Isothermal changes, reversible

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. 

Then, either no change at all can occur, or all possible changes are reversible. Hence, if we imagine any isothermal change in the state of the system, and calculate the value of Tco for that change, this value will be positive or zero if the former state is an equilibrium state. 

The equation (16) shows that the increase of bound energy in a reversible isothermal change is equal to the increase of entropy multiplied by the absolute temperature, so that the entropy may be regarded as the capacity for bound energy in such changes. B will evidently contain the arbitrary term / IT. 

Our problem is to determine how the changes of total and free energy, AU and A P, or, what are the same, the heat absorption at constant configuration and the maximum work, Qx and At, of an isothermal and reversible process, alter with the temperature of execution of the process. 

If we draw the two isochores imiii, i 2< 2, on the indicator diagram (Fig. 14), all the paths of change must lie within these limits, which fix the range or amplitude of the process. Let the initial state ( i, T) be a, and let the gas expand isothermally and reversibly to a the maximum work AT is represented by the area aa vtf i, which is shown later to be ... 

Now suppose N2 mols of pure liquid [2] are isothermally and reversibly distilled into Ni mols of pure liquid [1]. The change of free energy for distillation of 8N2 mols of [2] into a mixture over which the partial pressure is p2 is, as we have shown ... 

Thus, the reversible work is a limiting maximum value for the magnimde of work obtainable in an isothermal change, with the equality applying to the limit when the process becomes reversible. 

Let us consider a homogeneously, but not hydrostatically, stressed solid which is deformed in the elastic regime and whose structure elements are altogether immobile. If we now isothermally and reversibly add lattice molecules to its different surfaces (with no shear stresses) from the same reservoir, the energy changes are different. This means that the chemical potential of the solid is not single valued, or, in other words, a non-hydrostatically stressed solid with only immobile components does not have a unique measurable chemical potential [J. W. Gibbs (1878)]. 

First a reversible and isothermal process is carried out. Suppose 1 mole of A is transferred isothermally and reversibly from the first vessel to the second. This is done by means of the wall permeable to A. If pA and p A represent the initial and final pressures, respectively, then the change in free energy (AG) is given by,... 

The free energy change in the transfer of 1 mole of C from the box isothermally and reversibly, when the pressure of the gas C is reduced from p c to pc is given by,... 

For a system at constant temperature, this tells us that the work done is less than or equal to the decrease in the Helmholtz free energy. The Helmholtz free energy then measures the maximum work which can be done bv the system in an isothermal change For a process at constant temperature, in which at the same time no mechanical work is done, the right side of Eq. (3.5) is zero, and wo see that in such a process the Helmholtz free energy is constant for a reversible process, but decreases for an irreversible process. The Helmholtz free energy will decrease until the system reaches an equilibrium state, when it will have reached the minimum value consistent with the temperature and with the fact that no external work can be done. 

On the contrary, the absolute temperature T is alwa3rs positive the non-compensated transformation P, equal to 0 for a reversible transformation, is positive for every realizable transformation therefore the quantity of non-compensated heat liberated by any realizable isothermal change is a quantity essentially positive it is equal to Q for a reversible change. 

Each of these two quantities of work possesses naturally the same characteristics as the quantity of heat to which it is equivalent. In particular we may state this fundamental proposition Every real isothermal change engenders positive nonrcompensated work this work is zero for a reversible isothermal modification. 

If we remember that the non-compensated work r, equal to zero for every reversible isothermal change, is positive for eveiy realizable isothermal change, we obtain without dilBSculty the following proposition ... 

The real isothermal change considered is composed of a series of states of the system imagine that by means of external actions suitably chosen, whose nature it is not necessary to specify, we may keep the system in equilibrium in each of these states. The series of these states of equifibrium would form a reversible isothermal modification bringing the system from the same initial state 0 to the same final state 1 as the real modification considered for this reverEdble modification the non-compensated transformation P would have the value 0 if, therefore, we denote by W/ the work done by the external forces which act upon the system during this reversible modification, equation (28) would give, for this change,... 

Suppose, for instance, that for this change the quantity r(So Si) and consequently the quantity Q are positive the isothermal change considered liberates heat. This very slightly intense modification is very nearly reversible that is to say, in conditions very slightly different from those that determine the modifications considered, there is produced a modification in the opposite direction, absorbing about as much heat as the first sets free. 

Similar considerations apply to reversible isothermal changes of volume. These also can only be realised approximately. An isothermal compression or dilatation cannot be completely reversible, imless the pressure which causes the piston to move differs only by an infinitesimal quantity from the pressure in the interior of the gas, for in this case only is the driving force, and hence the velocity of the change, infinitesimally small. If the piston were to move rapidly, we should have in the first... 

On the other hand the reversible work for an isothermal change is obtained by substituting (3.21) in (3.17) with dT — 0 ... 

In the case of non-isothermal changes difficulties arise with Schottky s method. The reversible work done between two given states depends, by virtue of (3.37) on the path taken, and it cannot be defined un-equivocably. ... 

The method developed here is in many ways analogous to that employed by Schottky, Ulich and Wagner. Both methods emphasize the criterion for establishing the irreversibility of a chemical reaction and for deciding whether the reaction will proceed spontaneously in a particular direction. In De Bonder s method this criterion appears immediately the production of entropy must be positive. On the other hand Schottky, Ulich and Wagner employ as the criterion of irreversibility the loss of useful work associated with the real process when compared with a hypothetical reversible process. As is shown in chap. V, these criteria are equivalent for isothermal changes. For non-isothermal changes, however, the concept of loss of useful work... 

COMPRESSION/EXPANSION OF AN IDEAL GAS Consider an ideal gas enclosed in a piston-cylinder arrangement that is maintained at constant temperature in a heat bath. The gas can be compressed (or expanded) reversibly by changing the position of the piston to accomplish a specified change in volume. In Section 12.6, the heat transferred between system and bath when the gas is expanded (or compressed) isothermally and reversibly from volume Vi to Vi is shown to be... 

The discussion given above has referred in particular to isothermal changes, but reversible processes are not necessarily restricted to those taking place at constant temperature. A reversible path may involve a change of temperature, as well as of pressure and volume. It is necessary, however, that the process should take place in such a manner that the system is always in virtual thermodynamic equilibrium. If the system is homogeneous and has a constant composition, two thermodynamic variables, e.g., pressure and volume, will completely describe its state at any point in a reversible process. 

In general, the work that can be obtained in an isothermal change is a maximum when the process is performed in a reversible manner. This is true, for example, in the production of electrical work by means of a voltaic cell. Cells of this type can be made to operate isothermally and reversibly by withdrawing current extremely slowly ( 331) the e.m.f. of a given cell then has virtually its maximum value. On the other hand, if large currents are taken from the cell, so that it functions in an irreversible manner, the E.M.F. is less. Since the electrical work done by the cell is equal to the product of the e.m.f. and the quantity of electricity passing, it is clear that the same extent of chemical reaction in the cell will yield more work in the reversible than in the irreversible operation. 

According to equation (25.24), for an infinitesimal, reversible stage of an isothermal change involving work of expansion only. 

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