to be transformed into A. This is the law of mass action. By "amount is meant the number of grams of B per unit volume.

Let a denote the amount of B originally present, and a the amount which has already been transformed at any given instant, t; then a will denote the amount of B yet remaining to be transformed into A. The law of mass action may now be expressed in symbols, the velocity of the reaction

V = k(α- x),

where k is a numerical constant, whose value depends on the conditions of the experiment, the magnitude of the passive resistance, the temperature, etc.

It is easy to see that the velocity of the transformation must slacken down as time goes on. To fix

the idea, let k be unity, and the amount of a be 100 grams. At the beginning of the action the velocity will be V = 100 grams per hour; but when, say, 10 per cent. has been transformed, V will be 90 grams per hour; when 10 per cent. of B remains untransformed, the velocity of the reaction will be at the rate of 10 grams per hour. When we say that relaxed so that 10

at 230° the passive resistance is

per cent. of B passes into A," the meaning is that when 10 per cent. of B has passed into A the rate of transformation of the remainder of B is too slow to affect the temper very materially when the metal is heated for a short time at the given temperature. It is reasonable to suppose that a more or less prolonged exposure at 100° would anneal hardened steel just as effectually as a shorter exposure at a higher temperature, and that, if the heating at 100° were continued long enough, the whole of B would pass into A.

Those who are familiar with the calculus will see that the velocity of the reaction should be written

When all B has passed into A, we have a = x; but this can only happen after the elapse of an indefinite length of time. See J. W. Mellor, Chemical Statics and Dynamics, London, 1904.

§ 28. Theories of Annealing and Hardening

The constitution of steel may thus be viewed from two important aspects—

I. The Allotropic Changes of the Iron itself.—The explanation which emphasizes the allotropic changes in the iron is known as the allotropic theory.1

1 H. M. Howe, Metallographist, 1. 150, 1898.

Osmond1 has dealt particularly with this phase of the work. The hardening of suddenly cooled steel is supposed to be due to the presence of hard y- or B-iron. This state of things is favoured by the presence of foreign substances, like carbon, nickel, etc. But we do not know if other elements, in the absence of carbon, will effect similar changes in iron. Carbon seems to play an essential part in the action. But other explanations have been suggested.

II. The Relations between Iron and Carbon.-According to the carbon theory,2 the whole of the facts observed during the hardening of steel can be explained on the assumption that carbon exists in the two states-hardening carbon and cement carbonalready described. The cause of hardening by sudden cooling is due to the retention of carbon in the hardening state. This view does not explain the critical points in the cooling curve of pure iron, and the accompanying changes in, say, the magnetic properties of the metal.

It has also been suggested that the hardening of suddenly cooled steel is due to the presence of hard carbides of y- or ẞ-iron, which are decomposed at the critical points if the steel be cooled slowly; but, if cooled quickly, passive resistance sets in before the carbides have time to decompose. This is the socalled carbo-allotropic theory of Howe.3

1 F. Osmond and J. Werth, Compt. Rend., 100. 450, 1885; Annales des Mines, [8], 8. 5, 1885; F. Osmond, ibid., [8], 14. 1, 1888; F. Osmond, Transformations du Fer et du Carbone dans les Fers, les Aciers et Fontes Blanches: Paris, 1888.

2 A. Ledebur, Journ. Iron and Steel Inst., 44. ii. 53, 1893; Stahl und Eisen, 14. 523, 1894; 17. 302, 436, 1897.

3 H. le Chatelier, Metallographist, 1. 52, 1898.

J. O. Arnold1 has developed an interesting explanation, which is known as the subcarbide theory. The points of this theory are as follows: In eutectic or saturated steel there is only one critical point, Arı, which marks the passage of pearlite into an homogeneous mass corresponding with the empirical formula Fe24C. There is no evidence to show that this substance changes at higher temperatures, or that carbon separates from combination with the iron and passes into a solid solution of elementary carbon in iron. On dissolving this material in acid, practically the whole of the carbon is evolved as hydrocarbon gas.

In hypereutectic or supersaturated steel, with, say, 14 per cent. of carbon, the pearlite changes into hardenite at about 700°, but the cementite only dissolves in the hardenite above 900°.

With unsaturated steel containing, say, 0.2 per cent. of carbon at the Arı, 700°, "the pearlite areas pass into hardenite;" at the Arg, 750°, point "the hardenite areas dissolve in the beta ferrite;" at the Ars point "there is a dilation of iron like that of water at 4° C."

The terms "beta" and "gamma" are used in the sense of a range of temperature, and not of allotropic modifications of iron. Arnold lays no stress on the Arg and the Arg points. Tensile tests made on bars of iron containing 0.2 per cent. of carbon, and quenched in iced brine at temperatures ranging from atmospheric up to 1000° in an atmosphere of nitrogen, showed that the tenacity increased from about 500° to 950° proportionally with the quenching temperatures. There was no marked increase in the tenacity at the

1 J. O. Arnold, Journ. Iron and Steel Inst., 45. i. 314, 1894.

Ar, or the Arg critical points. The hardening of steel by sudden cooling is supposed to be due to the retention of hard subcarbide.

Let us, then, compare the explanation offered by the allotropic solution theory with Arnold's subcarbide theory for the condition of carbon in cooling iron.

I have frequently laid stress upon the fact that we can apply the ordinary laws of liquid solutions to the solidified solutions of carbon in iron. This has led to the use of the term "solid solution," as previously mentioned. We are indebted to Roberts-Austen for developing the subject on this side. I have treated the carbo-allotropic theory from the point of view of the theory of solutions, and summarized the results in Fig. 25. It would be an easy matter to reset the diagram so as to summarize Arnold's interpretation of the facts. I do not suppose for one moment that any of these hypotheses is dressed up in its final form. Each one has its weak and its strong points. All are, or ought to be, agreed as to the facts. But metallurgists are yet only groping for the true explanation.

Many attempts have been made to calculate the molecular weight of carbon, or of the carbides dissolved in iron. It is shown in Elements of Physical Chemistry that the lowering of the freezing-point of any

F