The system we are considering (Figure 1) is a gas contained in a cylinder permeable to heat (i.e. diathermal), and equipped with a mobile frictionless piston, which is also diathermal. In state 1, the external pressure P_e and the internal pressure P_i (pressure of the gas) have the same value P₁. Then, due to the deposition of a mass of sand on the piston, P_e passes from the value P₁ to a value P₂, and the piston moves down with the consequence that, in state 2, P_i also reaches the value P₂.

The sand can be deposited very rapidly (i.e. all at once) or very slowly (i.e. in several successive small batches).

In the first case, P_e reaches immediately the value P₂, while P_i reaches this value only when the piston stops. It is well known that this situation is designated irreversibility in the sense that P_e and P_i are always different during the process. In such conditions, the work done on the gas by the piston, for an elementary change in volume noted dV, must be written

d W i r r = − P e d V (1)

In the second case, by contrast, the pressure exerted by the piston passes very gradually from the initial value P₁ to the final value P₂, with the consequence that, at each moment of the process, P_e and P_i can be considered as equal. This situation is designated reversibility and, in such conditions, the elementary work done on the gas, by the piston, is given by the relation:

d W r e v = − P i d V (2)

For a given value of dV, the difference between dW_irr and dW_rev takes the form:

d W i r r − d W r e v = d V ( P i − P e ) (3)

that can also be written:

d W i r r = d W r e v + d V ( P i − P e ) (4)

Referring to Figure 1, it appears that we have dV > 0 when P_i > P_e and dV < 0 when P_i < P_e. (this second situation being the one illustrated by the diagram). As a consequence, the term d V ( P i − P e ) obeys in all cases the condition:

d V ( P i − P e ) > 0 (5)

meaning itself that we have always:

d W i r r > d W r e v (6)

Another possible expression of Equation (4) is therefore:

d W i r r = d W r e v + d W a d d (7)

where dW_add can be regarded as an additional energy representing the difference between dW_irr and dW_rev and obeying the condition:

d W a d d > 0 (8)

In practice, it is well known that the additional energy, noted here dW_add, mainly appears in the form of heat, classically designated dQ, that is evacuated towards the surroundings. This heat evacuation explains that the final temperature of the gas has in fact the same value as its initial temperature, that is also the temperature of the surroundings.

For the matter that will be discussed in this paper, the information we need about the first law is summarized in Equations (9), (10), (11), (12), and (13) given below.

The first law of thermodynamics is a postulate which consists in admitting that when a system passes from a determined state 1 to a determined state 2, its internal energy variation, noted dU, always obeys the relation:

d U i r r = d U r e v (9)

If the energy exchanges are limited to work (W) and heat (Q), the detailed meaning of Equation (9) takes the form:

d W i r r + d Q i r r = d W r e v + d Q r e v (10)

If the system taken in consideration is isolated, the complementary precision given by the first law is that Equations (9) and (10) become respectively:

d U i r r = d U r e v = 0 (11)

d W i r r + d Q i r r = d W r e v + d Q r e v = 0 (12)

Knowing that an isolated system has no exchanges of energy with its surroundings, all the terms of Equations (11) and (12) refer to energy exchanges occurring inside the system.

We have seen with Equation (6) that dW_irr and dW_rev obey the general relation:

d W i r r > d W r e v (R.6)

Combining this data with the information given by Equations (9) and (10), leads to the conclusion that the terms dQ_irr and dQ_rev must themselves obey the relation:

d Q r e v > d Q i r r (13)

In thermodynamic textbooks, the considerations concerning dQ are generally discussed in the chapters dealing with the second law. We will now enter in this context in order to see if the relation dQ_irr < dQ_rev given by Equation (13) can really be verified.

As was the case for the first law, the concepts of reversibility and irreversibility remain of fundamental importance for the understanding of the second law, but they are associated here with the complementary concept called entropy. Its presentation is often done in the following condensed form.

In conditions of reversibility, a change in entropy (noted dS) is linked to a change in heat (dQ) and to the absolute temperature (T) by the relation:

d S = d Q / T (14)

In conditions of irreversibility the corresponding relation becomes:

d S > d Q / T (15)

When we discover these equations for the first time and receive the complementary information that dS has the same value in both cases (because S is a state function, as well as V, the volume), we are tempted to think that dQ represents dQ_rev in Equation (14) and dQ_irr in Equation (15). With this idea in mind, we are led to deduce that these terms obey the relation:

d Q i r r < d Q r e v (R. 13)

which is the one suggested by Equation (13), at the end of the previous section.

At this stage of the presentation, the reasoning seems coherent and easily understandable. The problem is that the relation dQ_irr < dQ_rev is not confirmed when we examine the numerical applications presented as examples of the use of these equations (one of them, very simple, is mentioned below in paragraph 2.2).

