To analyze letters on my article I am going to answer first comments that are common across all letters and then comments that are specific for each author.

A general comment related to feasibility considers that the process described in Figure 1 of the article is not permitted by thermodynamic laws. In this respect thermodynamics suggests that for a process to be feasible temperature must be greater than 0 K, energy must be conserved, and the total entropy change of the universe should be equal to or greater than zero (1­5). I have demonstrated in the article that the process sketched in Figure 1 fits the above requirements. Therefore, its operation meets general thermodynamic requisites and it should be feasible.

In any case, an intuitive view of the dynamics of the process suggests that once the adiabatic film covering the metal partition is removed, transition starts with a spontaneous heat transfer between tank A and B, caused by a temperature gradient across the metal separation. Simultaneously, compression begins at a controlled rate to keep isothermal conditions in A. Then transformation may continue to reach the final state according to prediction.

It is convenient to say that the process shows a set of conditions where internal entropy is simultaneously created and destroyed in different parts of the universe. This is an interesting behavior suggesting the existence of an internal entropy coupling not seen before in common systems. Under these conditions the process is more efficient than a conventional reversible operation.

Now, the important fact here is that this unexpected universe may exist because it meets general thermodynamic requirements. Otherwise, articulation of thermodynamic laws should be reviewed to consider this case.

Another common comment considers that the process of Figure 1 is not allowed by thermodynamics because it is impossible that the isothermal compression process with internal entropy coupling requires less work than a conventional reversible isothermal compression for the same initial and final state.

To this objection, it is interesting to detect that by linking together a nonreversible isothermal compression with a heat transfer between two tanks it is possible to find a feasible set of conditions where the nonisothermal compression work input is less than the work required by a common reversible isothermal compression for the same initial and final states.

This result is unexpected from the point of view of classical thermodynamics, but irreversible thermodynamics suggests that such a behavior may occur as a consequence of the simultaneous production and destruction of internal entropy in different parts of the universe. Here an oriented heat transfer between tank A and tank B produces or creates enough internal entropy to drive a simultaneous nonreversible isothermal compression in tank A with destruction of internal entropy. Some authors believe that production of internal entropy causes a loss in capacity to do work (1, 3). Then, by analogy, destruction of internal entropy may increase the ability of the system to produce work. In this context the net result of simultaneous production and destruction of internal entropy is a gain in capacity of the system to do work relative to the corresponding reversible isothermal compression. During the operation energy is conserved and the total entropy change of the universe is greater than zero, fitting general thermodynamic requirements.

Behavior exhibited by this process implies that irreversibility under internal entropy coupling conditions may enhance the ability of a system to do work relative to an equivalent reversible operation for the same change of state. I have further confirmed this by designing a feasible thermodynamic cycle with internal entropy coupling resulting in a cycle of greater efficiency than an equivalent Carnot cycle operating between the same temperature levels (6, 7, J. I. Belandria, unpublished). This finding is unusual and reveals an extraordinary feature of internal entropy coupling systems that suggests the possibility of designing feasible thermodynamic cycles more efficient than conventional classical ones by introducing steps involving simultaneous production and destruction of internal entropy.

All the letters estimate the final pressure reached by a conventional reversible isothermal compression at 1500 K using as work input the value required in Figure 1 and find 3.086 atm. They argue correctly that a reversible compression cannot reach the final pressure of 4 atm obtained by the system sketched in Figure 1. This is true because the process described in the article is more efficient than a conventional reversible isothermal compression as a consequence of the simultaneous production and destruction of internal entropy, as I explained earlier.

Some letters express opinions in relation to specification of the final state. For example, Nash wonders "where the author gets his figure for the final pressure in tank A". Tykodi says "he assumes an impossible condition in the final state for his illustrative process".

Olivares and Colmenares state that "the final pressure of 405.32 kPa used by Belandria is unattainable". And Freeman considers that "he has simply postulated initial and final conditions without providing data about the change".

To get the final pressure I set up a thermodynamic model for the whole process using eqs 33 to 42 and investigated the changes of state permitted by thermodynamic laws, keeping energy constant and the total entropy change of the universe equal to or greater than zero. Surprisingly, I detected a set of conditions allowed by general thermodynamic restrictions where internal entropy is simultaneously created and destroyed in different regions of the universe, and the work required for the nonreversible isothermal compression is less than the value expected from classical thermodynamics. Then I selected at random one of these exceptional changes of state and designed the process shown in Figure 1 of the article. I have found that there is an infinite set of such states and several transformations that permit an internal entropy coupling process.

