THERMODYNAMICS. — Thermodynamics is the most important chapter of physics, far surpassing the limits implied by its name, as it studies the nature, properties, and transformations of energy in its broadest sense, abstracting from the secondary particulars of its various forms. It thus fundamentally concerns all other chapters of physics, physical astronomy, cosmology, and now even the natural biological sciences. Thermodynamics has attained such singular eminence by elaborating and generalizing two fundamental principles, originally conceived and partly formulated by Sadi Carnot and J. R. Mayer; principles that possess 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 around the mid-19th century, heat was regarded as a mysterious imponderable fluid that could be absorbed and emitted by bodies, though there had been some indications of its affinity with work, among which Rumford’s observations in 1798 are particularly notable. He astutely observed that with the mechanical devices then in use for cannon boring, unlimited quantities of heat could be generated provided only that the necessary work was supplied. However, the great merit of clearly recognizing the dynamic nature of heat and of indicating its fundamental constant relationship with work undoubtedly belongs to J. R. Mayer (v.). Considering any device undergoing a cyclic evolution—i.e., returning to its initial state after a certain time—he was the first to assert that if Q denotes the algebraic sum of the quantities of heat absorbed and emitted by the device during its cyclic evolution (measured in calories), and L denotes the algebraic sum of the quantities of work received and performed by it (measured in kilogram-meters), then the relationship between them is given by the equation:Q + L = 0,
where the constant coefficient J, now called the mechanical equivalent of heat, has the value confirmed by numerous determinations:
J = 427.22 kilogram-meters per calorie;
a value that Mayer could only approximate with the limited data available to him.
II. INTERNAL ENERGY AND THE FIRST LAW OF THERMODYNAMICS
The equivalence expressed by the preceding equation is undoubtedly valid for every cyclic transformation of any system or machine. But of far greater interest is the more general law corresponding to a non-cyclic transformation, i.e., between two possible states S₁ and S₂ of the system. The natural idea (also entertained by Mayer) is to consider the deviation of the value of Q + L from zero (which holds only for cyclic transformations) as equivalent to the difference in the internal energies of the system corresponding to its initial and final states S₁ and S₂; a difference conventionally denoted by ΔU = U₂ − U₁. The desired general law would thus take the form:Q + L = ΔU = U₂ − U₁.
However, no prior physical knowledge legitimizes this position a priori. Indeed, it seems to be contradicted by the well-known fact that the transition of any system from an initial state S₁ to a final state S₂ through the absorption or emission of heat Q represents a phenomenon that depends not only on these two states but also on the manner in which the transition occurs. To confirm this, one need only recall the necessity of considering a multiplicity of specific heats (v. TERMINOLOGY). A similar indeterminacy arises when the transition occurs through the performance or absorption of work. In other words, neither the quantities of heat Q nor the quantities of work absorbed or emitted by bodies are uniquely tied to their energetic states; hence, they cannot a priori be equated to differences in quantities that, like their internal energies, are by definition characteristic of such states.
Fortunately, however, experience intervenes to resolve this difficulty. It allows us to affirm that although neither Q nor L can individually be considered functions of the energetic state of a body, the quantity JQ + L must be so considered. This justifies the preceding equation, and the resulting first law of thermodynamics can be expressed in words, as Mayer himself intuited: “Whenever 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
Although the preceding principle represents only a partial aspect of thermodynamics, it already yields innumerable consequences. The most well-known is the impossibility of constructing a mechanism capable of indefinitely generating energy without receiving an equivalent from the outside—commonly, if imprecisely, referred to as the impossibility of constructing a perpetual motion machine. Among the many other consequences—too numerous to list—some examples may be recalled. Applied to ideal gases, conceived as composed of rapidly moving molecules, the first law enables the development of much of their kinetic theory and the rational derivation of empirical laws such as those of Mariotte and Gay-Lussac, and even permits the determination of the number of molecules in a mole (i.e., a quantity of gas whose total weight in grams equals its molecular weight). Avogadro had predicted that this number would be the same for all gases, and Loschmidt calculated it to be the enormous figure N = 0.6 × 10²⁴.The same principle has led to the recognition—of great theoretical and practical importance for the very constitution of thermodynamics—that the internal energy of perfect gases depends solely on their temperature, so that equal quantities of gas at equal temperatures possess the same internal energy regardless of volume. This fact underlies the identification of the gas thermometer scale with the thermodynamic Kelvin scale and justifies its continued widespread scientific use.
