Analogy to help learn and understand the second law of thermodynamics,entropy,entropy generation,irreversibility,and power cycles

Abstract

Teaching and learning Thermodynamics have challenges. Energy and the First Law as the Energy Conservation Principle are usually well accepted. This is not the case with the Second Law, entropy, entropy generation, and the consequences of the Second Law. Analogies can be used to help learn and understand the Second Law of Thermodynamics, entropy, entropy generation, irreversibility, and power cycles. The present work proposes a Thermodynamics-Economics analogy. The analog of heat transfer (flowing in the decreasing temperature direction, driven by a temperature difference) is the traded merchandise transfer (flowing in the increasing unit price direction) in normal trading operations. Developments start with the economic analog of Clausius inequality. The analog of entropy is the merchandise financial value, and the analog of the entropy generation is the financial value generation. Irreversibility is associated with entropy generation, and economic irreversibility is related to economic entropy (financial value) generation. The reversible and irreversible power cycles also have their economic analogs. The irreversibility associated with heat transfer through a finite temperature difference also has its economic analog. The economic Second Law developments are more familiar to students and more easily understandable, which can help them learn the Second Law of Thermodynamics and its implications.

Introduction

Teaching and learning Thermodynamics have their problems and challenges. Work in ¹ analyzes why Thermodynamics is usually considered a difficult subject. That motivates research and experimental/classroom approaches in the search for improved methodologies and solutions to solve those issues. Mulop et al. ² analyze the main difficulties of Thermodynamics learning and present a review and analysis of different approaches that can help. If the referred problems are general, they increase in relevance when related to entropy. ³ Brown and Sing ⁴ report how a survey indicated significant challenges for students’ understanding of Thermodynamics, thermodynamic processes, the First and Second Laws, and entropy. O’Connel ⁵ points out the main challenges of Thermodynamics teaching and learning. In, ⁶ Bain et al. review the research on Thermodynamics teaching and learning. A recent survey on research works on Thermodynamics teaching and learning can be found in. ⁷ In, ⁸ Yang et al. proposed some methods to overcome the challenges of heat transfer and Thermodynamics teaching and learning. Other efforts are made to simplify the students’ tasks, such as the recent work by Çengel and Kanoglu ⁹ proposing a new function for helping students deal with entropy calculations of ideal gases whose specific heats depend on temperature, and the work of Wang et al. ¹⁰ proposing a helping learning analytic system. In ¹¹ Gaggioli gives his opinions based on his extensive experience in Engineering Thermodynamics teaching. Smith ¹² emphasizes how students’ previous experiences can be intuitively related to energy, the Second Law, and entropy. With this in mind, analogies, invoking different but familiar domains, are also important to increase the students’ curiosity and interest, facilitating Thermodynamics learning and understanding.

The one-way direction set by the Second Law is usually well accepted, but not the Kelvin-Plack statement of the Second Law, contrary to the Clausius statement which seems obvious. It can be proved the equivalence between these two statements of the Second Law;^13,14 however, the developments leading to the Clausius Inequality and the definition of the property entropy are based on the less understood and less accepted Kelvin-Planck Statement. ¹⁵ Even if energy is not tangible, its effects are well perceived and felt. Its consumption can be measured using counters, its consumption is paid, and the Energy Conservation Principle is accepted without major problems. This is not the case of entropy, a strange thing that can be generated from nothing. Once installed the entropy strangeness, discomfort, and disgust, these feelings extend to everything related to the Second Law and entropy.

Several approaches can be followed to attract students to Thermodynamics making it an interesting and challenging subject to learn, study, and apply.^1,13,14 The applicability of Thermodynamics to virtually all domains and all kinds of problems^13,14 is one of its major strengths, yet usually, students see that as an additional problem, as no typical problems or well-established typical solutions exist for all problems. References 13 and 14 (both 9^th editions) result from successive pedagogical improvements based on the extensive experience of their authors in Engineering Thermodynamics teaching. Introduced novelties are mainly related to the pedagogical approach, trying to make Engineering Thermodynamics appealing and interesting for the students, highlighting how it is strongly related to, and helping understand, almost everything, technical and/or present in the student's everyday life. The usual Engineering Thermodynamics treatment in Mechanical Engineering, including that of the Second Law, entropy, entropy generation, irreversibility, and power cycles, are those proposed and followed in the referred textbooks.^13,14

Analogies can be used to help Thermodynamics teaching and learning, using more familiar or even common-sense analogs to help understand the less familiar, and harder-to-understand, Thermodynamics issues. ¹⁶ This always bearing in mind that some care is needed when teaching/learning using analogies. ¹⁷ It is interesting to mention in this context the work of Norton, ¹⁸ dissertating on how ‘analogy helped create the new science of Thermodynamics’, and the seminal Carnot work ¹⁹ based on the caloric theory, analogously to what happens with the falling flowing fluids. If analogies may be useful in science and engineering teaching and learning, there are challenges when using analogies in teaching. As examples we may have situations when students do not identify the used analogy and its usefulness, they might not recognize or understand the analogy of the auxiliary used domain with the source domain effectively under study, or they are not able to understand (simultaneously) the power and the weaknesses of the used analogies. Due to that, analogies are double-edged swords. Instructors/teachers need to be aware that any analogy may fail at some point, opening the way for miscomprehensions. Even thus, given the difficulties usually related to Thermodynamics teaching and learning, it is tempting to try new approaches, in this case through the proposed analogy, as it may increase interest and understanding of the Second Law and entropy as analogs to a more familiar domain for everyone.

Costa^20,21 proposed a parallel structure for looking at Economics through the eyes of Thermodynamics. These are far from being the first studies proposing an analogy for looking at Economics through the eyes of Thermodynamics, as pointed out in ²⁰ and the references therein. However, to the best of the author's knowledge, they are the first ones based on the analogy between heat transfer (flowing in the decreasing temperature direction, driven by a temperature difference) and the traded merchandise transfer (flowing in the increasing unit price direction) in normal trading operations. The works by Costa^20,21 emphasize the setting, analysis, and discussion of a parallel structure for looking at Economics using the eyes of Thermodynamics, and not in using the Thermodynamics-Economics analogy for helping the teaching and learning of the Second Law, entropy, and related subjects. This is the objective of the present paper: as there is an interesting and useful analogy between Thermodynamics and Economics, and as everyone is familiar with merchandise transfer and profit generation in trading operations, that familiarity can be advantageously used for a better understanding of the Second Law of Thermodynamics, entropy, and related subjects. An analogy exists not only for variables but also for the relevant mathematical expressions.

This work presents the developments for each subject, in each section, firstly in Thermodynamics (its main motivation), and secondly in Economics (in the present context, a more familiar domain for those learning Thermodynamics). Section 2 introduces and discusses the Clausius Inequality, entropy definition, and entropy generation. Section 3 analyzes the Carnot cycle, and Section 4 analyzes the power cycles arbitrary in terms of reversibility. In Section 5, the previous developments that apply to cycles are extended to apply to general processes, cyclic or not. Section 6 revisits the differential equation defining entropy and highlights the conditions in which it applies, and Section 7 is devoted to the irreversibility of the transfer through a finite temperature difference.

As many ideas, concepts, variables, and relations appear in the text, in two different domains, the units of the involved variables are indicated inside square brackets for accuracy and easier reading and understanding.

