Abstract: The end of the 18th century saw the beginnings of the use of steam engines, such as the James Watt steam engine [1] (Fig. 1A). They soon initiated the development of thermodynamics – the science of heat engines. The works of Carnot [1] and Clapeyron [4] initiated the development of thermodynamics and constituted the first approach to the second law and the Carnot cycle (Fig. 2B). The first complete formulation of the second law was made by Clausius [3], who also introduced the classical definition of entropy S, eq. (1). Thompson (Lord Kelvin) formulated the second law in a similar way [5]. These works formed the basis of so-called classical thermodynamics, assuming thermodynamic equilibrium during all processes. Real processes cannot occur under equilibrium conditions, so a "trick” was used in the form of the concept of quasi-static states, infinitely close in temperature. Surprisingly, these equations often describe real industrial processes with quite satisfactory accuracy.
The well-known fact that heat flows in the direction of decreasing temperature sparked scientific interest following Clausius' work. The statistical energy distribution of gas molecules was already known. This prompted Maxwell to propose his famous "thought experiment", later known as Maxwell's demon.
Maxwell proposed a small, intelligent being that could violate the Second Law, referred to as Maxwell's demon by Thompson [8]. (Fig. 2A) A vessel filled with a temperature-balanced gas is divided by a partition with a small hole. The demon sits at the partition and allows particles with higher-than-average energy to pass to the left, and those with lower energy to pass to the right. Everything occurs quasi-statically, without the input of energy. As a result, the energy of the particles on the left is greater than that on the right, creating a temperature difference and the possibility of doing work. Heat flows towards the higher temperature. This negates the Second Law.
A mathematical description was developed for the so-called monomolecular Szilard engine [9] (Fig. 2B). The state of one-gas molecule is given by eq. (2), the work of by eq. (3). This line of thinking was used by Bennett [10] and Landauer [11] to determine the energy of one bit of information (E1); the entropy of one bit is given by eq. (4). Entropy is increased by erasing information. Even when all the demon's operations are quasi-static, it must erase the memory of the molecules' energy, which causes an increase in entropy (disorder of the system) [13]. The demon's operation is explained in a comic book form in Fig. 3 (Bennett [13]): 1 – the demon remembers which side the molecule is on (memory occupied); 2 – it moves the piston on the empty side; 3 – it removes the partition; 4 – the molecule moves the piston and performs work; 5 – inserts the partition and ERASES the MEMORY, which causes the entropy to increase by k×ln2.
The debate continues, with over a thousand papers published in the last 20 years. However, the second law is based on experience; heat flow toward higher temperatures has never been observed. Note the direction of heat flow is not specified in the first law.
The description of phenomena far from equilibrium is dealt with by non-equilibrium thermodynamics, where entropy production is often calculated, i.e., the increase in entropy of the system due exclusively to irreversible phenomena [14]. Entropy production is calculated as the product of the flux Jx and the driving force (causing this flux), divided by the absolute temperature T [14, 15], eq. (5-7). Entropy in chemical reactors is produced by heat (H), mass (D), chemical reaction (R), and fluid flow (F) (Table 1); the total entropy is given by eq. (8). Fig. 4 shows the dependence of the components of the produced entropy on temperature for catalytic combustion of propane. The total entropy usually exhibits a minimum, which is the optimal operating point of the reactor, closest to thermodynamic equilibrium; this is the so-called entropic optimization.
The thermodynamics of irreversible processes deals with processes that are distant from equilibrium, but not particularly far. Industrial processes and physical transformations in the universe are often slow, relatively close to equilibrium. Relating energy and entropy production to unit mass reveals surprising relationships.
Figure 5 shows the relationship between the energy production (per unit mass) of various objects and their mass. Living organisms emit the greatest energy; humans emit about 104 times more energy per kilogram of mass than the Sun, despite the enormous temperature of the star's interior, approximately 1.6∙107 K. Living organisms have become the subject of research. Prigogine, the founder of biophysics, introduced the concept of dissipative structures, significantly deviated from equilibrium [18, 19]. These are living organisms that, at a significant distance from equilibrium, must absorb and release significant amounts of matter and energy.
The icons in Fig. 5 represent the locations of bacteria, insects, humans, a jet plane, and the Sun. The small graph shows entropy production over the entire lifespan, from birth to natural death. According to Hershey [20], the entropy Smax produced by a human over their entire lifetime is 1.006 107 [J/kg K] (men) and 1.070 107 [J/kg K] (women), with a lifespan of 83–103 years for men and 96–109 years for women. Maintaining vital functions requires a minimum energy and entropy production (dS/dt)min of 3.45 10-3 [W/kg K] (men) and 3.03 10-3 [W/kg K] (women) [20]. Death is identified with reaching a critical value of the system's entropy Smax (critical level of disorder). A decrease in the instantaneous entropy production (dS/dt) (and energy production) below a certain minimum (dS/dt)min also indicates death. In Figure 6, the end of the graph indicates death from natural causes.
We certainly still have a long way to go in addressing issues related to the Second Law and the concept of entropy. Entropy—an abstract concept—has found numerous applications in science and technology. One such "eternal" problem is the "heat death of the universe," as does the problem of the finiteness (or otherwise) of the universe. We know more and more about the universe—the thermodynamics of black holes, dark matter, and energy await research. We have drilled 12 kilometers into the Earth's crust, with a planetary radius of 6,370 km; we have not explored the depths of the oceans—and we are far from omniscient. The results of thermodynamic studies of various, as yet unknown, systems and phenomena are the work of future generations of researchers.
