Disgregation

He then states what he calls the “theorem respecting the equivalence-values of the transformations” or what is now known as the second law of thermodynamics, as such:

Quantitatively, Clausius states the mathematical expression for this theorem is as follows. Let dQ be an element of the heat given up by the body to any reservoir of heat during its own changes, heat which it may absorb from a reservoir being here reckoned as negative, and T the absolute temperature of the body at the moment of giving up this heat, then the equation:

${\displaystyle \int {\frac {dQ}{T}}=0}$

must be true for every reversible cyclical process, and the relation:

${\displaystyle \int {\frac {dQ}{T}}\geq 0}$

must hold good for every cyclical process which is in any way possible.

Clausius then points out the inherent difficulty in the mental comprehension of this law by stating: "although the necessity of this theorem admits of strict mathematical proof if we start from the fundamental proposition above quoted, it thereby nevertheless retains an abstract form, in which it is with difficulty embraced by the mind, and we feel compelled to seek for the precise physical cause, of which this theorem is a consequence." The justification for this law, according to Clausius, is based on the following argument:

To elaborate on this, Clausius states that in all cases in which heat can perform mechanical work, these processes always admit to being reduced to the “alteration in some way or another of the arrangement of the constituent parts of the body.” To exemplify this, Clausius moves into a discussion of change of state of a body, i.e. solid, liquid, gas. For instance, he states, “when bodies are expanded by heat, their molecules being thus separated from each other: in this case the mutual attractions of the molecules on the one hand, and external opposing forces on the other, insofar as any such are in operation, have to be overcome. Again, the state of aggregation of bodies is altered by heat, solid bodies rendered liquid, and both solid and liquid bodies being rendered aeriform: here likewise internal forces, and in general external forces also, have to be overcome.”

Clausius then goes on to introduce the term “disgregation”:

In the cases first mentioned, the arrangements of the molecules is altered. Since, even which a body remains in the same state of aggregation, its molecules do not retain fixed in varying position, but are constantly in a state of more of less extended motion, we may, when speaking of the arrangement of the molecules at any particular time, understand either the arrangement which would result from the molecules being fixed in the actual position they occupy at the instant in question, or we may suppose such an arrangement that each molecule occupies its mean position. Now the effect of heat always tends to loosen the connection between the molecules, and so to increase their mean distances from one another. In order to be able to represent this mathematically, we will express the degree in which the molecules of a body are separated from each other, by introducing a new magnitude, which we will call the disgregation of the body, and by help of which we can define the effect of heat as simply tending to increase the disgregation. The way in which a definite measure of this magnitude can be arrived at will appear from the sequel.

Ice melting on a beach in Iceland

Clausius then discusses the example of the melting of ice, a classic example which is used in almost all chemistry books to this day, and shows how we might represent the mechanical equivalent of work related to this energetic change mathematically:

The forces exerted upon one another by the molecules are not of so simple a kind that each molecule can be replaced by a mere point; for many cases occur in which it can be easily seen that we have not merely to consider the distances of the molecules, but also their relative positions. If we take, for example, the melting of ice, there is no doubt that interior forces, exerted by the molecules upon each other, are overcome, and accordingly increase of disgregation takes place; nevertheless the centers of gravity of the molecules are on the average not so far removed from each other in the liquid water as they were in the ice, for the water is the denser of the two. Again, the peculiar behaviour of water in contracting when heated above 0°C., and only beginning to expand when its temperature exceeds 4°, shows that likewise in liquid water, in the neighbourhood of its melting-point, increase of disgregation is not accompanied by increase of the mean distances of its molecules.

As it is difficult to obtain direct measures of the interior forces that the molecules of the body exert on each other, Clausius states that an indirect way to obtain quantitative measures of what is now called entropy is to calculate the work done in overcoming internal forces:

In the case of the interior forces, it would accordingly be difficult—even if we did not want to measure them, but only to represent them mathematically—to find a fitting expression for them which would admit of a simple determination of the magnitude. This difficulty, however, disappears if we take into calculation, not the forces themselves, but the mechanical work which, in any change of arrangement, is required to overcome them. The expressions for the quantities of work are simpler than those for the corresponding forces; for the quantities of work can be all expressed, without further secondary statements, by the numbers which, having reference to the same unit, can be added together, or subtracted from one another, however various the forces may be to which they refer.

It is therefore convenient to alter the form of the above law by introducing, instead of the forces themselves, the work done in overcoming them. In this form it reads as follows:

This description is an early formulation of the concept of entropy.