The Clausius Inequality, entropy, and entropy generation

The starting point of the presented developments is the Clausius Inequality, which can be obtained in both Thermodynamics^13–15 and (its economic counterpart) in Economics, based on the observations from the thermal engines and the economic activity (specifically from the observations of normal merchandise trading),^20,21 respectively. The developments leading to the Clausius Inequality start with the Kelvin-Plack statement based on those observations.^{13–15,20,21}

The thermodynamic system under analysis is separated from the rest of the Universe by a boundary, an imaginary frontier separating what is under analysis from what is out of analysis. The thermodynamic system may exchange heat and work with its neighboring through the impermeable parts of the boundary, and mass and energy through the permeable parts of the boundary if it is an open system.^13–15 These are the mass and energy interactions of the system with its neighboring. Exchanges may occur through the boundary, and processes may occur in the system, inside the boundary.

Analogously, the economic system under analysis is separated from the rest of the Universe by a boundary, an imaginary frontier separating what is under analysis from what is out of analysis. The economic system may exchange merchandise, wealth, and money with its neighboring through the boundary, which can be crossed by material (tangible, hardware) or immaterial (intangible, software, ideas, information, knowledge) species. The economic system can be an individual, a family, a set of individuals, an organization, a city, a county, a country, or a set of countries. Exchanges may occur through the boundary, and processes may occur in the economic system, inside the boundary.

In Thermodynamics

The developments are based on heat and temperature, and on the fact that heat flows spontaneously from higher to lower temperatures (which is an observed fact, imposed by nature).

The Clausius Inequality sets that, in a cycle ¹⁵

$$\oint {\displaystyle \frac{\delta Q}{T}}\le 0[\mathrm{J}/\mathrm{K}]$$

(1a)

$\delta Q$ [J] is the infinitesimal heat crossing the closed system's boundary, where the local temperature is T [K]. For a better interpretation of the message of the Clausius Inequality, let's divide the portions of the boundary where heat is entering the system (positive heat transfer interactions) and where heat is leaving the system (negative heat transfer interactions)

$$\underset{in}{\int }\frac{\delta Q_{in}^{+}}{T}+\underset{out}{\int }\frac{\delta Q_{out}^{-}}{T}\le 0[\mathrm{J}/\mathrm{K}]$$

(2a)

superscripts indicating the positivity or the negativity of the heat transfer interactions.

Considering now the heat entering the system in one cycle, $Q_{in}^{+}={\oint }_{in}\delta Q_{in}^{+}$ [J], the average temperature of the system's boundary where heat is entering the system being ${\overline{T}}_{in}=Q_{in}^{+}/\underset{in}{\int }(\delta Q_{in}^{+}/T)$ [K], and the heat leaving the system in one cycle, $Q_{out}^{-}={\oint }_{out}\delta Q_{out}^{-}$ [J], the average temperature of the system's boundary where heat is leaving the system being ${\overline{T}}_{out}=Q_{out}^{-}/\underset{out}{\int }(\delta Q_{out}^{-}/T)$ [K], as illustrated in Figure 1a, Eq. (2a) can be rewritten as

$${\displaystyle \frac{Q_{in}^{+}}{{\overline{T}}_{in}}+{\displaystyle \frac{Q_{out}^{-}}{{\overline{T}}_{out}}\le 0[\mathrm{J}/\mathrm{K}]}}$$

(3a)

from which it can be concluded that

$${\displaystyle \frac{|Q_{out}^{-}|}{{\overline{T}}_{out}}\ge {\displaystyle \frac{Q_{in}^{+}}{{\overline{T}}_{in}}[\mathrm{J}/\mathrm{K}]}}$$

(4a)

Figure 1.

Auxiliary figure for a better understanding of the Clausius inequality: (a) In thermodynamics; and (b) in economics.

This result sets that, in a cycle, the output ratio $|Q_{out}^{-}|/{\overline{T}}_{out}$ [J/K] can only be higher or in the limit equal, to the inlet ratio $Q_{in}^{+}/{\overline{T}}_{in}$ [J/K]. In practice, what happens is that as heat flows spontaneously from the higher temperatures to the lower temperatures, it must be ${\overline{T}}_{in}>{\overline{T}}_{out}$ [K], and additionally if the system operates irreversibly, with associated heat generation in the system, that contributes to increase $|Q_{out}^{-}|$ [J]. These are two factors contributing to the validity of Eq. (4a).

A reversible cycle can be reversed, that is, operated in the reverse direction, reverting its mass and energy interactions, without requiring other additional effects, namely without investing additional external energy. On the contrary, an irreversible cycle can be operated in the reverse direction only if other additional effects are provided.

The limiting equality situation of the Clausius Inequality corresponds to the equality in Eq. (4a), which must correspond to the absence of heat generation in the system due to irreversibility (lowest heat $|Q_{out}^{-}|$ [J] leaving the system), that is, it must correspond to the reversible situation. The reversible situation is the one for which the cycle is perfect and can be reversed by inverting its energy (heat and work) transfer interactions.

It is well-known from Calculus that if the cyclic integral is null the integrand function is an exact differential. This is the basis for the definition of the new property entropy, through its differential expression, as

$$dS\equiv {\left({\displaystyle \frac{\delta Q}{T}}\right)}_{\mathrm{int}\mathrm{rev}}[\mathrm{J}/\mathrm{K}]$$

(5a)

subscript ‘int rev’ remembering that this definition prevails for an internally reversible process (no internal heat generation due to irreversibility).

When the system operates irreversibly, some heat generation occurs in the system associated with that irreversibility, thus leading to entropy generation in the system. For the situation for which Eq. (3a) prevails, the entropy generation is defined as the strength of inequality, as

$$S_g\equiv -{\displaystyle \frac{Q_{out}^{-}}{{\overline{T}}_{out}}-{\displaystyle \frac{Q_{in}^{+}}{{\overline{T}}_{in}}\ge 0[\mathrm{J}/\mathrm{K}]}}$$

(6a)

which prevails for a cycle executed by a system when in contact with two heat reservoirs at temperatures ${\overline{T}}_{in}$ [K] and ${\overline{T}}_{out}$ [K].

If no heat leaves the system in one cycle, $Q_{out}^{-}=0$ [J] and $S_g=-Q_{in}^{+}/{\overline{T}}_{in}<0$ [J/K], a negative entropy generation, which is impossible to occur given Eq. (6a). If no heat enters the system in one cycle, $Q_{in}^{+}=0$ [J] and $S_g=-Q_{out}^{-}/{\overline{T}}_{out}>0$ [J/K], a result compatible with Eq. (6a). However, in a cycle no energy storage is possible, and the unique possibility of having heat leaving the system is to have mechanical work, $W_{in}^{-}$ [J] entering the system in that cycle. The work transfer interactions W [J] are taken as positive when released by the system, assuming that the positive effect of a thermal engine is the delivered mechanical work.^13–15 Due to internal irreversibility, this mechanical work $|W_{in}^{-}|$ [J] entering the system for the situation when $Q_{in}^{+}=0$ [J] is converted into heat $|Q_{out}^{-}|$ [J], which leaves the system as $Q_{out}^{-}$ [J].

The mechanical work transfer interaction W [J] can be seen as heat crossing the system's boundary at an infinite temperature. In terms of energy, it continues being the mechanical work, W [J], and from the entropy viewpoint it has the associated null entropy $(W/T{)}_{T\to \mathrm{\infty }}=0$ [J/K]. In this case, it is like heat $|Q_{out}^{-}|=|W_{in}^{-}|$ [J] flowing from the infinite temperature to the finite temperature ${\overline{T}}_{out}$ [K] at the system's boundary where it ( $Q_{out}^{-}$ [J]) leaves the system.

