The second law of thermodynamics is one of the most highly respected concepts in the pantheon of Science. Despite many attempts to overthrow or subvert it, there has, as yet, been no documented, verifiable, repeatable example of that having been accomplished. That fact does not prove that the second law can never be broken, but it does send the clear message that anyone who proposes a scheme for doing so should expect thorough dissection of the scheme and should be prepared to supply full information about the details of the scheme.

José Belandria (JB) has failed to provide those detailsboth in his original paper (1) and in his response (2) to his critics (3). Various thermodynamic deficiences in his paper (1) have been described (3) but JB has offered essentially nothing in his reply (2) that helps to resolve the issues.

In the interest of brevity, consensus, and closure, the Editor has asked me to prepare, on behalf of all the critics (3) a reply to JB's response (2). In doing so, I have attempted to find an approach that focuses on the critical point and that elicits additional imput from Belandria. Note that since JB has explicitly challenged the second law, logically one can not appeal to the second law to show that JB is wrong.

All of the critics question how JB arrived at the indicated final pressure in tank A (1). In response (2) he says: "To get the final pressure I set up a thermodynamic model for the whole process using eqs 33 to 42 and investigated". However, if one examines eqs 33 to 42, one finds that the final pressure in tank A is fixed at 405.32 kPa, and there is no indication of how that pressure was obtained.

Since the final pressure in A is a crucial point in the criticism, it is useful to examine JB's "thermodynamic model" in some detail; for reasons disclosed later, it is convenient to refer to this example as the "connected tanks" case. In the reconstruction of his eqs 33-42 below, I have used parameters slightly different from those given in JB's paper, so as to minimize confusion between the two calculations. I assume exactly the same arrangement and process as that described by JB (1), with these two changes: (i) the initial temperatures are 1400 and 400 K in A and B, respectively, and (ii) the initial pressure is 1 bar (100 kPa) in both. Note that the final temperatures are fixed at 1400 K by the specifications of the process, and that the final pressure in B is readily obtained from the known increase in temperature at constant volume. The only parameter not fixed by the initial temperatures and pressures and by the process specifications is the final pressure in A. With these new parameters, JB's eqs 33­42 become:

Delta U_A= 0 ; [A is isothermal] (33b)

(34b)

(35b)

(36b)

w_B = 0 [isochoric] (37b)

w_A = Q_A [Delta U_A = 0] (38b)

(39b)

(40b)

(41b)

(42b)

A comment is required with regard to eq 42b. Given JB's concept of entropy creation/destruction, his eq 19 for Delta S_gA is logically consistent and eq 21 follows; however, his eq 27 for Delta S_gM is logically inconsistent, and is arthmetically incompatible with eq 29. If in his eq 27 a minus sign is inserted before each integral sign, these difficulties are eliminated. JB has acknowledged the problem with his eq 27 (see below) and attributes it to a "transcription error"a typo; eq 42b has the needed minus signs before the integral signs. Note also that 42b is included in the list of equations above only to provide 1:1 comparison with JB's eqs 33­42; Olivares and Colmenares (3) provide a detailed discussion of the errors in JB's eq 42.

In eqs 33b­42b, numerical values have been obtained for all thermodynamic quantities except Delta S_A and those which depend on it. Delta S_A, in turn, depends on the final pressure in A. The analysis is straightforward to this point, but it can not be completed without a value for P_Af. How can one get that value? From what is given above, one can not. To obtain the correct value for the final pressure in A one must (i) measure it, or (ii) specify the compression process in sufficient detail that the final pressure may be determined. JB's specification (1) of the compression process as "isothermally in a nonreversible way" is not sufficient.

