Reversible path

There are now various adiabatic reversible paths because one can choose to vary dF or dF in any combination of steps. The paths can cross and intercoimect. The question of integrability is tied to the question... 

It suffices to carry out one such experiment, such as the expansion or compression of a gas, to establish that there are states inaccessible by adiabatic reversible paths, indeed even by any adiabatic irreversible path. For example, if one takes one mole of N2 gas in a volume of 24 litres at a pressure of 1.00 atm (i.e. at 25 °C), there is no combination of adiabatic reversible paths that can bring the system to a final state with the same volume and a different temperature. A higher temperature (on the ideal-gas scale Oj ) can be reached by an adiabatic irreversible path, e.g. by doing electrical work on the system, but a state with the same volume and a lower temperature Oj is inaccessible by any adiabatic path. 

Equation (A2.1.15) involves only state fiinctions, so it applies to any infinitesimal change in state whether the actual process is reversible or not (although, as equation (A2.1.14) suggests, dS is not experimentally accessible unless some reversible path exists). 

For if AMB, ANB are any two such reversible paths (Fig. 12), these taken together constitute a reversible cycle AMBN, for which... 

Just as the intrinsic energy of a body is defined only up to an arbitrary constant, so also the entropy of the body cannot, from the considerations of pure thermodynamics, be specified in absolute amount. We therefore select any convenient arbitrary standard state a, in which the entropy is taken as zero, and estimate the entropy in another state /3 as follows The change of entropy being the same along all reversible paths linking the states a and /3, and equal to the difference of the entropies of the two states, we may imagine the process conducted in the following two steps ... 

Since these processes are reversible and isothermal, Q and (Q + SQ) depend only on the initial and final states, and are the same for different isothermal reversible paths (Moutier s theorem, 86). 

In addition, we have established that there is a sense of direction to the location of the inaccessible states. State 2, the state reached from 1 by a reversible adiabatic path, represents the division between the states on the second isotherm that are accessible and inaccessible from state 1. We represent this schematically in Figure 2.1 lb, where the reversible adiabatic path separates states that are accessible from state 1 from those that are inaccessible. The observation that the reversible path serves as the boundary between the two sets of states will be useful later when we show the direction of allowed processes in terms of the sign of A5(universe). 

The change may be brought about in an infinite variety of ways. Consider for this problem the following two reversible paths (I and II), each consisting of two parts (1 and 2) ... 

These equations can be used to derive the four fundamental equations of Gibbs and then the 50,000,000 equations alluded to in Chapter 1 that relate p, T, V, U, S, H, A, and G. We should keep in mind that these equations apply to a reversible process involving pressure-volume work only. This limitation does not restrict their usefulness, however. Since all of the thermodynamic variables are state functions, calculation of AZ (Z is any of these variables) by a reversible path between two states gives the same value as would be obtained for all other paths between those states. When other forms of work are involved, additions can be made to the equations to account for the additional work. The... 

FIGURE 6.16 (a) On the reversible path, the work done in Example 6.5 is relatively large (w = -2.12 kj) because the change in internal energy is zero, heal flows in to maintain constant temperature and constant internal energy. Therefore, q =... 

Because entropy is a state function, the change in entropy of a system is independent of the path between its initial and final states. This independence means that, if we want to calculate the entropy difference between a pair of states joined by an irreversible path, we can look for a reversible path between the same two states and then use Eq. 1 for that path. For example, suppose an ideal gas undergoes free (irreversible) expansion at constant temperature. To calculate the change in entropy, we allow the gas to undergo reversible, isothermal expansion between the same initial and final volumes, calculate the heat absorbed in this process, and use it in Eq.l. Because entropy is a state function, the change in entropy calculated for this reversible path is also the change in entropy for the free expansion between the same two states. 

To calculate a change in entropy for a process we find a reversible path between the initial and final states. It is immaterial whether the actual process is irreversible or reversible. Because entropy is a state function, the change for that path will be the same as that for the irreversible path. 

STRATEGY Because the entropy is a state function, we can calculate the change in entropy by choosing a reversible path that results in the same final state. In this case, consider the following ... 

Note that on the right-hand side, the work W is that of the reversed path [i.e., has the opposite sign see (5.33)]. Equation (5.50) has to be solved numerically (e.g., by using a Newton-Raphson solver) for the free-energy difference A A. 

Note that the funnel requirement is attenuated to the degree that the path approaches reversibility. Actually for a reversible path, the phase spaces of successive states along the path will be almost identical (cf. Fig. 6.1b). 

Figure 6.6 is a representation of two possible reversible paths for reaching State b from State a. We have just shown that over a reversible closed path, the entropy... 

Figure 6.6. Two reversible paths, acb and adb, from state a to state b.

Notice that the order of limits is reversed in the right equality. Equation (6.67) emphasizes that the entropy change is the same for all arbitrary reversible paths from a to b. Thus, the entropy change AS is a function only of State a and State b of the system. 

The definition of entropy requires that information about a reversible path be available to calculate an entropy change. To obtain the change of entropy in an irreversible process, it is necessary to discover a reversible path between the same initial and final states. As S is a state function, AS is the same for the irreversible as for the reversible process. 

To determine the entropy change in this irreversible adiabatic process, it is necessary to find a reversible path from a to b. An infinite number of reversible paths are possible, and two are illustrated by the dashed lines in Figure 6.7. 

As V > Vfl, the entropy change for the gas is clearly positive for the reversible path and, therefore, also for the irreversible change. 

Figure 6.8. Schematic diagram of general irreversible change. The dashed line represents one possible reversible path between State a and State b.

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