Only for a reversible cycle (this whose sense can be reversed, inverting its energy (work and heat) transfer interactions with no other additional effects)) it is

$$S_g=-{\displaystyle \frac{Q_{out}^{-}}{{\overline{T}}_{out}}-{\displaystyle \frac{Q_{in}^{+}}{{\overline{T}}_{in}}=0[\mathrm{U}/\text{€}]}}$$

(7a)

In Economics

The developments are based on traded merchandise and economic temperature, and on the fact that traded merchandise flows spontaneously in the increasing unit price direction in natural trading operations.^20,21

The economic temperature $T_E$ [U/€] is defined as the inverse of the merchandise unit price p [€/U], as^20,21

$$T_E={\displaystyle \frac{1}{p}[\mathrm{U}/\text{€}]}$$

(8)

Unit [U] is used for the merchandise units. The fact that traded merchandise flows spontaneously in the increasing unit price direction in natural trading operations is a factual observation, and imposed by the states and the regulatory authorities, as in natural trading operations it is forbidden to sell merchandise at a unit price lower than its purchasing unit price. Trading operations in the decreasing unit price direction may occur, depending on arbitrary decisions of the economic system controller, but, in the present context, such trading operations are not considered to be natural trading operations. Eq. (40) clarifies why the economic temperature is defined as the inverse of the unit price.

Temperature is a property of a thermodynamic system,^13–15 and all the chemical species in the thermodynamic system share the same temperature value.^13–15 The proposed economic temperature is not a property of the economic system but a property of one traded merchandise species in the economic system, as it depends on the context, which influences its unit price when crossing the economic system's boundary.^20,21 Different merchandise species in the same economic system have different economic temperatures. A more complete and accurate definition of the economic temperature must refer to merchandise species i, $T_{E,i}=1/p_i$ [U/€].^20,21 However, once this is made clear, for conciseness, only economic temperature $T_E$ [U/€] will be used whenever possible. If the economic system is composed of different merchandise species, each with its unit price, and thus with its own economic temperature, analysis must be conducted considering each merchandise species separately and then adding the contributions of all merchandise species.

Additionally, the presented developments apply to situations when there is no generation or destruction of merchandise species in the economic system.

The economic Clausius Inequality sets that, in a cycle ²¹

$$\oint {\displaystyle \frac{\delta M}{T_E}}\le 0[\text{€}]$$

(1b)

$\delta M$ [U] is the infinitesimal traded merchandise crossing the economic system's boundary, where the local economic temperature is $T_E$ [U/€]. For a better interpretation of the message of the economic Clausius Inequality, let's divide the portions of the boundary where traded merchandise is entering the economic system (positive traded merchandise transfer interactions) and where traded merchandise is leaving the system (negative traded merchandise transfer interactions)

$$\underset{in}{\int }\frac{\delta M_{in}^{+}}{T_E}+\underset{out}{\int }\frac{\delta M_{out}^{-}}{T_E}\le 0[\text{€}]$$

(2b)

Considering now the traded merchandise units entering the system in one cycle, $M_{in}^{+}={\oint }_{in}\delta M_{in}^{+}$ [U], the average economic temperature of the system's boundary where traded merchandise is entering the system being ${\overline{T}}_{E,in}=M_{in}^{+}/\underset{in}{\int }(\delta M_{in}^{+}/T_E)$ [U/€], and the traded merchandise units leaving the system in one cycle, $M_{out}^{-}={\oint }_{out}\delta M_{out}^{-}$ [U], the average economic temperature of the system's boundary where traded merchandise is leaving the system being ${\overline{T}}_{E,out}=M_{out}^{-}/\underset{out}{\int }(\delta M_{out}^{-}/T_E)$ [U/€], as illustrated in Figure 1b, Eq. (2b) can be rewritten as

$${\displaystyle \frac{M_{in}^{+}}{{\overline{T}}_{E,in}}+{\displaystyle \frac{M_{out}^{-}}{{\overline{T}}_{E,out}}\le 0[\text{€}]}}$$

(3b)

from which it can be concluded that

$${\displaystyle \frac{|M_{out}^{-}|}{{\overline{T_E}}_{out}}\ge {\displaystyle \frac{M_{in}^{+}}{{\overline{T_E}}_{,in}}[\text{€}]}}$$

(4b)

This result sets that, in a cycle, the output ratio $|M_{out}^{-}|/{\overline{T}}_{E,out}$ [€] can only be higher or in the limit equal to the inlet ratio $M_{in}^{+}/T_{E,in}$ [€]. In practice, what happens is that as traded merchandise flows spontaneously from higher economic temperatures to lower economic temperatures in normal trading operations, it must be ${\overline{T}}_{E,in}>{\overline{T}}_{E,out}$ [U/€] (from lower unit prices to higher unit prices, it must be ${\overline{p}}_{in}<{\overline{p}}_{out}$ [U/€]). Additionally, suppose the economic system operates irreversibly, with associated financial value generation (merchandise unit price increase) in the system. In that case, there are two factors contributing to the observation of Eq. (4b). In any natural cyclic trading operation, the financial value of the leaving (sold) traded merchandise is higher than, or in the limit equal to, the financial value of the entering (purchased) traded merchandise. Anyone (the economic system controller) expects a higher financial value of the merchandise leaving the system (the merchandise being sold) than the financial value of the merchandise entering the system (the merchandise being purchased), to make money in the trading operation. In the limiting reversible situation, the financial value of the merchandise leaving the system (the merchandise being sold) equals the financial value of the merchandise entering the system (the merchandise being purchased), and no profit is generated in such a trading operation.

A reversible economic cycle can be reversed, that is, operated in the reverse direction, reverting its merchandise, wealth, and money interactions, without requiring other additional effects, namely without investing additional external merchandise, wealth, or money. On the contrary, an irreversible economic cycle can be operated in the reverse direction only if other additional effects are provided.

The limit equality situation of the economic Clausius Inequality corresponds to the equal sign in Eq. (4b), which must correspond to the absence of merchandise financial value generation in the economic system due to irreversibility, that is, it must correspond to the reversible situation. The reversible situation is the one for which the cycle is perfect and can be reversed by inverting its merchandise (traded merchandise and merchandise wealth) transfer interactions.

It is well-known from Calculus that if the cyclic integral is null the integrand function is an exact differential. This is the basis for the definition of the new property economic entropy, through its differential expression, as ²¹

$$d{S}_E\equiv {\left({\displaystyle \frac{\delta M}{T_E}}\right)}_{\mathrm{int}\mathrm{rev}}[\text{€}]$$

(5b)

subscript ‘int rev’ remembering that this definition prevails for an internally reversible process (no internal financial value generation due to irreversibility).

From Eq. (5b) it is anticipated that the economic entropy of an economic system is its merchandise financial value. Thus, if the economic system is composed of $N_M$ [U] merchandise units whose economic temperature is $T_E$ [U/€] (whose unit price is p [€/U]), it is

$$S_E={\displaystyle \frac{N_M}{T_E}=N_M\cdot p[\text{€}]}$$

(9)

When the system operates irreversibly, some financial value generation exists in the economic system associated with that irreversibility, thus leading to economic entropy generation in the economic system. For the situation for which Eq. (3b) prevails, the economic entropy generation is defined as the strength of the inequality as

$$S_{E,g}\equiv -{\displaystyle \frac{M_{out}^{-}}{{\overline{T_E}}_{out}}-{\displaystyle \frac{M_{in}^{+}}{{\overline{T_E}}_{in}}\ge 0[\text{€}]}}$$

(6b)

which prevails for a cycle executed by an economic system when in contact with two merchandise reservoirs at the economic temperatures ${\overline{T}}_{E,in}$ [U/€] and ${\overline{T}}_{E,out}$ [U/€]. The concept of merchandise reservoir is defined similarly to a heat reservoir in Thermodynamics,^13–15 whose temperature remains unchanged regardless of the heat it receives or releases. Thus, a merchandise reservoir is an economic system whose economic temperature remains unchanged and is independent of the traded merchandise units it receives or releases.^20,21