However, one can easily place some limits on the final pressure in A. If the compression is assumed to be reversible, PAf may be calculated from:

w_A = -RT ln(P_Af/P_Ai) (1)

The result is 2.92 bar, and Delta S_A is -8.91 J/K, whence Delta S_gA in eq 40b is zero - as expected for a reversible isothermal process. One might also ask what value of P_Af makes Delta S_u equal to zero. From eq 39b, with Delta S_u set to 0, one obtains P_Af = 6.55 bar (4); for higher values, Delta S_u is negative. For any assumed value of P_Af greater than 2.92 bar and less than 6.55 bar, Delta S_u will be positive, Delta S_gA will be negative ("entropy is destroyed") and Delta S_gM will be positive ("entropy is created"); that is, JB's criteria are met. But which of these - if any (4) - is the correct value of P_Af?

In early December, I wrote to Belandria, presented the reconstructed version of his example [i.e., the material in the paragraph surrounding (33b - 42b) and half of the following paragraph], and solicited his answer to these questions: (i) What is the value of P_Af in eq 39b? and (ii) How did you arrive at your value? In his reply, Belandria commented on the transcription error in his eq 27 (see above), then presented some calculations that essentially duplicate those in the preceding paragraph (which was not included in my letter to him), and then provided these comments:

"According to this [JB's] view, there is an infinite set of final pressures between 2.92 and 6.55 bar where internal entropy coupling exists and the compression process is more efficient than a reversible compression for the same change of state. In this region total entropy change is greater than zero and energy is conserved, and [the] process describes trajectories not predicted by classical thermodynamics or Caratheodory's theorem. Finally, to answer your question there is an infinite set of PAf values that satisfy eqs 33b - 42b according to the above description.

In relation to the process depicted in [JB's original paper (1)] I detected that internal entropy coupling occurs in a range of final pressures between 312.75 and 811.09 kPa. In the article I selected one final-pressure-value of this set, equal to 405.32 kPa, and designed the system published by the Journal."

These comments provide the first indication from JB about how he obtained the final pressure 405.32 kPa in his example (1). It seems clear from these comments that JB believes that any value of P_Af is legitimate if his criteria are satisfied, that is, if Delta S_u is positive, Delta S_gA is negative ("entropy is destroyed"), and Delta S_gM is positive ("entropy is created"). Parenthetically, if JB is correct, one wonders how Nature "knows" which of the "infinite set of P_Af values" to chooseor is the value of P_Af a matter of chance?

In subsequent correspondence with JB, I posed another example, the "separated tanks" case. The scenario is very similar to the "connected tanks" case, above, and the initial conditions are the same. However, now tank A and tank B are separated, and partition M is replaced by M_A in the end of tank A, and M_B in the end of tank B. Suppose that these two tanks are immersed in an isothermal "thermal reservoir" or "heat bath" at 1400 K, and are widely separated in that bath. To execute the process, manipulate M_B to permit heat to flow from the reservoir into tank B until the temperature in B is 1400 K; "close" M_B. Then manipulate M_A to permit heat flow from A to the reservoir, and compress A until Q_A = -Q_B; "close" M_A.

For this "separated" example, the changes in tank B are the same as in eqs 33b-41b, and the heat bath has undergone no change (it lost 12.47 kJ to B, but the same amount was restored from A). Further, Q_A and w_A are the same. In fact, all of eqs 33b-41b apply to this new example.

My question to JB was, in this "separated tanks" case, what is the final pressure in tank A, and how did you obtain the answer? JB agrees (letter dated March 11) that eqs 33b-41b apply to this case, and states: "For the specified process there is not a temperature gradient between tank A and the surrounding heat reservoir, and to carry out the compression process under such a condition requires that heat should be released from tank A to the heat reservoir in differential amounts to avoid finite temperature gradients [i.e., the process must be carried out in a way], similar to a conventional isothermal compression. For this condition, Delta S_gA is zero and the final pressure for the separated tank A is 292 kPa [2.92 bar]." He also states that "'isothermally' means that the temperature is constant or almost constant throughout the process". He does not elaborate on the meaning of "almost" in this context (e.g., ± 0.1 or ± 1.0 or ± 10.0 or ± ?? kelvin). Note that JB's value for the final pressure in A is the same as that given above, following eq 1, for a reversible, isothermal compression.

