THERMODYNAMICS. — Thermodynamics is the most important chapter of physics which, far surpassing the limit that its name would seem to impose, studies the nature, properties, and transformations of energy, understood in its broadest sense, that is, abstracting from the secondary details of its various forms. It therefore fundamentally concerns all the other branches of physics, physical astronomy, cosmogony, and by now even the biological natural sciences themselves. Thermodynamics has attained such singular eminence by elaborating and generalizing two fundamental principles conceived with genius and partly formulated by Sadi Carnot and J. R. Mayer—principles that have the singular prerogative of remaining unchanged even after the advent of the revolutionary criteria of modern physics.
I. HEAT AND WORK
It is well known that until approximately the middle of the nineteenth century heat was regarded as a mysterious, imponderable fluid that could be absorbed and emitted by bodies, although there had been no lack of indications of some affinity between it and work, among which the best known were those of Rumford, who in 1798 astutely observed that, with the mechanical devices then used for polishing cannons, unlimited quantities of heat could be produced whenever the necessary work was supplied. But the great merit of having clearly recognized the dynamic nature of heat and of having indicated its fundamental constant relationship with work belongs, without any doubt, to J. R. Mayer (v.). Considering any device that operates cyclically—that is, one which, starting from a given state, returns to it after a certain period of time—he was the first to affirm that, if Q denotes the algebraic sum of the quantities of heat, measured in calories, absorbed and emitted by the said device during its cyclic operation, and, analogously, L denotes the algebraic sum of the quantities of work, measured in kilogram-metres, received or performed by it, the aforementioned ratio is given by the equation: JQ + L = 0, where the constant coefficient J, today called the mechanical equivalent of the calorie, has the value now confirmed by many and varied determinations: J = 427.22 kilogram-metres/calorie; a value that Mayer had been able to give only with the limited accuracy permitted by the data available to him.II. INTERNAL ENERGY AND THE FIRST PRINCIPLE OF T
The equivalence expressed by the preceding equation is undoubtedly valid for every cyclic transformation of any system, machine, etc. But it is much more important to know the more general law corresponding to a transformation that is no longer cyclic, that is, between two possible states S₁ and S₂ of the system. The idea that arises spontaneously (and which Mayer also had) is to regard the difference between the value of the energy JQ + L and zero (which would pertain to the case of cyclic transformations alone) as equivalent to the difference between the internal energies of the system corresponding to its initial and final states S₁ and S₂: a difference usually denoted by ΔU = U₂ - U₁. The general law sought would thus take the form JQ + L = ΔU = U₂ - U₁. But no prior physical knowledge legitimizes this assumption a priori. Indeed, it seems to be opposed by consideration of the well-known fact that the passage of any system from an initial state S₁ to a final state S₂ as a result of positive or negative supplies of heat Q represents a phenomenon that depends not only on the said two states, but also on the manner in which this passage occurs. As confirmation, one may recall the necessity of taking into consideration a multiplicity of specific heats (v. TERMOLOGIA). A similar indeterminacy also arises when the said passage takes place as a result of positive or negative supplies of work. In other words, neither the quantities of heat Q nor the quantities of work absorbed or emitted by bodies are uniquely linked to their energetic states; hencethey cannot, a priori, be equated with differences in quantities which, like the internal energies of bodies, are, by definition, characteristics of those states.
Fortunately, however, experience can intervene to resolve the difficulty. It makes it possible to affirm that, although neither Q nor L can individually be regarded as functions of the energetic state of a body, the quantity JQ + L must instead be regarded as such. This justifies the preceding equation and the resulting first principle of t., also called the principle of the conservation of energy, a principle customarily expressed in words in the form already glimpsed by Mayer himself: « When a quantity of energy of any kind disappears during any process, an equivalent quantity of energy of some other kind is always produced, and vice versa ».