If no traded merchandise leaves the economic system in one cycle, $M_{out}^{-}=0$ [U] and $S_{E,g}=-M_{in}^{+}/{\overline{T}}_{E,in}<0$ [€], a negative economic entropy generation, which is impossible given Eq. (6b). This corresponds to purchasing traded merchandise but not selling any of the purchased merchandise, which corresponds to a decrease in the financial value of the economic system. If no traded merchandise enters the system in one cycle, $M_{in}^{+}=0$ [U] and $S_{E,g}=-M_{out}^{-}/{\overline{T}}_{E,out}>0$ [€], compatible with Eq. (6b). However, in a cycle no merchandise storage in the system is possible, and the unique possibility of having traded merchandise leaving the system is to have merchandise wealth $W_{M,in}^{-}$ [U] entering the system in that cycle. Due to economic irreversibility, the merchandise wealth $|W_{M,in}^{-}|$ [U] entering the system is converted into traded merchandise $|M_{out}^{-}|$ [U], which leaves the system as $M_{out}^{-}$ [U].

Merchandise wealth is merchandise that is not in the market and is thus not transferred by trading reasons/motivations. Consider, for example, a loaded truck, the load being (tradable) merchandise but not the truck itself (which, in this sense, is merchandise wealth).^20,21 In a different context, the truck can itself be tradable merchandise. In common language, it is usual to say that (true) wealth corresponds to things that are not on the market, are not to be purchased or sold, and have no price, or, in the present context, have a null unit price, which is the same as saying that they have an infinite economic temperature. Analogously as for mechanical work in Thermodynamics,^13–15 the merchandise wealth transfer interactions $W_M$ [U] are taken as positive when released by the economic system, assuming that the positive effect of the cyclically operating economic engine is the delivered merchandise wealth.^20,21

The merchandise wealth $W_M$ [U] can be seen as traded merchandise crossing the system's boundary at an infinite economic temperature (at a null unit price). In terms of merchandise, it continues to be the merchandise wealth, $W_M$ [U], and from the economic entropy viewpoint, it has the associated null economic entropy $(W_M/T_E{)}_{T_E\to \mathrm{\infty }}=(W_M\cdot p{)}_{p\to 0}=0$ [€].^20,21 In this case, it is like traded merchandise $|M_{out}^{-}|=|W_{M,in}^{-}|$ [U] flowing from the infinite economic temperature to the finite economic temperature ${\overline{T}}_{E,out}$ [U/€] (from the null unit price to the finite unit price ${\overline{p}}_{out}$ [€/U]) at the system's boundary where it ( $M_{out}^{-}$ [U]) leaves the system.

Only for an economic reversible cycle (this whose sense can be reversed, inverting its merchandise (traded merchandise and merchandise wealth) transfer interactions with no other additional effects)) it is

$$S_{E,g}=-{\displaystyle \frac{M_{out}^{-}}{{\overline{T_E}}_{out}}-{\displaystyle \frac{M_{in}^{+}}{{\overline{T_E}}_{in}}=0[\text{€}]}}$$

(7b)

That is, only the economic reversible cycle is the one for which there is no internal economic entropy generation (no internal financial value generation).

The Carnot cycle

The Carnot cycle is crucial in Thermodynamics.^13–15 In this section, it is analyzed as usual, and its analog, the economic Carnot cycle, is proposed and analyzed.

In Thermodynamics

For the (reversible) Carnot cycle in Figure 2a, it is null the entropy generation

$$S_g=-{\displaystyle \frac{Q_1^{+}}{T_1}-{\displaystyle \frac{Q_{2,C}^{-}}{T_2}=0[\mathrm{J}/\mathrm{K}]}}$$

(10a)

Figure 2.

Graphical representation of the Carnot cycle: (a) In thermodynamics; and (b) In economics.

For a given $Q_1^{+}>0$ , from the energy balance equation^13–15 it is

$$W_C^{+}=Q_1^{+}+Q_{2,C}^{-}[\mathrm{J}]$$

(11a)

and using Eq. (10a) it is obtained

$$W_C^{+}=Q_1^{+}\left(1-{\displaystyle \frac{T_2}{T_1}}\right)[\mathrm{J}]$$

(12a)

$$Q_{2,C}^{-}=-Q_1^{+}\left({\displaystyle \frac{T_2}{T_1}}\right)[\mathrm{J}]$$

(13a)

The efficiency of the Carnot cycle is defined as the ratio between its useful effect, the released mechanical work $W_C^{+}$ [J], and the heat input $Q_1^{+}$ [J] required for that, that is

$${\eta }_C={\displaystyle \frac{W_C^{+}}{Q_1^{+}}=1-{\displaystyle \frac{T_2}{T_1}[\text{-}]}}$$

(14a)

We would, in principle, expect to have a performance even better than that of the Carnot cycle, and in that case, for a fixed $Q_1^{+}>0$ [J], from the energy balance equation^13–15 the increase in $W^{+}$ [J] would appear as a decrease in $|Q_2^{-}|$ [J]. As it is

$$W^{+}=W_C^{+}-(W_C^{+}-W^{+})[\mathrm{J}]$$

(15a)

in that case with $W^{+}>W_C^{+}$ [J], and

$$Q_2^{-}=Q_{2,C}^{-}-(Q_{2,C}^{-}-Q_2^{-})[\mathrm{J}]$$

(16a)

with $Q_2^{-}>Q_{2,C}^{-}$ [J], obeying $(W_C^{+}-W^{+})=(Q_{2,C}^{-}-Q_2^{-})<0$ [J], it would be

$$S_g=-{\displaystyle \frac{Q_1^{+}}{T_1}-{\displaystyle \frac{Q_2^{-}}{T_2}=-{\displaystyle \frac{Q_1^{+}}{T_1}-{\displaystyle \frac{Q_{2,C}^{-}-(Q_{2,C}^{-}-Q_2^{-})}{T_2}={\displaystyle \frac{(Q_{2,C}^{-}-Q_2^{-})}{T_2}<0[\mathrm{J}/\mathrm{K}]}}}}}$$

(17a)

what is impossible to occur as $S_g<0$ [J/K], and the Carnot thermal engine, for which $S_g=0$ [J/K], has the best performance when operating in contact with the same temperatures $T_1$ [K] and $T_2$ [K]. As given by Eq. (10a), the entropy entering the system $Q_1^{+}/T_1$ [J/K] is equal to the entropy leaving the system $|Q_{2,C}^{-}|/T_2$ [J/K], and thus no entropy generation occurs in the Carnot cycle.

At this point, it can be easily understood why even the Carnot thermal engine, the best thermal engine, is unable to integrally convert heat into mechanical work. If that would be possible, for such a situation it would be $Q_{out}^{-}=0$ [J] and $S_g=-Q_{in}^{+}/{\overline{T}}_{in}<0$ [J/K], a situation impossible to occur at it would lead to $S_g<0$ [J/K].