The model indicates that the total entropy change of the universe is less than zero for temperatures in tank A below 940.14 K, keeping constant other variables. Therefore, the process is not allowed by thermodynamics in this range of temperature. Here, internal entropy destruction in tank A is greater than production of internal entropy by heat transfer.

For temperatures in tank A above 940.14 K the total entropy change of the universe is greater than zero and the process should be allowed by the second law of thermodynamics. It is possible to see that for temperatures between 940.14 K and 4948.20 K the process shows an interesting and unexpected behavior suggesting an internal entropy coupling. In this range, internal entropy is simultaneously produced and destroyed in different regions of the universe, and the nonreversible compression in tank A is more efficient than a reversible isothermal compression. At 940.14 K the production of internal entropy is equal to destruction of internal entropy and efficiency reaches its maximum value. A thermodynamic cycle operating in the region of almost cancellation of internal entropy shows a greater efficiency than an equivalent Carnot cycle working between the same temperature levels (6, 7, J. I. Belandria, unpublished).

For temperatures in tank A greater than 4948.20 K, internal entropy is created in all regions of universe and the system operates according to classical thermodynamics expectations.

From this analysis it seems that in some relatively simple interconnected systems, general thermodynamic restrictions may allow the theoretical existence of a region with simultaneous creation and destruction of internal entropy. Under certain conditions close to equal production and destruction of internal entropy, the process exhibits a superefficiency, as Tykodi expresses. In this transformation, thermal death is retarded or avoided by the internal entropy coupling process, keeping variation of the total entropy of the universe as low as possible.

In some letters Nash, Tykodi, and Olivares and Colmenares comment about entropy destruction. Tykodi opines that "there is never entropy destruction". Nash asks "if we should give credit to entropy destruction" and Olivares and Colmenares follow Prigogine's statement that "we can therefore say that absorption of entropy in one part, compensated by a sufficient production in another part of the system is prohibited".

To this question, irreversible thermodynamics suggests the possibility of simultaneous creation and destruction of internal entropy in relatively complex systems. These transformations have been detected in multireaction and biological systems, thermodiffusion, and active transport of ions, and in thermomechanical closed systems (5­8). In these systems there is at least a process that creates internal entropy coupled to a simultaneous transformation that destroys internal entropy. The destruction of internal entropy is not expected to take place by itself in a single process but can be made to occur by coupling it with another simultaneous process that creates enough internal entropy to compensate internal entropy destruction.

Freeman, Nash, and Olivares and Colmenares discuss the creation of internal entropy across a metallic partition. Freeman says that reference to creation of entropy in the metal partition "smacks of witchcraft" because an object with no mass and no heat capacity cannot create entropy. He explains that there is an increase in entropy as a result of energy flow through the partition, but the increase arises from the difference in temperature on the two sides of the partition. Nash indicates that "the increase in S_univ is a simple consequence of the flow of heat from a hotter body A to a cooler B". And Olivares and Colmenares calculate the creation of internal entropy due to heat flow through the metallic partition, concluding incorrectly that there is no creation of internal entropy in this transition.

Now, in my article, system M is the metal partition of negligible mass surrounded by an imaginary surface representing the boundaries of system M. Surrounding system M are tanks A and B at different initial temperatures. Therefore, there is a heat flow from tank A to B through limits of system M. System M is receiving heat Q_a at a constant temperature T_A and expelling it to a source at a variable temperature T_B without accumulation of energy or entropy. It is evident, as Nash and Freeman say, that there is a creation of entropy that may be attributed to heat flow from A to B arising as a consequence of the difference in temperature on the two sides of the partitions. I would like to say that the entropy balance expressed by eq 27 of the article is based on this consideration and I have assumed for convenience that internal entropy creation is located within the boundaries of system M.

On the other hand, Olivares and Colmenares try to make rigorous demonstrations to estimate the creation or production of internal entropy in the process of Figure 1 and find erroneously that there is no production of internal entropy by heat flow through the partition.

