THERMODYNAMICS. - Thermodynamics is the most important chapter of physics which, going far beyond the limit that its name would seem to impose, studies the nature, properties and evolutions of energy, understood in its broadest sense, that is, abstracting from the secondary particulars of its various forms. It therefore fundamentally concerns all the other chapters of physics, physical astronomy, cosmogony and by now even the biological natural sciences themselves. Thermodynamics has reached such a singular loftiness by elaborating and generalizing two fundamental principles, brilliantly conceived and in part formulated by Sadi Carnot and J. R. Mayer, principles that have the singular prerogative of being the only ones to remain unchanged even after the advent of the revolutionary criteria of modern physics.
I. HEAT AND WORK
It is well known that until around the middle of the 19th century heat was considered a very mysterious imponderable fluid that could be absorbed and emitted by bodies, although there had been some hints of some affinity between it and work, among which the very well-known ones of Rumford, who in 1798 acutely observed that with the mechanical devices then in use for the grinding of cannons one could provoke the development of unlimited quantities of heat whenever one had only the requisite work available. 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 evolves cyclically, that is, that, starting from one of its states, returns to it after a certain 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 evolution, and, similarly, L denotes the algebraic sum of the quantity of work, measured in kilogram-meters, received or performed by it, the aforementioned relationship is given by the equation: JQ + L = O, where the constant coefficient J, today called the mechanical equivalent of the calorie, has the value by now confirmed by many and various determinations: J = 427.22 kilogram-meters/calories; a value that Mayer had been able to give only with the scant approximation that the data at his disposal allowed.
II. INTERNAL ENERGY AND THE I PRINCIPLE OF THERMODYNAMICS
The equivalence expressed by the preceding equation is undoubtedly valid for every cyclic transformation of any system, machine, etc. But far more important is 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 Mayer also had it) is that of considering the difference of the value of the energy JQ + L from zero (which would apply to the case of cyclic transformations alone) as equivalent to the difference of the internal energies of the system corresponding to its initial and final states S₁ and S₂: a difference that is customarily denoted by ΔU = U₂ - U₁. The desired general law would thus assume the form JQ + L = ΔU = U₂ - U₁. But no previous physical knowledge legitimizes this position a priori. Indeed, the 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, seems to oppose it. In confirmation, let us recall the necessity of taking into consideration a multiplicity of specific heats (v. TERMOLOGIA). An analogous indeterminacy arises also when the said passage occurs 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 univocally linked to their energy states; therefore they cannot be equated a priori to differences of 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 in fact allows us to affirm that although neither Q nor L can individually be considered functions of the energy state of a body, the quantity JQ + L must instead be considered such. This justifies the preceding equation and the consequent first principle of thermodynamics, also called the principle of the conservation of energy, a principle that is customarily expressed in words in the form glimpsed by Mayer himself: "When a quantity of energy of any type disappears during any process, an equivalent quantity of energy of some other type is always produced, and vice versa."
III. CONSEQUENCES
The preceding principle, although it presents only a partial aspect of thermodynamics, already brings with it innumerable consequences. The principal one is the very well-known one of the impossibility of constructing a mechanism capable of developing energy indefinitely without receiving the equivalent from outside, or, as it is customarily put in imprecise form, of the impossibility of constructing a perpetual motion machine. Of the others, which it is impossible even to list, a few examples are recalled. The first principle, applied on average to ideal gases, conceived as made up of molecules in rapid motion, makes it possible to develop a large part of their kinetic theory and to recover rationally the empirical laws of Mariotte, of Gay-Lussac, etc., and even to determine the number of molecules contained in a mole (that is, in a quantity of gas whose total weight in grams equals the molecular weight), a number that Avogadro had predicted to be equal for all gases, and that Loschmidt calculated to be the enormous figure of 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 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 scale of the gas thermometer with Kelvin's thermodynamic one and justifies the wide scientific use that is still made of it.
IV. FORMULATION ACCORDING TO CARNOT OF THE II PRINCIPLE OF THERMODYNAMICS
The preceding principle informs us about the relations between quantities of heat and the corresponding quantities of work when any transformation of one into the other takes place, but not about the conditions under which such transformations can actually begin to unfold. This is instead provided for, in its various formulations, by the second principle of thermodynamics, whose earliest origins it is also interesting to consider.