In Economics

For the (reversible) economic Carnot cycle in Figure 2b, it is null the economic entropy generation

$$S_{E,g}=-{\displaystyle \frac{M_1^{+}}{T_{E,1}}-{\displaystyle \frac{M_{2,C}^{-}}{T_{E,2}}=0[\text{€}]}}$$

(10b)

For a given $M_1^{+}>0$ [U], from the merchandise units balance equation^20,21

$$W_{M,C}^{+}=M_1^{+}+M_{2,C}^{-}[\mathrm{U}]$$

(11b)

and using Eq. (10b) it is obtained

$$W_{M,C}^{+}=M_1^{+}\left(1-{\displaystyle \frac{T_{E,2}}{T_{E,1}}}\right)=M_1^{+}\left(1-{\displaystyle \frac{p_1}{p_2}}\right)[\mathrm{U}]$$

(12b)

$$M_{2,C}^{-}=-M_1^{+}\left({\displaystyle \frac{T_{E,2}}{T_{E,1}}}\right)=-M_1^{+}\left({\displaystyle \frac{p_1}{p_2}}\right)[\mathrm{U}]$$

(13b)

The efficiency of the economic Carnot cycle is defined as the ratio between its useful effect, in this case the released merchandise wealth $W_{M,C}^{+}$ [U], and the traded merchandise input $M_1^{+}$ [U] required for that, that is^20,21

$${\eta }_{E,C}={\displaystyle \frac{W_{M,C}^{+}}{M_1^{+}}=1-{\displaystyle \frac{T_{E,2}}{T_{E,1}}=1-{\displaystyle \frac{p_1}{p_2}[\text{-}]}}}$$

(14b)

We would, in principle, expect to have a performance even better than that of the economic Carnot cycle, and in that case, for a fixed $M_1^{+}>0$ [U], from the merchandise units balance equation,^20,21 the increase in $W_M^{+}$ [U] would appear as a decrease in $|M_2^{-}|$ [U]. As it is

$$W_M^{+}=W_{M,C}^{+}-(W_{M,C}^{+}-W_M^{+})[\mathrm{U}]$$

(15b)

with $W_M^{+}>W_{M,C}^{+}$ [U], and

$$M_2^{-}=M_{2,C}^{-}-(M_{2,C}^{-}-M_2^{-})[\mathrm{U}]$$

(16b)

with $M_2^{-}>M_{2,C}^{-}$ [U], obeying $(W_{M,C}^{+}-W_M^{+})=(M_{2,C}^{-}-M_2^{-})<0$ [U], it would be

$$S_{E,g}=-{\displaystyle \frac{M_1^{+}}{T_{E,1}}-{\displaystyle \frac{M_2^{-}}{T_{E,2}}=-{\displaystyle \frac{M_1^{+}}{T_{E,1}}-{\displaystyle \frac{M_{2,C}^{-}-(M_{2,C}^{-}-M_2^{-})}{T_{E,2}}={\displaystyle \frac{(M_{2,C}^{-}-M_2^{-})}{T_{E,2}}<0[\text{€}]}}}}}$$

(17b)

what is impossible to occur as $S_{E,g}<0$ [€], and the economic Carnot engine, for which $S_{E,g}=0$ [€], has the best performance when operating in contact with the same economic temperatures $T_{E,1}$ [U/€] and $T_{E,2}$ [U/€] (when operating in contact with the same unit prices $p_1$ [€/U] and $p_2$ [€/U]).

At this point, it can be easily understood why even the economic Carnot engine, the best economic engine, is unable to integrally convert traded merchandise into merchandise wealth in normal trading operations. If that were possible, in such a situation it would be $M_{out}^{-}=0$ [U] and $S_{E,g}=-M_{in}^{+}/{\overline{T}}_{E,in}<0$ [€], a situation impossible to occur at it would lead to $S_{E,g}<0$ [€]. No one is interested in trading operations leading to negative economic entropy generation (negative financial value generation). As given by Eq. (10b), the economic entropy (the financial value) entering the system $M_1^{+}/T_{E,1}$ [€] is equal to the economic entropy (the financial value) leaving the system $|M_{2,C}^{-}|/T_{E,2}$ [€], and thus no economic entropy generation (no financial value generation) occurs in the economic Carnot cycle.

At this point, it can be understood how the economic Carnot cycle partially converts traded merchandise into merchandise wealth,^20,21 which helps understand how the Carnot cycle partially converts heat into mechanical work.^13–15 Considering Figure 2b, the traded merchandise $M_1^{+}$ [U] is obtained (purchased) at the higher economic temperature $T_{E,1}$ [U/€] (at the lower unit price $p_1$ [€/U]), entering the economic system with the associated merchandise economic entropy transfer $M_1^{+}/T_{E,1}$ [€] (with the associated financial value transfer $M_1^{+}\cdot p_1$ [€]). The traded merchandise $|M_{2,C}^{-}|$ [U] leaves the economic system (is sold) at the lower economic temperature $T_{E,2}$ [U/€] (at the higher unit price $p_2$ [€/U]), with the associated merchandise economic entropy transfer $|M_{2,C}^{-}|/T_{E,2}$ [€] (with the associated financial value transfer $|M_{2,C}^{-}|\cdot p_2$ [€]), obeying to $M_1^{+}\cdot p_1=|M_{2,C}^{-}|\cdot p_2$ as no economic entropy generation (no financial value generation) occurs in the (reversible) economic Carnot cycle. Under these economically reversible conditions (null merchandise economic entropy generation), it suffices to sell the $|M_{2,C}^{-}|$ [U] merchandise units at the higher unit price $p_2$ [€/U] to obtain the financial value necessary to purchase the $M_1^{+}$ [U] merchandise units at the lower unit price $p_1$ [€/U], and the wealth merchandise units $W_{M,C}^{+}=M_1^{+}-|M_{2,C}^{-}|=M_1^{+}(1-p_1/p_2)$ [U] can be seen as obtained for free, at a null unit price (at an infinite economic temperature). This is why $W_{M,C}^{+}$ [U] is merchandise wealth (which, if seen as traded merchandise, has a null unit price).

Cycles arbitrary in terms of reversibility

The irreversibility concept and entropy generation are crucial in Thermodynamics.^13–15 In this section are analyzed cycles arbitrary in terms of reversibility and the associated entropy generation, and their economic analogs, the economic cycles arbitrary in terms of reversibility and the associated economic entropy generation.

In Thermodynamics

For a fixed $Q_1^{+}>0$ [J], for an irreversible cycle as illustrated in Figure 3a, from the energy balance equation^13–15 the decrease in $W^{+}$ [J] appears as an increase in $|Q_2^{-}|$ [J], Eqs. (15a) and (16a) continue applying, and

$$S_g=-{\displaystyle \frac{Q_1^{+}}{T_1}-{\displaystyle \frac{Q_2^{-}}{T_2}=-{\displaystyle \frac{Q_1^{+}}{T_1}-{\displaystyle \frac{Q_{2,C}^{-}-(Q_{2,C}^{-}-Q_2^{-})}{T_2}={\displaystyle \frac{(Q_{2,C}^{-}-Q_2^{-})}{T_2}>0[\mathrm{J}/\mathrm{K}]}}}}}$$

(18a)

from where it is obtained

$$(Q_{2,C}^{-}-Q_2^{-})=(W_C^{+}-W^{+})=T_2{S}_g\ge 0[\mathrm{J}]$$

(19a)

Figure 3.

Graphical representation of a cycle arbitrary in terms of reversibility: (a) In thermodynamics; and (b) In economics.

As given by Eq. (18a), the entropy entering the system $Q_1^{+}/T_1$ [J/K] is lower than the entropy leaving the system $|Q_{2,C}^{-}|/T_2$ [J/K], and thus entropy generation occurs in the irreversible cycle.

In turn, the efficiency of the irreversible thermal engine, $\eta =W^{+}/Q_1^{+}$ [-], can be expressed as

$$\eta ={\eta }_C-{\displaystyle \frac{T_2{S}_g}{Q_1^{+}}[\text{-}]}$$

(20a)

evidencing how irreversibility, accounted for through the entropy generation $S_g$ [J/K], decreases the performance of the thermal engines.