IV. CARNOT’S FORMULATION OF THE SECOND LAW OF THERMODYNAMICS
The preceding principle informs us about the relationships between quantities of heat and corresponding quantities of work when any transformation between them occurs, but not about the conditions under which such transformations can actually begin or proceed. This is instead addressed by the second law of thermodynamics in its various formulations, whose early origins are also of interest.Sadi Carnot, even before the recognition of the energetic nature of heat, had set himself the theoretical study of the efficiency of steam engines, which were then beginning to spread. In his work *Reflections sur la puissance motrice du feu...* (1824), he considered the quantity of heat \(Q_1\) contained in the steam introduced into the engine at the absolute temperature \(T_1\) and the same quantity returned by the engine at the temperature \(T_2\), concluding: 1) that the power of that heat (today we would say energy) could not be transmitted integrally to the engine, which consequently could never achieve 100% efficiency; 2) that only if the engine operated according to a particular cycle (now called the Carnot cycle) could its efficiency reach the maximum possible, expressed by the ratio \(u = (T_1 - T_2)/T_1\); 3) that this ratio was identical for any type of engine operating between the two extreme temperatures \(T_1\) and \(T_2\) and independent of the nature of the fluid used. Carnot had conducted his calculations with reference to suitably chosen particular cases, but then generalized his results as a principle, whose validity rested on absolute experimental confirmation. From the more modern point of view regarding the energetic constitution of heat, the foregoing considerations can be summarized as follows: the heat \(Q_1\) introduced into a heat engine at temperature \(T_1\) is never completely transformed into work, but part of it is returned still in the form of heat \(Q_2\), but at the lower temperature \(T_2\). Consequently: i) the efficiency of every heat engine, by definition equal to \((Q_1 - Q_2)/Q_1\), is always less than unity; 2) its maximum value, corresponding to operation according to the Carnot cycle, is \(\mu = (T_1 - T_2)/T_1\), independently of the type of engine and the nature of the fluid used. It is therefore evident that to increase the efficiency of heat engines, one should seek to make them operate as closely as possible according to a Carnot cycle, as well as to raise the higher temperature \(T_1\) and lower the lower temperature \(T_2\). It is precisely on the basis of such criteria that modern technology has achieved notable practical results.
V. THE NOTIONS OF ENTROPY AND FREE ENERGY
The innate human tendency toward generalization and the extraordinary importance of the principles of thermodynamics in all fields of physics led to many formulations more general and abstract than the preceding ones. Without being able to go into detail on this elevated and difficult subject, we shall recall only the two fundamental notions of entropy and free energy, now widely used.Toward the middle of the 19th century, R. Clausius considered an isolated system with a given internal energy that, by the first principle, had to remain constant, and that, through internal transformations, by the second principle, could be made to develop a limited amount of work and thus kinetic energy. He observed that while there was no possibility of increasing, through internal transformations, the said amount of work, there was instead the possibility—one might even say a notable tendency—to reduce II. The system, while conserving its energy, tended to lose the faculty of producing work, tended to age until reaching a uniformity equivalent to death. And consequently he asked whether one could conceive of a physical quantity capable of expressing this natural evolution. He thus arrived at the concept of "entropy" (S), defined as the sum of all the quantities of heat of the system divided each by its absolute temperature: \(S = \sum Q/T\).
With no internal process can such entropy be diminished; but it can be increased with extreme ease, even by spontaneous natural processes, e.g., thermal conduction. To clarify the difficulty, it is convenient to consider an extremely simple example. Suppose the system consists of two containers of 1 m³ each, filled with water at absolute temperatures of 273 and 373 degrees. Let \(E_0\) measure the heat, the energy of the container at the lower temperature. That of the other is then \(E_0 + 100,000\) cal. The total energy is \(E = 2E_0 + 100,000\) cal. The entropy of the system is \(S = 2E_0/273° + 100,000\) cal/373°. Under the assumed conditions, the system, by the second principle, would be able to transform part of its thermal energy into an equivalent amount of work. Now suppose the two containers are placed in contact. Gradually they will naturally both reach the average temperature of 323°. The total energy remains equal to the previous value, but the entropy becomes \(S = 2E_0/273° + 2 \times 50,000\) cal/373°, i.e., notably greater than before; it has thus reached its maximum value and the system, at uniform temperature, is no longer able to provide any work.
The notion of free energy is arrived at 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, can be thought of as being obtainable 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 to 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 THERMODYNAMICS
The increase of entropy in any system left to itself, despite the undoubted experimental confirmations, greatly concerned physicists in the second half of the last century. They asked whether this was an absolute law, of 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. Consider a container divided into two compartments by a diaphragm with a very small hole. A certain quantity of gas is introduced. Its rapidly moving molecules, passing back and forth through the small hole, will first eliminate any initial difference in pressure between the two compartments (with an increase in entropy) and then maintain it constant. But if the system included a tiny demon capable of seeing the molecules and allowing them to pass in only one direction by appropriately opening and closing the hole, a decrease in entropy would certainly be produced. It was at this point that L. Boltzmann made the radical proposal, soon accepted, of considering the second principle as a law only extremely probable. It is now necessary to realize the degree of these probabilities. They are such that practically they amount to absolute certainty. To illustrate such probabilities, many examples have been imagined. What is the probability that a monkey, pounding on a typewriter with continuous paper, will reproduce the *Divine Comedy*? Anyone answers: none; and yet there is a very small probability that can even be calculated. The observable exceptions to the second principle are to be considered equally unlikely.The probabilistic interpretation of the second principle of thermodynamics represents the first entry of probability into physical laws. Today, probabilistic laws dominate, at least provisionally, in all modern physics. Even the biological sciences interpret certain evolutions of evident finalistic aspect through innumerable microscopic deviations from the second principle.