In the absence of explicit details, one must analyze JB's claims from the available evidence; even with two papers (1, 2) and two or three letters available, it is still difficult to be sure about the basis for JB's statements; this lack of precision and detail has been a major problem from the beginning (1). The preceding paragraph seems to imply rather strongly that, in JB's view, the final pressure in tank A is different for the two casesseparated vs. connected tanksbecause of a difference in the temperature gradient between tank A and its surroundings. Clearly, JB is correct in saying that there is no (or an infinitesimal) gradient in the separated-tanks case. However, the implication that there is a gradient in the connected-tanks case contradicts the stated specifications (1), namely, the compression of the gas in A occurs isothermally. In the connected-tanks case, the gases in the two tanks are at different temperatures; there must be a temperature gradient somewhere. But if the gas in A is compressed isothermally, as JB specifies, there can be no gradient within the gas, and the surface of the partition M must be at the temperature of the gas. Hence, the gradient must lie totally within the partition M and can have no effect on the final pressure of the gas. Further, in an isothermal compression, the path of the state of the gas, as represented on a P - V plot, is - obviously - along an isotherm; as Battino and Wood have noted (3), this is precisely the same path followed by the system (gas A) during a reversible (and isothermal) compression; that is, isothermal compressions and expansions are inherently reversible. Therefore, JB's specification (1) that the compression of A occur "isothermally in a nonreversible way" is self contradictory.

If the gas in A is compressed isothermally, as specified, then the final pressure in A is the same for both cases, connected or separated. JB's value for the final pressure in A for the connected case (1, his Dec. 13 letter) is based on at least two contradictions: (i) temperature gradients during an isothermal process; and (ii) an isothermal compression occurring in a "nonreversible way". He also ignores the fact that, independent of his "criteria" about "entropy creation and destruction, there are constraints on the possible final pressure in Acontraints that can not be hand-waved away with appeals to "entropy coupling"; Tykodi (3) and Olivares and Colmenares (3) have given simple, mechanical, nonthermodynamic arguments that show JB's value for the final pressure in A (1) to be impossible. Finally, Olivares and Colmenares (3) have argued persuasively that JB's application (entropy "creation and destruction") of Prigogine's formulation of "entropy production" is not valid.

Until Belandria provides a legitimate scheme for confirming his value of P_Af, the status of his "exceptional theoretical process" must be placed somewhere between "simply wrong" and "unproved and highly unlikely to be proved"; the critics believe very strongly that the proper description is "simply wrong".

Finally, on behalf of all the critics (3), who are all experienced thermodynamicists, I express our concern about the review process that resulted in publication of JB's original paper (1). Our unanimous opinion is that it should not be been publishedcertainly not in its present form. An acceptable alternative form might have been foundfor example, "Provacative Opinion"­type, but even that is highly questionable.

Literature Cited

1. Belandria, J. I. J. Chem. Educ. 1995, 72, 116­188.

2. Belandria, J. I., J. Chem. Educ. 1997, 74, 286.

3. Batino, R.; Wood, S. E.; Freeman, R. D.; Nash, L. K.; Olivares, W.; Colemenares, P. J.; Tykodi, R. J. J. Chem. Educ. 1997, 74, 256, 281­286.

4. Calculation of the value P_Af = 6.55 bar when Delta S_u = 0 has been done solely to clarify what JB has done. Note that in the calculation it has been implicitly assumed that all other parameters remain the same as in the reversible isothermal compression case (P_Af = 2.92 bar; eqs 33b-41b and 1). Clearly, this can not be, for it violates the first law. Compression of the gas in A to a pressure higher than 2.92 bar requires additional work; therefore, w_A increases, which means the -Q_A must increase (not stay the same), which means that Q_B must increase, which means that the final temperature in B must be higher than 1400 K (which also violates the specifications of the process).