In Economics

For a fixed $M_1^{+}>0$ [U], for an economic irreversible cycle as illustrated in Figure 3b, from the merchandise units balance equation,^20,21 the decrease in $W_M^{+}$ [U] appears as an increase in $|M_2^{-}|$ [U], Eqs. (15b) and (16b) continue applying, and it is

$$S_{E,g}=-{\displaystyle \frac{M_1^{+}}{T_{E,1}}-{\displaystyle \frac{M_2^{-}}{T_{E,2}}=-{\displaystyle \frac{M_1^{+}}{T_{E,1}}-{\displaystyle \frac{M_{2,C}^{-}-(M_{2,C}^{-}-M_2^{-})}{T_{E,2}}={\displaystyle \frac{(M_{2,C}^{-}-M_2^{-})}{T_{E,2}}>0[\text{€}]}}}}}$$

(18b)

from where it is obtained

$$(M_{2,C}^{-}-M_2^{-})=(W_{M,C}^{+}-W_M^{+})=T_{E,2}{S}_{E,g}\ge 0[\mathrm{U}]$$

(19b)

As given by Eq. (18b), the economic entropy (the financial value) entering the system $M_1^{+}/T_{E,1}$ [€] is lower than the economic entropy (the financial value) leaving the system $|M_{2,C}^{-}|/T_{E,2}$ [€], and thus economic entropy generation (financial value generation) occurs in the irreversible economic cycle. The economic entropy generation occurring in such a cycle is $S_{E,g}=-M_1^{+}/T_{E,1}-M_2^{-}/T_{E,2}=|M_2^{-}|p_2-M_1^{+}{p}_1>0$ [€], the profit generation (the financial value generation) in the normal (cyclic) trading operation. This profit generation (this financial value generation) is a measure of the economic irreversibility of the considered economic cycle.

In turn, the efficiency of the irreversible economic engine, ${\eta }_E=W_M^{+}/M_1^{+}$ [-], can be expressed as

$${\eta }_E={\eta }_{E,C}-{\displaystyle \frac{T_{E,2}{S}_{E,g}}{M_1^{+}}[\text{-}]}$$

(20b)

evidencing how economic irreversibility, accounted for through the economic entropy generation $S_{E,g}$ [€], decreases the performance of the economic engines.

The normal economic activity is to create merchandise wealth and introduce it into the market, giving it a unit price, to generate economic entropy (to generate financial value). The economic normal is not to have an economic engine operating cyclically to obtain the maximum merchandise wealth from normal merchandise trading.^20,21 However, the economic cycles and the associated concepts are of major relevance to the Second Law developments and understanding.^{13–15,20,21}

Processes in general

Sections 2–4 were devoted to cyclic processes. The analysis is now extended to general processes, cyclic or not. As illustrated in Figure 4, any cycle can be decomposed into two processes, process 1–2 being arbitrary in terms of reversibility and process 2–1 being reversible.

Figure 4.

A cycle as composed of two processes, process 1–2 being arbitrary in terms of reversibility and process 2–1 being reversible.

In Thermodynamics

Given Figure 4, Eq. (1a) can be written as

$$\oint \frac{\delta Q}{T}=\underset{1}{\overset{2}{\int }}\frac{\delta Q}{T}+\underset{2}{\overset{1}{\int }}\frac{\delta Q}{T}\le 0[\mathrm{J}/\mathrm{K}]$$

(21a)

As process 2–1 is reversible, Eq. (21a) can be rewritten as

$$S_g=\left(S_2-S_1\right)-\underset{1}{\overset{2}{\int }}\frac{\delta Q}{T}\ge 0[\mathrm{J}/\mathrm{K}]$$

(22a)

where $S_g$ [J/K] is the entropy generation in the 1–2 process (a measure of the irreversibility of the 1–2 process).

For example, if we have the reversible heating of a body, Eq. (22a) gives

$$S_g=\left(S_2-S_1\right)-\underset{1}{\overset{2}{\int }}\frac{\delta Q}{T}\ge 0[\mathrm{J}/\mathrm{K}]$$

(23a)

and the final entropy of the body is higher than its initial entropy, as from Eq. (23a)

$$S_2=S_1+\underset{1}{\overset{2}{\int }}{\displaystyle \frac{\delta Q_{in}^{+}}{T}}[\mathrm{J}/\mathrm{K}]$$

(24a)

If we have the reversible heat release (cooling) of a body, from Eq. (22a) one obtains that the final entropy of the body is lower than its initial entropy, as it is

$$S_2=S_1+\underset{1}{\overset{2}{\int }}{\displaystyle \frac{\delta Q_{out}^{-}}{T}}[\mathrm{J}/\mathrm{K}]$$

(25a)

It is clarified that the entropy of a system could decrease in a process, as given by Eq. (25a) for the heat release (cooling) process under analysis. In turn, the entropy generation can only be positive in a process, or null in the limit reversible process.

Consider now the irreversible process of water heating in an electric water heater, where electrical resistance generates heat $Q_g=W_e$ [J] (converting the electric energy $W_e$ [J] into heat $Q_g$ [J]), used to heat the mass m [kg] of water from temperature $T_1$ [K] to temperature $T_2$ [K]. Assuming that the water behaves as an incompressible liquid of constant specific heat c [J/(kg.K)], and that the water heater is thermally insulated, the energy balance equation^13–15 allows obtaining the final water temperature as

$$T_2=T_1+{\displaystyle \frac{Q_g}{mc}[\mathrm{K}]}$$

(26)

and the entropy balance equation^13–15 allows obtaining the entropy generation in the process as

$$S_g=mc\ln \left(1+{\displaystyle \frac{Q_g}{mc{T}_1}}\right)>0[\mathrm{J}/\mathrm{K}]$$

(27)

We are familiar with temperature and energy, in this case with heat generation by electrical resistance, but not with entropy generation. However, we could imagine a situation in which we were familiar with entropy and entropy generation, but not with heat generation, and in that case once known the entropy generation in the process we could obtain its associated heat generation as

$$Q_g=mc{T}_1\left[\exp \left({\displaystyle \frac{S_g}{mc}}\right)-1\right][\mathrm{J}]$$

(28)

In general, we prescribe energy transfer interactions of the thermodynamic systems and the entropy generation results from the so-induced process. We act based on the First Law, the results from the Second Law being a consequence of that. However, we could imagine the situation in which we prescribe entropy generation to the thermodynamic processes, the energy transfer interactions being a result of that. In that case, we would act based on the Second Law, the First Law results (the energy transfer interactions) being a consequence of that.

In Economics

Given Figure 4, Eq. (1b) can be written as

$$\oint \frac{\delta M}{T_E}=\underset{1}{\overset{2}{\int }}\frac{\delta M}{T_E}+\underset{2}{\overset{1}{\int }}\frac{\delta M}{T_E}\le 0[\text{€}]$$

(21b)

As process 2–1 is reversible, Eq. (21b) can be rewritten as

$$S_{E,g}=\left(S_{E,2}-S_{E,1}\right)-\underset{1}{\overset{2}{\int }}\frac{\delta M}{T_E}\ge 0[\text{€}]$$

(22b)

where $S_{E,g}$ [J/K] is the economic entropy generation in the 1–2 process (a measure of the economic irreversibility of the 1–2 economic process).