They attempt to evaluate entropy flow through the metal partition using eq 17 of their comments. They did not realize that their procedure evaluated the variation of entropy inside the metal partition rather than the entropy flow generated by heat transfer from tank A to tank B. Next, they concluded that eq 19 represents entropy flow across the metal partition. Indeed, eq 19 takes into account the variation of internal entropy inside the partition and does not represent the entropy flow caused by heat transfer from tank A to tank B through barrier boundaries. This is demonstrated when they calculate the entropy change of the metal partition and obtain eq 20, which is identical to eq 19, confirming my hypothesis. When eqs 19 and 20 are introduced in the entropy balance expression given by eq 21, they obtain zero production of internal entropy. Obviously, Olivares and Colmenares are not correct because everybody knows that internal entropy is produced when heat transfer takes place across a finite temperature difference. Freeman, Nash, and conventional engineering thermodynamic textbooks confirm opinion suggesting that entropy should be produced by the flow of heat from A to B through a metal partition (4, 5).

From this wrong conclusion, Olivares and Colmenares consider that no entropy coupling exists and the process is not allowed by thermodynamic laws because the internal entropy destroyed in tank A is not compensated by production of internal entropy. This is not true, and whatever conception we select to analyze the process of Figure 1, we find that creation of internal entropy should occur as a consequence of heat transfer across a finite temperature difference. Also, when they calculate the total entropy production of the universe with eq 25 they find 5.83 JK^-1. By a simple balance if -2.16 JK^-1 is destroyed in tank A, then 7.99 JK-1 should be created in some other place. According to the discussion, such creation must be attributed to heat flow across the metal partition because nothing else produces entropy in my process. Therefore they do not solve the "paradox" affecting general validity of Prigogine's formulation as they say.

Next, they comment that my eq 27 used to estimate creation of internal entropy is not correct and does not have thermodynamic foundations. To this I would say that such an equation comes from a simple entropy balance in a metal partition and its surroundings. Some thermodynamics textbooks present similar entropy balances in related cases, demonstrating the validity of my calculations (4, 5).

Tykodi comments that for system A to stay isothermal during compression means that the work interaction with the surroundings must be mechanically reversible. Similarly Battino and Wood say that "the only way the compression in A may occur isothermally is via a reversible process". In relation to this, most laboratory and industrial isothermal operations are irreversible. Some conventional textbooks show examples of irreversible isothermal processes invalidating the above argument (9, 10). By controlling compression force and heat transfer we may reach irreversible isothermal conditions without serious difficulties.

Now, I am going to discuss specific comments of each letter. Tykodi points out that my notation could be improved by using IUPAC rules. This may be true, but when I wrote this paper I had in mind engineering convention and I used it for simplicity and customary reasons. In any case, results and consequences of my work are independent of any arbitrary or conventional notation system.

In a third point Tykodi assumes that I treat the uptake of heat by system B as a reversible process. Indeed, when I analyze tank B I do not make any previous assumption about reversibility, but entropy balance suggests that transition in B occurs without production or destruction of internal entropy. Since production of internal entropy is zero in B, then it appears that an event in tank B occurs as if it were reversible as Tykodi thinks.

Tykodi indicates that the only irreversible part of my process is the heat transfer across the finite temperature between systems A and B. This picture is not correct because the process in tank A is also an attainable nonreversible isothermal compression as explained previously.

Then he lists results for an imagined process consisting of a reversible compression in tank A, a reversible heating in tank B, and an irreversible heat transfer between A and B. Such values are correct for his assumption but not for the process described in my article, composed of a nonreversible isothermal compression in tank A, a constant volume heating in tank B, and an irreversible heat transfer between A and B across a metal barrier. His calculation does not consider the internal entropy coupling occurring in the cited process. Tykodi believes that entropy can only be produced and never can be destroyed. This statement is true in all systems described by classical thermodynamics, but my process is an interesting exception of this behavior.

He argues that final state cannot be reached adiabatically from initial state for my selected path. To this consideration I have explained that the process fits general thermodynamic requirements. Therefore, the overall adiabatic path selected is allowed by thermodynamics and may represent an exception of Caratheodory's theorem for adiabatic processes (11).

At the end of his letter Tykodi opines that referees did not review the article well. For me it is difficult to think that referees of a universally known journal did not read my work carefully and critically. I guess that reviewers felt that the article was interesting enough to be published independent of notation and nonconventional ideas that may generate an stimulating discussion about thermodynamic topics.