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 at that time were beginning to spread. In his work Réflexions sur la puissance motrice du feu... (1824), considering the quantity of heat Q₁ contained in the steam always 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 the energy) could not be transmitted entirely to the engine, which consequently could never reach 100% efficiency; 2) that only if the operation of the engine took place according to a particular cycle (today called the Carnot cycle) could its efficiency rise to the maximum attainable, expressed by the ratio μ = (T₁ - T₂)/T₁; 3) that this ratio proved 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 by referring to suitably chosen fluids, but had then generalized his results in the manner of a principle, which bases its validity on absolute experimental confirmation. From the more modern point of view regarding the energetic constitution of heat, the preceding considerations can be summarized as follows: the heat Q₁ admitted into a thermal engine at the temperature T₁ never transforms completely into work, but a part of it is returned still in the form of heat Q₂, but at the lower temperature T₂. Consequently: 1) the efficiency of every thermal 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 character of the engine and of the nature of the fluid employed. It is therefore evident that in order to increase the efficiencies of thermal engines it is advisable to try to make them operate as approximately as possible according to a Carnot cycle, as well as to raise the upper temperature T₁ and to reduce the lower temperature T₂. And it is precisely on the basis of these criteria that modern technology has been able to achieve notable practical results.
V. THE NOTIONS OF ENTROPY AND FREE ENERGY
The innate tendency of the human mind 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. Since it is evidently impossible to go into particulars on this lofty and difficult subject, only the two fundamental notions of entropy and free energy, by now in widespread use, are recalled here.
Toward the middle of the 19th century R. Clausius considered an isolated system, endowed with a given internal energy which, by the first principle, had to remain constant, and which, by means of internal transformations, by the second principle, had to be able to allow it to develop a limited quantity of work and hence of kinetic energy. He observed that while there was no possibility of increasing, by means of internal transformations, the said quantity of work, there existed instead the possibility, one might even say a notable tendency, of reducing it. The system, while conserving its energies, tended to lose the faculty of producing work, tended to age toward a uniformity that is equivalent to death. And consequently he asked himself whether one could conceive 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, each divided by its absolute temperature: S = Σ Q/T.
By no internal process can this entropy be diminished; but it can with extreme ease be increased, possibly even by spontaneous natural processes, e.g. of thermal conduction. To clarify this difficult question it is worth considering an extremely simple example. Let the system be supposed to consist of two vessels of 1 m³ each, filled with water at the absolute temperatures of 273 and 373 degrees. Let E₂, measured in calories, denote the energy of the vessel 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 assumed conditions the system, by the second principle, would be capable of transforming a part of its thermal energy into the equivalent quantity of work. Let it now be supposed that the two vessels are placed in contact. Little by little both will naturally be reduced to the mean temperature of 323°. The total energy is conserved equal to the preceding one, but the entropy now becomes S = 2 E₂ / 273° + 2 × 50,000 Cal / 323°, that is, notably greater than the preceding one; it has thus reached its maximum value, and the system, now at uniform temperature, is no longer capable of supplying 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, one may think of being able to obtain in the form of work, and which is called free energy, and that which inexorably cannot manifest itself except in the form of heat degraded to a lower temperature, and which is called bound energy. In the simple preceding example 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 chemical physics 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 worried physicists in the second half of the last century. They asked themselves whether it 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 eluded. Consider a vessel divided into two compartments by a diaphragm with a very small hole. A certain quantity of gas is introduced into it. Its molecules in rapid motion, passing in one direction and the other through the little hole, will first eliminate any possible initial difference of pressure in the two compartments (with an increase of entropy) and then keep it constant. But if a little demon capable of seeing the molecules were part of the system and allowed them to pass in only one direction, opportunely covering and uncovering the little hole, a decrease of entropy would at once be produced. It was at this point that L. Boltzmann made the radical proposal, which soon prevailed, of considering the second law as only an extremely probable law. It is now necessary to realize the degree of those probabilities. They are such that in practice they are equivalent 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, reproduces the Divine Comedy? Anyone answers: none; and yet there exists a very small probability that can even be calculated. The observable exceptions to the second law are to be considered just as improbable.
The probabilistic interpretation of the second law of thermodynamics represents the first entry of probability into physical laws. Today, on the other hand, probabilistic laws dominate, at least provisionally, in all of the most modern physics. The biological sciences themselves interpret certain evolutions of evident finalistic aspect through innumerable microscopic deviations from the second law.