The reason is due to the fact that, in this context, we discover that the precise meaning of Equation (14) is really:

d S = d Q r e v / T i (16)

but the precise meaning of Equation (15) is interpreted as being:

d S = d Q r e v / T e + d S i (17)

complemented by the information that the term dS_i, designated as the internal component of entropy, has always a positive value and therefore can always be written:

d S i > 0 (18)

A related piece of information is that the term dS = dQ/T_e is designated dS_e and called the external component of entropy, so that a condensed writing of Equation (17) is:

d S = d S e + d S i (19)

Although the term dQ of Equation (15) is sometimes designated dQ_irr in thermodynamics textbooks, the numerical value given to dQ in Equation (17) is really dQ_rev, as will be seen further with Equation (28) that refers to a typical numerical example.

The conclusion which follows from this observation is that the term dQ_irr is not present in the equations going from number 14 to number 19, so that its comparison with the term dQ_rev is not possible at the moment.

The situation was different for the comparison between the terms dW_irr and dW_rev introduced in paragraph 1 because their definitions, given by Equations (1) and (2), were directly presented in the form of energy equations.

To obtain the conversion of the entropy equations (16) and (17) into energy equations, the solution consists in multiplying both terms of Equation (16) by T_i, and the three terms of Equation (17) by T_e. The results are:

For Equation (16):

d Q r e v = T i d S (20)

For Equation (17)

T e d S = d Q r e v + T e d S i (21)

whose meaning is

d Q i r r = d Q r e v + T e d S i (22)

In the same manner as, in Equation (7), we have designated dW_add the term representing the difference dW_irr – dW_rev, we can here adopt the designation dQ_add for the term T_edS_i, so that an alternative writing of Equation (22) is:

d Q i r r = d Q r e v + d Q a d d (23)

Having noted with Equation (17) that dS_i is always positive and knowing that the same is true for T_i and T_e (because, in thermodynamics, temperatures are always expressed in Kelvin), the term T_edS_i, also noted dQ_add is itself positive. Consequently, it can be derived from Equation (23) that dQ_irr and dQ_rev are linked together, not by the relation dQ_irr < dQ_rev presented above as the expected one (Equation (13)), but by the inverted relation, that is:

If we accept the proposal dQ_irr > dQ_rev given by Equation (24) and at the same time the proposal dW_irr > dW_rev given by Equation (6), their addition leads to the proposal:

which disagree with the proposal presented in Equation (10) as the conventional understanding of the first law of thermodynamics.

The situation is similar if the addition of Equations (24) and (6) is entered in Equations (11) and (12), because the result obtained would mean that an isolated system can be concerned by a positive variation of its internal energy (dU_irr > 0).

Despite the fact that such a proposal appears impossible in the conventional approach of thermodynamics, it becomes easily understandable if the mass-energy relation is taken into account. Two reasons may explain why it has not been done previously. The first one is that at the time of the discovery of the laws of thermodynamics, the theory of relativity was not known. The second is that even without the help of the mass-energy relation, the efficiency of the thermodynamic tool is indisputable, so that no urgency was felt to revise the interpretation of its laws. Nevertheless, it seems that the theory gets more easily understandable when this complement is added and perhaps the scope of its applications may be extended.

For an easy approach of the matter, let us examine the situation below, presented in a basic thermodynamic course as an example of use of the concept of entropy. The system is defined as 1000 g of water (1 liter) that are heated from 20˚C (293 K) to 80˚C (353 K), by contact with a thermostat at 80˚C (353 K), under atmospheric pressure. The asked question concerns the determination of the terms ΔQ, ΔS, ΔS_e and ΔS_i, knowing that the specific heat capacity of water is cp_water = 4,184 JK⁻¹∙g⁻¹ that can be admitted constant over the temperature range.

The answers, equally given in the course, are as follows:

So far, these results do not raise ambiguity whether the hypothesis to which we refer is the classical interpretation or the new suggested one. The divergence begins when the asked question is in knowing whether the term ΔQ, mentioned in Equation (28) means ΔQ_irr or ΔQ_rev. The interest of the question comes from the fact that in the classical approach, the relation between both terms is the inequality dQ_irr < dQ_rev given by Equation (13), while in the new suggested hypothesis, it is the inequality dQ_irr > dQ_rev given by Equation (24).

In the first hypothesis, the question is considered useless for the example presently examined, because the formulation of the first law being dU_irr = dU_rev, it reduces to dQ_irr = dQ_rev when the energy exchange is limited to heat. This is practically the case here, since the work due to change in volume of the water heated from 20˚C to 80˚C can be considered negligible, compared to the change in heat, so that we can write dW = 0.

The situation is not the same in the second hypothesis because the conversion of the Entropy Equation (17) into the Energy Equation (21) introduces a real difference between dU_irr and dU_rev and consequently between ΔQ_irr and ΔQ_rev although the change in volume is always considered negligible.