Now, I will consider Nash's comments. He states in his first paragraph that "the change of state taking place in tank A is nothing but the isothermal compression of an ideal gas from 1 to 4 atm". To this I would like to say that the change of state taking place in tank A is something more than a conventional solitary isothermal compression. Indeed, the process illustrated in Figure 1 of my article represents an internal entropy coupling system in which an oriented heat transfer between tank A and tank B produces enough internal entropy to drive a simultaneous reversible isothermal compression in tank A with destruction of internal entropy. During the process heat is released to tank B, where the temperature varies from 373 to 1500 K and both tanks are covered externally by an adiabatic wall. From this outline it is possible to visualize that the process described in Figure 1 is not equivalent to a conventional isothermal compression as Nash considers.

It is obvious, as he explains, that the minimum work required "by nothing but an isothermal compression at 1500 K from 1 to 4 atm" is -17.3 kJ. This performance corresponds to a conventional reversible isothermal compression releasing heat to a constant-temperature heat reservoir at the same temperature of the system equal to 1500 K. Classical thermodynamics postulates that the above value is the minimum work required for the best isothermal compression system designed by man for the given change of state.

However, if we link an irreversible isothermal compression with a heat transfer between two tanks as described in Figure 1, it is possible to find a feasible set of conditions where the work input is less than the work required by a conventional reversible isothermal compression. In this sense, I have demonstrated that under internal entropy coupling it is possible to design a feasible irreversible isothermal compression with a work input of -14,054 J, which is less than the -17,289 J required by a reversible isothermal compression for the same initial and final states. This result is unexpected from the point of view of classical thermodynamics as Nash claims, but irreversible thermodynamics suggests that this behavior may occur as a consequence of the internal entropy coupling process as explained earlier.

Nash tries to use the process in a closed cycle to conclude that the process described in my article violates the second law of thermodynamics, but he makes a wrong assumption that invalidates his reasoning. He writes "imagine tank A as the cylinder of a Carnot engine in thermal contact with an immense heat reservoir at 1500 K. Beginning at 1 atm, let the author's notional irreversible isothermal compression proceed to 4 atm with work input of -14.05 kJ. Let the gas then resume its original state by a reversible isothermal expansion yielding a work output of 17.3 kJ". Here, Nash is imagining a process different from the one described in Figure 1 of my article. It is possible to see that tank A cannot be used as the cylinder of a Carnot engine in thermal contact with a heat reservoir at 1500 K as Nash thinks. It can be seen that during the process heat is transferred from tank A to tank B, which is a nonisothermal heat sink and its temperature varies from 373 to 1500 K. Also, both tanks A and B are covered externally by an adiabatic wall. Therefore, the cycle imagined by Nash does not fit the requirements of the geometry of the process described in the paper and his cycle does not work. It is evident that the process does not violate the second law of thermodynamics because the total entropy change of the whole universe is greater than zero.

He continues and calculates the entropy change for his conventional reversible compression and finds a value of -9.369 J K^-1 and concludes that the difference between the value given in my article and his value is the entropy destroyed, equivalent to -2.15 J K^-1. He calculates the total entropy change for the conventional isothermal compression using the work input of Figure 1 and a final pressure of 3.086 atm and obtains a value of 7.99 J K^-1 for his compression model, which is different from the system represented in Figure 1, where the total entropy change of the universe is equal to 5.83 J K^-1. Therefore, there are not discrepancies in my conclusion, as he asserts, because all my calculations are correct and consistent with the system expressed in Figure 1; and his results are valid only for his system, which represents a different situation.

Other specific comments appear in Battino and Wood's letter. They assume a reversible compression in tank A and a reversible heating process in tank B and calculate correctly the required pressure and total entropy change of such a system. Then, they estimate the heat transferred from tank A considering a reversible isothermal compression and find 17,289 J. Next they compare this value with 14,054 J taken by tank B and ask "what happens to the excess heat?" Well, the numbers are incompatible because they compare heat intake of tank B with heat release from a reversible isothermal compression in tank A. They should make comparison using the actual heat released by the nonreversible isothermal compression described in the article, which is 14,054 J. In this case, the numbers are compatible.

They said finally that there are no surprises here and no exceptions to the laws of thermodynamics. In relation to this, I have explained my ideas in the beginning of this rebuttal letter.

