Uncompensated heat

The magnitude on the left is the heat absorbed in the isothermal change, and of the two expressions on the right the first is dependent only on the initial and final states, and may be called the compensated heat, whilst the second depends on the path, is always negative, except in the limiting case of reversibility, and may be called the uncompensated heat. From (3) we can derive the necessary and sufficient condition of equilibrium in a system at constant temperature. 

In contrast to Clausius, Gibbs did not discuss uncompensated heat, as he started directly with the total differential of entropy. Gibbs presentation appealed very much to De Donder, However, he wanted to find the meaning of this mysterious uncompensated heat. He considered a system whose physical conditions, such as pressure and temperature, were uniform and which was closed to the flow of matter. Chemical reactions, however, could go on inside the system. De Donder first introduced what he called the degree of advancement, , of the chemical reaction so that the reaction rate v is the time derivative of . 

Now, it is easy to see that under these conditions the uncompensated heat can be written as the differential form... 

As a result, the uncompensated heat must be the same whatever the set of physical variables, pressure and temperature or volume and temperature, is used to describe the state of the system. Indeed, we have... 

Irreversible processes correspond to the time evolution in which the past and the future play different roles. In processes such as heat conduction, diffusion, and chemical reaction there is an arrow of time. As we have seen, the second law postulates the existence of entropy 5, whose time change can be written as a sum of two parts One is the flow of entropy deS and the other is the entropy production dtS, what Clausius called uncompensated heat, ... 

In an advancing irreversible process such as a mechanical movement of a body, dissipation of energy for instance from a mechanical form to a thermal form (frictional heat) takes place. The second law of thermodynamics defines the energy dissipation due to irreversible processes in terms of the creation of entropy Slrr or the creation of uncompensated heat Qirr. 

This equation 3.10 defines the creation of uncompensated heat Qlrr and the creation of entropy Sirr ... 

For a closed system with reversible transfer of heat dQrev where an irreversible process occurs creating uncompensated heat Q, these transferred and created parts of entropy are thus given, respectively, in Eq. 3.13 ... 

As an irreversible process advances in a closed system, the creation of entropy inevitably occurs dissipating a part of the energy of the system in the form of uncompensated heat. The irreversible energy dissipation can be observed, for instance, with the generation of frictional heat in mechanical processes and with the rate-dependent heat generation in chemical reactions different from the reversible heat of reaction. In general, the creation of entropy is always caused by the presence of resistance against the advancement in irreversible processes... 

According to irreversible thermodynamics [Ref. 2.], the rate of the creation of uncompensated heat, which equals the rate of the creation of entropy times the absolute temperature, is equivalent to the driving force A times the rate v = d /dt of the irreversible reaction as shown in Eq. 3.14 (vid. Eq. 3.39) ... 

Equation 3.39 holds valid for the system in which only a single process or reaction is occurring. In a system in which multiple chemical reactions are simultaneously occurring, Eq. 3.27 for the uncompensated heat can be expressed by the sum of the products of all independent affinities and their conjugated reaction rates as given in Eq. 3.40 ... 

No creation of entropy and uncompensated heat occurs in the reversible heat engine and pomp, and hence Eq. 3.45 gives the maximum efficiency theoretically attainable for heat engines and heat pumps. This equation also shows that thermal energy (heat) can not be... 

The affinity of irreversible processes is a thermodynamic function of state related to the creation of entropy and uncompensated heat during the processes. The second law of thermodynamics indicates that all irreversible processes advance in the direction of creating entropy and decreasing affinity. This chapter examines the property affinity in chemical reactions and the relation between the affinity and various other thermodynamic quantities. 

The term (Pgas - Pvlston)dV is the uncompensated heat as named by Clausius. 

Clausius called dQ the uncompensated heat, which is always positive or zero in classical thermodynamics it played a purely qualitative part. It was used to delimit reversible changes for which dQ = 0, and when dealing with non-equilibrium states it was sufficient to write dQ X) without attempting an explicit calculation of its value. 

The entropy created in the system is thus equal to the Clausius uncompensated heat divided by the absolute temperature this gives the uncompensated heat a physical significance. 

Suppose T and p are chosen as physical variables, then since the uncompensated heat is uniquely related to the increase in the chemical variable d, and is independent of the increments dp and dT which occur concurrently, dQ will be the same whatever may be the values of dp and dT during the change under consideration. In other words we need not limit our consideration of affinity to conditions of constant temperature or of constant pressure. The function A defined by (3.21) does not depend upon the kind of transformation considered, but depends solely on the state of the system at a particular instant. In general we may express the state of the system in. terms of physical variables x and y e.g. T, p or T, V. ..) and... 

We now proceed to show that in this system the uncompensated heat is necessarily of the form (3.21). Equation (2.13) may be substituted in (3.5), and differentiated with respect to t to give... 

The results of the preceding paragraphs are readily extended to the case in which several simultaneous chemical reactions occur in the system. De Bonder s inequality gives for the uncompensated heat... 

The loss of work is thus simply equal to the uncompensated heat, and we obtain De Lender s fundamental inequality. Schottky, Ulich and Wagner s method thus leads to the same results as ours, but requires the association of a hypothetical reversible process with each real irreversible change. 

The average affinity of a change which corresponds to one equivalent of reaction, is thus equal to the uncompensated heat of that reaction. 

Let us consider a system in a state P characterized by physical variables x and y (e.g. T and p), together with the extent of reaction p. Suppose a change takes place such that the system moves to a state P characterized by Then the uncompensated heat corresponding to this change (c/. 3.21) is... 

It may happen, however, that at P, all the successive differentials of A with respect to up to the n - 2)th are zero. The uncompensated heat is therefore given according to (15.5) by... 

The discussions of stability given by Gibbs and Duhem are based upon the behaviour of the thermodynamic potentials U, H, F and G. The link between their methods and that adopted here is seen immediately, for we know that for a change at constant T and p, the uncompensated heat is equal to the decrease in free energy G cf. 3.20). Hence a study of the uncompensated heat during a perturbation at constant T and p reduces essentially to a study of the behaviour of G,... 

Consider a process during the course of which the affinity remains less than a given value A. Now let Aq tend to zero. The limiting case defined in this way is called an equilibrium process or equilibrium transformation. The uncompensated heat of this transformation is zero because of (3.21), and so equation (3.5) reduces to... 

We see that the chemical potentials of the individual ions do not appear separately either in the equation for the uncompensated heat, or in the conditions for equilibrium. 

Theophile De Donder showed that this paradox could be resolved elegantly by the explicit calculation of the uncompensated heat, or better of the entropy production, resulting from a chemical reaction. To do this it is necessary to introduce a new function of state, the affinity, characteristic of the reaction and closely related to its irreversibility. In a series of papers since 1920, De Donder has developed a new formulation of chemical thermodynamics by combining the fundamental features of both the Gibbs method and those of the vanT Hoff-Nernst school. 

Here U, is an internal energy of the skeleton, S, -entropy of solid phase, <50 - external stream of heat, SQ >0 - uncompensated heat, JA " -elementary work of the internal surface strength. 

Noticing the importance of relating entropy to irreversible processes, Pierre Duhem (1861-1916) began to develop a formalism. In his extensive and difficult two-volume work, Energetique [8], Duhem explicitly obtained expressions for the entropy produced in processes involving heat conductivity and viscosity [9]. Some of these ideas for calculating the uncompensated heat also appeared in the work of the Polish researcher L. Natanson [10] and the fiennese school led by G. Jaumann [11-13], where the notions of entropy flow and entropy production were developed. 

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