III. CONSEQUENCES
The foregoing principle, although presenting only a partial aspect of thermodynamics, already yields innumerable consequences. The best known is the impossibility of constructing a mechanism capable of producing energy indefinitely without receiving an equivalent amount from outside, or, as it is imprecisely expressed, the impossibility of constructing a perpetual-motion machine. Of the others, which it is impossible even to list, a few examples may be recalled. The first principle, applied on average to ideal gases, conceived as consisting of rapidly moving molecules, makes it possible to develop a large part of their kinetic theory and to recover rationally the empirical laws of Mariotte, Gay-Lussac, etc., and even to determine the number of molecules contained in a Mol (that is, in a quantity of gas whose total weight in grams is equal to its molecular weight), a number that Avogadro had predicted to be the same for all gases, and that Loschmidt calculated as the enormous figure N = 0.6 · 10²⁴.The same principle led to the recognition—of great theoretical and practical importance for the very constitution of thermodynamics—that the internal energy of perfect gases depends only on their temperature, and that therefore equal quantities of gas at equal temperature possess the same internal energy, whatever their volume. This is the fact that leads to identifying the gas-thermometer scale with Kelvin’s thermodynamic scale and justifies the extensive scientific use still made of II.
IV. CARNOT’S FORMULATION OF THE SECOND LAW OF THERMODYNAMICS
The foregoing principle provides information concerning the relations between quantities of heat and the corresponding quantities of work when any transformation of one into the other takes place, but not concerning the conditions under which such transformations can actually begin and proceed. This is instead provided, in its various formulations, by the second law of thermodynamics, whose early origins are also of interest.Even before the recognition of the energetic nature of heat, Sadi Carnot had undertaken the theoretical study of the efficiency of steam engines, which were then beginning to spread. In his work Réflexions sur la puissance motrice du feu... (1824), considering the quantity of heat Q₁ contained in the steam continually admitted into the engine at the absolute temperature T₁ and the same quantity returned by the engine at the temperature T₂, he concluded: 1) that the power of that heat (today we would say its energy) could not be transmitted wholly to the engine, which consequently could never attain 100% efficiency; 2) that only if the engine operated according to a particular cycle (today called the Carnot cycle) could its efficiency rise to the maximum attainable value, expressed by the ratio μ = (T₁ - T₂)/T₁; 3) that this ratio was identical for any type of engine operating between the two extreme temperatures T₁ and T₂, and independent of the nature of the fluid employed. Carnot had carried out his calculations with reference to particular cases suitably chosen, but subsequently generalized his results into a principle, whose validity is founded on absolute experimental confirmation. From the more modern standpoint concerning the energetic constitution of heat, the foregoing considerations may be summarized as follows: the heat Q₁ supplied to a heat engine at the temperature T₁ is never transformed entirely into work, but part of it is returned still in the form of heat Q₂, though at the lower temperature T₂. Consequently: 1) the efficiency of every heat engine, by definition equal to (Q₁-Q₂)/Q₁, is always less than unity; 2) its maximum value, corresponding to operation according to the Carnot cycle, is μ = (T₁-T₂)/T₁, independently of the nature of the engine and of the nature of the fluid employed. It is therefore evident that, in order to increase the efficiency of heat engines, it is advisable to make them operate as nearly as possible according to a Carnot cycle, and also to raise the upper temperature T₁ and lower the lower temperature T₂. It was precisely on the basis of these criteria that modern technology was able to achieve significant practical results.