Olivares and Colmenares start their letter considering that the process is physically unfeasible because according to classical thermodynamics for a reversible isothermal compression in tank A, pressure should be equal to or less than 312.71 kPa. As I explained earlier, the process meets general thermodynamic requirements. Then it should be feasible, being more efficient than a conventional reversible compression. The internal entropy coupling process allows the existence of a path with a final pressure greater than the value expected from simple mechanical arguments, and the system does more work than a reversible process between the same states.

They continue to explain that a total entropy change greater than zero does not imply that a process will in fact occur because it depends on the dynamics of transformation. They suggest as an example that the formation of water from its elements at 25 °C and 1 atm is highly favored thermodynamically; however, it does not take place spontaneously unless a catalyst or a spark initiates reaction. Now, if we analyze intuitively the dynamics of the process described in Figure 1 of my article, it can be deduced that the process starts spontaneously when the adiabatic film on the metal partition is removed. The removal of the adiabatic film is the act or impulse that initiates a spontaneous heat transfer from tank A to tank B because of the temperature difference across the metal barrier. Simultaneously, compression starts at a controlled rate to keep isothermal conditions in A. Then, the system may continue to reach a final state according to model prediction. In any case, experimental evidence would be necessary to verify my hypothesis; but theoretically, the process starts.

Next, they said that I failed to recognize that entropy production is not an additive property as they show in eq A7. Although this may be generally true, for the process it is additive as I will now show. They find in eq 29 a value of 5.83 J K^-1 for total entropy production according to their reasoning. Now, if I assume additivity, the total entropy production for the whole universe will be obtained summing up my eqs 21, 25, and 27 and I get 5.83 J K^-1, which is the same value found by them. Since the values coincide my hypothesis is correct.

Other specific comments appear in Freeman's letter. He considers that the term creation/destruction has a number of connotations and should not be used in a scientific discipline. In this respect I think that any term used to describe a process must have a physical or intuitive feeling to understand better its behavior. I have selected for the article a name that reflects the nature of a process involving simultaneous creation and destruction of internal entropy. The term creation/destruction gives us a stimulating view of some special transformations of nature and describes at different levels the events taking place in the article, where many interpretations are possible.

Freeman states that another puzzling aspect is my claim that the nonreversible compression process is more efficient than a reversible isothermal compression for the same initial and final states. He says this is an invalid comparison because if such a reversible compression were done the heat generated would raise the temperature in tank B above 1500 K or raise the temperature of both chambers, invalidating the posed conditions of the process. To this argument I would say that my intention is to compare the process taking place in tank A in my model with a conventional reversible isothermal compression occurring in a tank surrounded by an isothermal heat reservoir. This tank is not connected to tank B as Freeman thinks. I found that the nonreversible isothermal compression in Figure 1 of my article requires a work input of -14,054 J. Now, if gas is compressed between the same states under isothermal reversible conditions in a tank immersed in an isothermal heat reservoir it would require a work input of -17,289 J. Evidently, the process taking place in tank A is more efficient than the corresponding isothermal reversible compression occurring in a tank releasing heat reversibly to isothermal surroundings. This seems to me a reasonable comparison to measure the efficiency of the process.

He comments that the question to be asked is "given the posed initial conditions, let the gas in A be compressed isothermally and reversibly to a final pressure such that the heat produced in A is exactly that required to produce the given results in B. What is the final pressure?" He finds 312.7 kPa, compared to 405.32 kPa found in the process. Obviously, these results mean that the compression process represented in Figure 1 is more efficient than a reversible isothermal compression. This behavior is a consequence of the internal entropy coupling process previously explained.

Freeman also says that it is not possible to accomplish the stated compression of the gas in A. However, I have demonstrated that the process fits general thermodynamic requirements and should be feasible through the path described.

Finally, Freeman states that there is no merit in this paper other than its being used as a debugging assignment. I would be happy if this paper were used as a debugging assignment, because I feel people will find a new vision of the universe and thermodynamic fundamentals. I consider that this paper has drawn attention from readers around the world generating an interesting discussion about the existence of internal entropy coupling processes and the implications of their extraordinary behavior. The article has interesting aspects and presents a stimulating situation that deserves to be discussed at different levels. I think this is relevant whatever interpretation we assign to it.

Literature Cited

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