For example, if we have the $M_{in}^{+}$ [U] traded merchandise units entering the economic system at the economic temperature $T_{E,in}$ [U/€] (at the unit price $p_{in}$ [€/U]), in a reversible process, Eq. (22b) gives that

$$0=\left(S_{E,2}-S_{E,1}\right)-\underset{1}{\overset{2}{\int }}\frac{\delta M_{in}^{+}}{T_{E,in}}[\text{€}]$$

(23b)

and the final economic entropy of the system is higher than its initial economic entropy, as from Eq. (23b) it is

$$S_{E,2}=S_{E,1}+{\displaystyle \frac{M_{in}^{+}}{T_{E,in}}=S_{E,1}+M_{in}^{+}\cdot p_{in}[\text{€}]}$$

(24b)

If we have the $|M_{out}^{-}|$ [U] traded merchandise units leaving the economic system at the economic temperature $T_{E,out}$ [U/€] (at the unit price $p_{out}$ [€/U]), in a reversible process, from Eq. (22b) one obtains that the final economic entropy of the system is lower than its initial economic entropy, as

$$S_{E,2}=S_{E,1}+{\displaystyle \frac{M_{out}^{-}}{T_{E,out}}=S_{E,1}+M_{out}^{-}\cdot p_{out}[\text{€}]}$$

(25b)

The economic entropy of an economic system could decrease in a process, as given by Eq. (25b) for the process under analysis. In turn, the economic entropy generation can only be positive in a normal trading operation, or null in the limit reversible economic process.

If we have the irreversible process of increasing the unit price of the $N_M$ [U] units of merchandise in the economic system from $p_1$ [€/U] to $p_2$ [€/U] (decreasing the economic temperature of the $N_M$ [U] units of merchandise in the economic system from $T_{E,1}$ [U/€] to $T_{E,2}$ [U/€]), Eq. (22b) gives that

$$S_{E,g}=(S_{E,2}-S_{E,1})+0>0[\text{€}]$$

(29)

a result that can be written in a more familiar form as

$$S_{E,g}=N_M\cdot p_2-N_M\cdot p_1=N_M(p_2-p_1)>0[\text{€}]$$

(30)

which is the financial value generation in the economic system associated with that process.

In general, we prescribe the economic entropy generation (the financial value generation) to be obtained from the economic systems’ operation, the merchandise interactions required for that being a consequence. In this case, we act based on the economic Second Law (financial value), the results (merchandise units transfer interactions) being an economic First Law consequence of that.

Entropy is generated (appearing from nothing) in irreversible processes. The same happens with the economic entropy (the financial value), which is generated (appearing from nothing) in irreversible economic processes, as is the case of the processes increasing the merchandise unit price.

Revisiting the equation defining entropy

The property entropy is defined through a differential equation for the special case of an internally reversible process.^13–15 Such an equation and the requirement of internal reversibility are revisited. Its economic analog, the differential equation defining the merchandise economic entropy, and the requirement of the internal economic reversibility involved in its definition, are detailed.

In Thermodynamics

The First Law of Thermodynamics (the energy balance equation) for a closed system sets that^13–15

$$\underset{inthesystem}{\underbrace{dU}}=\underset{\mathrm{through}\mathrm{the}\mathrm{boundary}}{\underbrace{\delta Q-\delta W}}[\mathrm{J}]$$

(31)

Heat transfer interaction $\delta Q$ [J] occurs through the system's boundary and the heat generation due to irreversibility $\delta Q_{irr}^{+}$ [J] (positive, as irreversibility is always associated with heat generation) occurs in the system. The change

$$\underset{inthesystem}{\underbrace{TdS}}=\underset{\mathrm{through}\mathrm{the}\mathrm{boundary}}{\underbrace{\delta Q}}+\underset{inthesystem}{\underbrace{\delta Q_{irr}^{+}}}[\mathrm{J}]$$

(32)

occurring in the system results from the algebraic addition of both heat transfer interaction $\delta Q$ [J] and the heat generation $\delta Q_{irr}^{+}$ [J], as illustrated in Figure 5, which legitimates Eq. (32). In turn, the mechanical work transfer interaction $\delta W$ [J] is equal to the change $PdV$ [J] occurring in the system minus the mechanical work $|\delta W_{irr}^{-}|$ dissipated as heat in the system due to irreversibility, as illustrated in Figure 5, thus leading to

$$\underset{\mathrm{through}\mathrm{the}\mathrm{boundary}}{\underbrace{\delta W}}=\underset{inthesystem}{\underbrace{PdV+\delta W_{irr}^{-}}}[\mathrm{J}]$$

(33)

Figure 5.

Graphical representation for a better understanding of the fundamental relation of thermodynamics.

Inserting Eqs (32) and (33) into Eq. (31) leads to

$$\underset{inthesystem}{\underbrace{dU=(TdS-PdV)-(\delta Q_{irr}^{+}+\delta W_{irr}^{-})}}[\mathrm{J}]$$

(34)

As $\delta Q_{irr}^{+}$ [J] is generated from $|\delta W_{irr}^{-}|$ it is

$$\underset{inthesystem}{\underbrace{(\delta Q_{irr}^{+}+\delta W_{irr}^{-})}}=0[\mathrm{J}]$$

(35)

and from Eq. (34) it results that

$$\underset{inthesystem}{\underbrace{dU=TdS-PdV}}[\mathrm{J}]$$

(36)

Eq. (36) is the Fundamental Relation of Thermodynamics, which applies to both irreversible and reversible processes.^13–15 Previous developments clarify that, and why and how the irreversibility terms $\delta Q_{irr}^{+}$ [J] and $\delta W_{irr}^{-}$ [J] cancel each other in the system. Consideration of these irreversibility terms is relevant also for a better understanding of Eq. (5a) defining entropy. The usual justification in the Thermodynamics textbooks does not consider that, invoking just that as Eq. (36) involves only thermodynamic properties, it applies to both irreversible and reversible processes.^13–15

The Fundamental Relation, Eq. (36), can be rewritten as

$$dS={\displaystyle \frac{1}{T}dU+{\displaystyle \frac{P}{T}dV={\displaystyle \frac{1}{T}[(\delta Q-\delta W)+PdV][\mathrm{J}/\mathrm{K}]}}}$$

(37)

which, considering the results in Eqs. (32) and (33), leads to

$$\underset{inthesystem}{\underbrace{dS}}=\underset{\mathrm{through}\mathrm{the}\mathrm{boundary}}{\underbrace{{\displaystyle \frac{\delta Q}{T}}}}-\underset{inthesystem}{\underbrace{{\displaystyle \frac{\delta W_{irr}^{-}}{T}}}}[\mathrm{J}/\mathrm{K}]$$

(38a)

Figure 6a is a graphical illustration of Eq. (38a).

Figure 6.

Graphical representation of the equation relevant to defining entropy: (a) In thermodynamics; and (b) In economics.

For an internally reversible process $\delta W_{irr}^{-}=0$ [J], and Eq. (5a) can be obtained from Eq. (38a). The previous developments clarify the meaning of the ‘int rev’ subscript in Eq. (5a).

In Economics

The proposed state equation (Eq. (9)) of the economic system composed of only one merchandise species is simpler than the state equation for thermodynamic systems.^13–15

Differentiation of Eq. (9) leads to

$$\underset{inthesystem}{\underbrace{d{S}_E}}=\underset{\mathrm{through}\mathrm{the}\mathrm{boundary}}{\underbrace{{\displaystyle \frac{\delta M}{T_E}}}}-\underset{inthesystem}{\underbrace{{\displaystyle \frac{N_M}{T_E^2}(d{T}_E{)}_{irr}^{-}}}}[\text{€}]$$

(38b)

as $d{N}_M=\delta M$ [U], and any economic process that changes the economic temperature of the economic system by increasing the unit price (decreasing the economic temperature) of its merchandise units generates economic entropy (generates financial value), which means economic irreversibility.^20,21

For an internally reversible process $-(N_M/T_E^2)(d{T}_E{)}_{irr}^{-}=0$ [€], and Eq. (5b) can be obtained from Eq. (38b). The previous developments clarified the meaning of the ‘int rev’ subscript in Eq. (5b).