V. LE NOZIONI DI ENTROPIA E DI ENERGIA LIBERA
The innate tendency of the human mind toward generalization and the extraordinary importance of the principles of t. in every field of physics led to many formulations more general and abstract than those given above. Since it is clearly impossible to enter into details on this elevated and difficult subject, only the two fundamental notions of entropy and free energy, now in widespread use, will be recalled.Around the middle of the nineteenth century, R. Clausius considered an isolated system endowed with a given internal energy which, according to the First Principle, had to remain constant and which, by means of internal transformations, according to the Second Principle, had to be capable of enabling it to develop a limited quantity of work and therefore of kinetic energy. He observed that, while there was no possibility of increasing the said quantity of work by means of internal transformations, there did exist the possibility—or one might even say a considerable tendency—of reducing II. Although conserving its energies, the system tended to lose its capacity to produce work; it tended to grow old until reaching a uniformity equivalent to death. Consequently, he asked whether one could conceive of a physical quantity capable of expressing this natural evolution. Thus he arrived at the concept of entropy (S), defined as the sum of all the quantities of heat in the system, each divided by its absolute temperature: S = Σ Q/T.
No internal process can diminish this entropy; but it can be increased with the greatest ease, possibly even through spontaneous natural processes, for example thermal conduction. To clarify this difficult question, it is useful to consider an extremely simple example. Let us suppose that the system consists of two containers, each of 1 m³, filled with water at absolute temperatures of 273 and 373 degrees. Let E₂, measured in calories, denote the energy of the container at the lower temperature. That of the other is then E₂ + 100,000 Cal. The total energy is E = 2 E₂ + 100,000 Cal. The entropy of the system is S = 2 E₂ / 273^° + 100,000 Cal / 373°. Under the supposed conditions, according to the Second Principle, the system would be capable of transforming part of its thermal energy into an equivalent quantity of work. Now suppose that the two containers are placed in contact. Little by little, both will naturally be reduced to the mean temperature of 323°. The total energy remains equal to the previous amount, but the entropy now becomes S = 2 E₂ / 273^° + 2 × 50,000 Cal / 323°, that is, considerably greater than before; it has thus reached its maximum value, and the system, now at a uniform temperature, is no longer capable of supplying any work.
The notion of free energy is reached by considering a priori the internal energy as distinguishable into two parts: that which, on the basis of the Second Principle and by means of perfect devices, may be considered capable of being obtained in the form of work, and which is called free energy; and that which inexorably can manifest itself only in the form of heat degraded at a lower temperature, and which is called bound energy. In the simple example above, the free energy would have been that transformable into work, and the bound energy the difference between the total energy and the free energy. In physical chemistry, the notion of free energy is essential.
VI. THE PROBABILISTIC INTERPRETATION OF THE SECOND PRINCIPLE OF T
The increase in the entropy of any system left to itself, despite its unquestionable experimental confirmation, greatly concerned physicists in the second half of the last century. They wondered whether it was an absolute law, on the order of that of the conservation of energy, or a law that could admit exceptions. Maxwell made a decisive contribution by imagining an isolated system in which the said law could be evaded. Let us consider a container divided into two compartments by a diaphragm with a very small hole. A certain quantity of gas is introduced into II. Its rapidly moving molecules, passing in one direction and the other through the little hole, will first eliminate any possible initial difference in pressure between the two compartments (with an increase in entropy) and then maintain it constant. But if the system included a little demon capable of seeing the molecules and allowing them to pass in only one direction by suitably covering and uncovering the hole, there would certainly be a decrease in entropy. It was at this point that L. Boltzmann made the radical proposal, which soon prevailed, that the second principle should be regarded as a law that is only extremely probable. It is now necessary to understand the degree of those probabilities. They are such that, in practice, they amount to the most absolute certainty. To illustrate such probabilities, many examples have been imagined. What probability is there that a monkey, pounding on a typewriter with continuous paper, would reproduce the Divine Comedy? Anyone would answer: none; yet there is a minuscule probability, which can even be calculated. Observable exceptions to the second principle are to be regarded as equally improbable.The probabilistic interpretation of the second principle of t. represents the first entry of probability into physical laws. Today, by contrast, probabilistic laws dominate, at least provisionally, throughout the most modern physics. The biological sciences themselves interpret certain developments of evident teleological character through innumerable microscopic deviations from the second principle.