To know the exact meaning of the term noted ΔQ, the criterion to take into account is that its value is independent of the degree of irreversibility of the heating process when it represents ΔQ_rev, while it is not the case when it represents ΔQ_irr. To make a test using this criterion, we may imagine a division of the heating process in two steps noted A and B, the first one from 293 K to 223 K, the second from 223 K to 253 K. Reproducing the calculation presented above and adding the obtained results, we are led to the following table:

Through these results, it can be effectively observed that the division of the heating process in two steps has no effect on the values of ΔQ and ΔS, but increases the value of ΔS_e (change from 711 to 744 JK⁻¹) and decreases that of ΔS_i (change from 69 to 36 JK⁻¹). If the number of steps is multiplied, we easily conceive that the value ΔS_i tends progressively towards zero, meaning that the heating process tends towards reversibility. The question asked above is therefore answered in the sense that ΔQ represents here ΔQ_rev and the corresponding term ΔQ_irr can be calculated by Equation (22), whose differential and integrated formulations are respectively:

Applying Equation (34) to the considered system leads to the following observations:

When the heating process is done in a single step, we have:

When it is done in two steps, we have:

Being lower than the previous one, this last value concerns effectively the term ΔQ_irr which decreases progressively and tends towards ΔQ_rev (itself remaining constant) when the number of steps is multiplied. Thanks to this difference of behavior, the problem of the identification of ΔQ_irr and ΔQ_rev is solved.

The reason why this procedure is not used in thermodynamic courses is evidently that the conclusion ΔQ_irr > ΔQ_rev to which it leads it is not in accordance with the usual understanding of the first law. As mentioned above, this last one implies the general equality ΔU_irr = ΔU_rev which becomes ΔQ_irr = ΔQ_rev when the energy exchange is limited to heat, as it is the case here. Referring to this situation, it can be observed that the strict obedience to the conventional understanding of the first law reduces the information given by the second law, because it does not allow the conversion of the entropy Equation (17) to the energy Equation (21). In close relation with this remark, some additional suggestions are discussed below.

If we group together Equations (20), (21) and (22) which are respectively:

we see that the terms dQ_irr and dQ_rev can be directly defined by the energy formulations:

The entropy being a state function, as the volume, the differential dS has the same value in both equations. Therefore we can write:

i.e.

The terms T_e and T_i being positive (absolute temperatures), we also see from Equations (37) and (38) that dS and dQ have the same sign. The term dQ being positive when T_e > T_i and negative when T_e < T_i, the same is true for dS. Consequently, the term is always positive, confirming the relation:

As already done for dW_irr, with Equation (7), we can give to this last equation the expression:

where the term dQ_add is positive.

Then adding the relations (Equation (41)) and, (Equation (7)), we get the global result:

whose meaning is:

Let us imagine now, the case of an isolated system made of two parts 1 and 2 separated by a diathermic wall. Supposing that part 1 is defined as 1 liter of water at 20˚C and part 2 as 1 liter of water at 80˚C, our objective is the analysis of their exchange of energy.

A first idea that comes to mind is that, just after the end of this exchange, both parts will be at the same final temperature T_f = 50˚C, (= 323 K). If a thermometer is placed in each of them, this result can be verified directly, but can also be predicted by the well-known equation:

which gives T_f = 323 K (taking into account that C₁ = m₁c₁ and C₂ = m₂c₂).

Designating T_i the initial temperature and T_f the final temperature, the main elements of the calculation are given below.

For the terms ΔQ_rev1, ΔQ_rev2 and ΔQ_{rev syst}, we get:

For the terms ΔS₁, ΔS₂ and ΔS_syst:

For the terms and representing the average temperatures of part 1 and part 2:

For the terms ΔQ_irr1, ΔQ_irr2 and ΔQ_{irr syst}:

In the conventional approach of the theory, the only results taken in consideration for the whole system are ΔQ_syst = 0 (Equation (47)), where ΔQ_syst means indifferently ΔQ_{rev syst} or ΔQ_{irr syst} and ΔS_syst = 36 JK⁻¹ (Equation (50)).

In the revised approach that is proposed here, we see that a distinction is made between the term ΔQ_{rev syst} = 0, and the term ΔQ_{irr syst} > 0, whose calculated value is ΔQ_{irr syst} = 23328 J.

The reader accustomed to the usual approach of the theory may be tempted to consider this last result as irrealist, because it seems impossible that an energy can be created inside an isolated system. Before examining below the possibility of a link between thermodynamics and relativity, the attention is called to the following remark, briefly evoked in the abstract.

Taking into account that thermodynamics is among the most synthetic theories of physics and the relation E = mc² among the most general equations of science, we may be astonished that this latter is not explicitly present in the laws of thermodynamics. Perhaps it is hidden behind our habit of giving the first law the meaning dU_irr = dU_rev instead of dU_irr > dU_rev? It is on the basis of this idea that the considerations discussed in the following paragraph are based.