Transfer through a finite temperature difference

The heat transfer through a finite temperature difference and the associated entropy generation are of major relevance in Thermodynamics.^13–15 They are analyzed in this section, as well as their economic counterparts of merchandise transfer through a finite economic temperature difference (through a finite unit price difference) and the associated economic entropy generation.

In Thermodynamics

The entropy balance equation for the thermodynamic system sandwiched between the heat reservoirs at temperatures $T_1$ [K] and $T_2$ [K] in Figure 7a leads to^13–15

$$S_g=-{\displaystyle \frac{Q^{+}}{T_1}-{\displaystyle \frac{Q^{-}}{T_2}=|Q|\left({\displaystyle \frac{1}{T_2}-{\displaystyle \frac{1}{T_1}}}\right)={\displaystyle \frac{|Q|}{T_2}\left(1-{\displaystyle \frac{T_2}{T_1}}\right)[U/\text{€}]}}}$$

(39a)

noting that this same result could have been obtained using Eq. (20a) for $\eta =0$ [-], as in this case no mechanical work is released from the system ( $W^{+}=0$ [J]). As the same heat $|Q|$ [J] enters and leaves the system, and $T_2<T_1$ [K], the entropy $|Q|/T_1$ [J/K] entering the system is lower than the entropy $|Q|/T_2$ [J/K] leaving the system, and thus entropy generation occurs in the system.

Figure 7.

Graphical representation of the transfer through a finite temperature difference: (a) In thermodynamics; and (b) In economics.

Taking Eq. (12a) in mind, and as $|Q|(1-T_2/T_1)=T_2{S}_g$ [J], $T_2{S}_g$ [J] is a measure of the lost opportunity of obtaining the maximum mechanical work that would otherwise be obtained using the Carnot thermal engine operating when in contact with the same $T_1$ [K] and $T_2$ [K] heat reservoirs, due to this fully irreversible heat transfer process. It is to be retained that this fully irreversible process generates the maximum entropy as given by Eq. (39a).

In Economics

The economic entropy balance equation for the economic system sandwiched between the merchandise reservoirs at the economic temperatures $T_{E,1}$ [U/€] and $T_{E,2}$ [U/€] in Figure 7b leads to^20,21

$$S_{E,g}=-{\displaystyle \frac{M^{+}}{T_{E,1}}-{\displaystyle \frac{M^{-}}{T_{E,2}}=|M|\left({\displaystyle \frac{1}{T_{E,2}}-{\displaystyle \frac{1}{T_{E,1}}}}\right)={\displaystyle \frac{|M|}{T_{E,2}}\left(1-{\displaystyle \frac{T_{E,2}}{T_{E,1}}}\right)[\text{€}]}}}$$

(39b)

noting that this same result could have been obtained using Eq. (20b) for ${\eta }_E=0$ [-], as in this case no merchandise wealth is released from the system ( $W_M^{+}=0$ [U]). As the same traded merchandise $|M|$ [U] enters and leaves the system, and $T_{E,2}<T_{E,1}$ [U/€] ( $p_2>p_1$ [€/U]), the economic entropy (the financial value) $|M|/T_{E,1}=|M|p_1$ [€] entering the system is lower than the economic entropy (the financial value) $|M|/T_{E,2}=|M|p_2$ [€] leaving the system, and thus economic entropy generation (financial value generation) occurs in the economic system.

Taking Eq. (12b) in mind, and as $|M|(1-T_{E,2}/T_{E,1})=|M|(1-p_1/p_2)=T_{E,2}{S}_{E,g}=S_{E,g}/p_2$ [U], $T_{E,2}{S}_{E,g}$ [U] is a measure of the lost opportunity of obtaining the maximum merchandise wealth that would otherwise be obtained using the economic Carnot engine operating when in contact with the same $T_{E,1}$ [U/€] and $T_{E,2}$ [U/€] merchandise reservoirs, due to this fully irreversible traded merchandise transfer process. It is to be retained that this fully irreversible process generates the maximum economic entropy (generates the maximum financial value) as given by Eq. (39b).

Equation (39b) can be rewritten as

$$S_{E,g}=|M|p_2\left(1-{\displaystyle \frac{p_1}{p_2}}\right)=|M|(p_2-p_1)[\text{€}]$$

(40)

and the financial value generation in the fully irreversible trading process can be seen as the financial value spent to purchase the $|M|(1-p_1/p_2)$ [U] traded merchandise units at the highest unit price $p_2$ [€/U]. In the reversible economic process, the $W_M^{+}=|M|(1-p_1/p_2)$ [U] merchandise wealth is obtained for free, without spending any financial value for that. The maximum obtainable merchandise wealth can be retained in the economic system or be placed in the market giving it the maximum unit price $p_2$ [€/U]. In the last case, if it is sold at that unit price, the monetary units obtained in the selling process can be taken as monetary wealth in the system or used to purchase merchandise units from the market at the lower unit price $p_1$ [€/U]. This is the meaning of economic imperfection, or economic irreversibility, and lost merchandise wealth in the present context. It is not lost merchandise wealth in the sense that it has been obtained and subsequently lost, but in the sense that it is the opportunity of obtaining it that has been lost.

Eq. (40) highlights also a very well-known result, even if it is not usually read as an economic entropy generation. Product $S_{E,g}=M(p_2-p_1)$ [€] is the profit generation (the financial value generation) resulting from the traded merchandise transfer from the lower unit price $p_1$ [€/U] to the higher unit price $p_2$ [€/U] (from the higher economic temperature $T_{E,1}$ [U/€] to the lower economic temperature $T_{E,2}$ [U/€]). According to the proposed analogy, the counterpart of the entropy generation is the profit generation in Economics resulting from normal merchandise trading. Eq. (40) is also the reason why the economic temperature is defined as the inverse of the unit price, as set by Eq. (8).

Conclusions

An analogy is proposed to help learn and understand the Second Law of Thermodynamics and related subjects. As a parallel structure of that of Thermodynamics can be set for Economics, and especially for normal trading operations, and everyone is familiar with the merchandise trading operations and the profit generation (the financial value generation) in normal trading operations, these economic processes can be taken as the analogs of thermodynamic processes. This analogy can be used advantageously to help learn and attract to the study of Thermodynamics, and especially to learn and study the Second Law of Thermodynamics and related subjects. The proposed and explored analogy exists for both variables and relevant mathematical expressions.

The main results of the proposed analogy include:

- The traded merchandise transfer (flowing in the increasing unit price direction) in trading operations as the analog of heat transfer (flowing in the decreasing temperature direction, driven by a temperature difference),

- The differential equation defining the economic entropy as the economic analog of the differential equation defining entropy,

- The merchandise financial value as the economic analog of entropy,

- The financial value generation in normal trading operations as the analog of entropy generation,

- The economic analog of irreversibility,

- The economic irreversibility related to economic entropy (financial value) generation in normal trading operations as the analog of entropy generation in irreversible processes,

- The (reversible)economic Carnot cycle, the analog of the Carnot cycle,

- The economic analogs of the irreversible power cycles, and the associated economic entropy generation (the associated financial value generation),

- The economic entropy generation in merchandise transfer through a finite unit price difference as the analog of the entropy generation in heat transfer through a finite temperature difference.

The proposed analogy is advantageous for better learning and understanding of the Second Law of Thermodynamics and related issues. It is also useful in giving new insights and promoting the use of new approaches for economic and financial analyses. It sets analogies of the (only apparently different and non-related) Thermodynamics and Economics domains.