In this note we would like to solve the "paradox" and to show that a careful analysis of Belandria's proposal demonstrates that such a process is not feasible and that no violation of Prigogine's formulation exists. Furthermore, we show the usefulness of Prigogine's formulation and clarify some relevant aspects of such formulation.

The Physical Unfeasibility of the Process

The process in discussion is well described in ref 1. Briefly, it consists in the heating of a constant-volume system B, consisting of one mole of an ideal gas initially at 373 K and 101.33 kPa, which ends up in a final state of 1500 K and 407.49 kPa. The heat is provided by the irreversible isothermal compression of one mole of an ideal gas, system A, from an initial pressure of 101.33 kPa to a final pressure of 405.32 kPa. The two systems are coupled through a dividing metal partition M of negligible mass.

In order to attain the equilibrium temperature of 1500 K, keeping its volume constant, system B should absorb a heat

(1)

The heat released from system A through the partition M is then Q_A = -Q_B = -14,053.69 J. Since the compression of the ideal gas in A is isothermal, the work that has to be done on it is w_A = -14,053.69 J. This work can be provided in one, two, or infinite steps. When carried out in one irreversible step, an external pressure P_ex has to be applied to a piston acting on system A. The work w_A is then

(2)

If we require this work to be -14,053.69 J and the final pressure of system A to be 405.32 kPa, as proposed by Belandria, a contradiction is reached, since then

(3)

For simple mechanical reasons it is impossible to compress a gas up to 405.32 kPa by applying an opposing pressure as low as 152.25 kPa. In fact, the higher pressure to which system A can be compressed in one step is obtained by letting the external pressure be equal to the final pressure P_ex = P_A2. Then

(4)

from which the maximum final pressure would be P_A2 = P_ex = 215.52 kPa instead of 405.32 kPa.

The other extreme would be to carry out the compression in infinite steps, that is, reversibly. Then

(5)

from which P_A2 = 312.71 kPa.

No matter how the gas in A is compressed isothermally, to destroy a work of 14,053.69 J, the final pressure should be in the interval between 215.52 kPa and 312.71 kPa. Therefore the final pressure of 405.32 kPa used by Belandria (1) is unattainable on simple mechanical, nonthermodynamic arguments. Thus, the proposed process is not physically feasible. If one compresses the gas reversibly to the final pressure of 405.32 kPa, then the needed work would be, from eq 5,w_A = -17,288.47 J. From a thermodynamical point of view, the system would have done (destroyed) more work than in a reversible process between the same states!

Entropy Considerations

In the frame of the classical formulation of the second law, the proposed process involves two coupled changes: the irreversible isothermal compression of system A and the heat transfer from the hot reservoir A to the constant-volume system B. The entropy changes corresponding to the initial and final states of ref 1 are simply

(6)

(7)

The massless metal partition M has a negligible heat capacity and therefore Delta S_M = 0. The entropy change of the Universe is then

(8)

That is, the entropy increase due to the heat absorption in system B dominates the entire process giving Delta S_U > 0, even though the final pressure in system A is unphysical. This is not a contradiction, since the second law says that a spontaneous process will always give Delta S_U > 0, and conversely, that a physically or chemically possible process with Delta S_U > 0 is spontaneous, but it does not imply or guarantee that if Delta S_U > 0 the process will in fact occur. The second law does not give information on the dynamics of physicochemical feasibility of a given process.

A classic example is the production of H₂O(l) from its elements H₂ (g) and O₂ (g) at 1 atm and 25 °C. Thermodynamically that reaction is highly favored because Delta S_U = Delta G_f /T > 0, but it is well known that the reaction does not occur spontaneously unless a catalyst or a spark initiates the reaction. Coupled processes, where a process not favored thermodynamically is coupled with one highly favored to make the first one to occur, are very important and well known: for example, in the metabolism of living beings. But, in those cases, the unfavored process must be physically or chemically possible.

In Prigogine's formulation the total entropy change of a given system is written in terms of the internal entropy production Delta _iS and the entropy flow Delta _eS:

(9)

where Delta _eS is defined as

(10)

and

(11)

where dQ is the actual heat transfer to the system, T its temperature, and dQ_rev is the heat transfer through a reversible process between the same initial and final states.

The second law becomes

(12)

This is nothing else but the Clausius inequality. For an isothermal process, this is equivalent to Q_rev Q and hence w_rev w. This says that no internal entropy destruction is possible in any region of space. In fact, Prigogine clearly states 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" (2).

The evaluation of the entropy creation in systems A and B follows as in eqs 16 to 25 of ref 1; that is,

(13)

(14)

(15)

(16)

However, the evaluation of Delta _iS for system M, the infinitesimal metal partition in ref 1, is erroneous. Its evaluation should be carried out carefully. By definition,

(17)

Here we have stressed that the temperature inside the integral is that of the metal M. It is straightforward to realize that if the partition's mass is negligible, heat capacity is also negligible and no heat would be absorbed, leading to Delta _eS_M = 0. Another way to look at this is to consider a system M with finite mass m_M and specific heat capacity c_M, and then let the mass tend to zero. We can separate the process occurring in M into two steps. The metal first absorbs heat -Q_A from system A, increasing its temperature from T_M1 to T'M. It then releases the heat completely to system B, decreasing its temperature to the final temparature T_M2. In that case

-Q_A = m_Mc_M (T_M2 - T_M1) + Q_B (18)

(19)

the total entropy change of the metal heated from T_M1 to T_M2 is

(20)

and the internal entropy creation is null

Delta _iS_M = Delta S_M - Delta _eS_M = 0 (21)

In the limit of m_M -> 0 all the entropy terms are negligible and Delta S_M = Delta _eS_M = Delta _iS_M = 0, as expected. Belandria fails to get this result because he uses, instead of eq 19,

(22)

an expression that has no thermodynamic foundation. Therefore, no entropy coupling is actually detected; the internal entropy destroyed in system A is not compensated by systems B and M. As we mentioned above, Prigogine's formulation clearly states that the process in A, with Delta _iS_A = -2.16 J K^-1, is not possible. This result is in agreement with our previous discussion where we showed that in fact the proposed final state for system A is unattainable on simple mechanical grounds.

Belandria's process is not exceptional; instead, it illustrates the beauty of Prigogine's local formulation: while the global classic formulation gives Delta S_U > 0 allowing the possibility of the coupled process, Prigogine's local formulation give Delta _iS_B = Delta _iS_M = 0, since the heat transference is done reversibly; but Delta _iS_A < 0, which says that the proposed irreversible compression is not possible!

To close this section, it is worth mentioning that if we used a correct mechanical value for P_A2 of 215.52 kPa for compression in only one step, the Delta _iS_A = 3.09 JK^-1 > 0. In fact, any value in the interval of 215.52 kPa to 312.71 kPa would give Delta _iS_A 0, as it should be.

Entropy Production of the Entire System

We would like to stress that, while entropy is an extensive property and therefore the entropy increase of the Universe is the sum of the entropy changes for each system, namely

(23)

the so-called entropy production is not an additive property. The detailed demonstration of this is given in the appendix. In fact, for the process under discussion, since there are no heat flows into the entire system

Delta _eS_U = 0 (24)

and the entropy production of the entire system is just equal to the entropy change

Delta _iS_U = Delta S_U = 5.83 JK^-1 > 0 (25)

while the sum of the entropy production of each system in the process in discussion is negative; that is,

(26)

If we apply the results of eq A7 (appendix) to evaluate Delta _iS:

(27)

where

(28)

Using this result in eq 28, we get

Delta _iS = (-2.16 + 7.99) JK^-1 = 5.83 JK^-1 (29)

which is identical to Delta S, as expected, because the entire system is isolated and Delta _eS = 0 with Delta S_U = Delta S = Delta _iS.

Belandria (1) failed to recognize that the entropy production is not an additive property, as we show in the appendix. In order to obtain the expected result in eq 29, he was then forced to use eq 22 for Delta _eS_M, which corresponds numerically to the right hand side of eq 28. This led that author to make the massless partition responsible for most of the entropy production of the entire system, a result at first sight striking.

Conclusions

We have carefully analyzed the apparently exceptional process proposed by Belandra (1) and solved the "paradox" that would have led to an irreversible process more efficient than its corresponding reversible process. We pointed out that the process in tank A, destroying entropy (i.e., Delta _iS_A = - 2.16 JK^-1), is not physically feasible, since the opposed external pressure needed to compress a gas must be higher than its internal pressure. Secondly, we have recalculated the internal entropy production in the metal partition to obtain Delta _iS_M = 0. Therefore, actually, no internal entropy coupling occurs in Belandria's process.

Finally, we have shown that entropy production is not an additive quantity; that is, the entropy production of an entire system is not simply the sum of the entropy production of its parts. Instead, as given in the appendix, an additional term due to the heat exchange among the subsystems must be added.

In conclusion, the theoretical process proposed by Belandria is very illustrative of the usefulness of Prigogine's formulation of the second law. While the global classic formulation gives Delta S_U > 0 for the coupled process in discussion, Prigogine's local formulation says that processes occurring in systems M and B with Delta _iS = 0 are reversible, and that the compression proposed for system A with Delta _iS_A < 0 is not feasible.

Appendix

Here we will show how to calculate the entropy production of an entire complex system. Let us consider a complex system composed of N coupled subsystems. For convenience, let us split the heat transferred to system, dQ_n into two parts

(A1)

where d_eQ_n is the heat transferred to subsystem n from the outside surrounding the entire system and dQ_nm is the heat transferred to subsystem n from its surrounding neighbor subsystem m. The sum is over all the rest of subsystems. Clearly, by conservation of energy,

dQ_nm =-dQ_mn

By definition, the entropy production of a given subsystem n is

(A2)

where dS_n is the infinitesimal entropy change of subsystem n and T_n its temperature.

Similarly, the entropy production of the entire complex system is

d_iS = dS - d_eS (A3)

where dS is the entropy change of the entire system, given by

(A4)

since the entropy is an extensive property.

Using the notation introduced in eq A1, the external entropy change d_eS is, by definition, given as

(A5)

Summing over all subsystems in both sides of eq A2 and using eqs A1 and A4

(A6)

Then using eqs A3 and A5 and integrating over all processes occurring, we have that the net entropy production of the entire system, Delta _iS, is

(A7)

This result is physically meaningful. It says that the net entropy production should be obtained by adding not only the internal entropy productions of the individual susbystems, but also the entropy fluxes among them.

Equation A7 is quite general and not well known. The second term on the right-hand side vanishes only when the temperature of the entire system is uniform and T_n = T for all subsystems. Therefore, in general

(A8)

That is, the entropy production is not an additive property.

Acknowledgment

We would like to thank Professor Belandria for making a copy of his manuscript available to us and for kindly discussing his proposed process.

Literature Cited

1. Belandria, J. I. J. Chem. Educ. 1995, 72, 116-118.

2. Prigogine, I. Thermodynamics of Irreversible Processes; Interscience: New York, 1997.