Entropy Changes in Irreversible Processes

The change in entropy for temperature changes at constant volume are analogous to those at constant pressure except that Cy replaces Cp. Thus, because PdV = 0, 

the entropy change for the system plus surroundings is zero. Strictly speaking. Equations (6.78), (6.79), and (6.80) are applicable only when no phase changes or chemical reactions occur. 

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. 

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The Caratheodory analysis has shown that a fundamental aspect of the Second Law is that the allowed entropy changes in irreversible adiabatic processes can occur in only one direction. Whether the allowed direction is increasing or decreasing turns out to be inherent in the conventions we adopt for heat and temperature as we will now show. 

We shall now show that mixing is not an irreversible process, and the entropy change, in the process depicted in figure H.l, is not due to the mixing process, but to expansion. Therefore, the reference to the quantity (H.10) or (H.12) as entropy of mixing is inappropriate and should be avoided. 

Concerning the use of Eq. (1.10.2c), in conformity with earlier statements, since Eq. (1.10.2c) relates only to reversible processes, while dS refers to the total entropy change. For irreversible processes Eq. (1.10.2a) or (1.10.2b) must be used. Similar caveats hold for the remaining differential equations for functions of state introduced below. 

Entropy is the Boltzmann constant multiplied by the logarithm of the number of states accessible to the system. In reversible processes an isolated system exhibits no change of entropy while in irreversible processes the entropy must always increase. Entropy contributions to the energy of systems become increasingly important at higher temperatures. 

The increase in the entropy of an irreversible process may be illustrated in the following manner. Considering the spontaneous transfer of a quantity of heat 8q from one part of a system at a temperature T, to another part at a temperature 7, then the net change in the entropy of the system as a whole is then ... 

Fiolitakis, E., Some Aspects on the Entropy Change in Onsa-ger s Sense for Irreversible Chemical Processes, to be published... 

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. 

Calculation of the entropy produced in systems undergoing different flow processes (called irreversible processes) is key for considering steady-state systems. In order to measure the entropy produced in the system, we think of it as transported to the surroundings in a reversible manner and measure the entropy changes in the surroundings. From Eqs. (5) and (7),... 

The concept of affinity introduced in the foregoing chapter (section 3.5) can apply to all the physicochemical changes that occur irreversibly. Let us now discuss the physical meaning of the affinity of chemical reactions. As mentioned in the foregoing, we have in Eq. 3.27 the fundamental inequality in entropy balance of irreversible processes as shown in Eq. 4.1 ... 

Equation (1.126) represents the change of entropy for an irreversible process in an adiabatic system as a function of the internal and external parameters. This may be an important property to quantify the level of irreversibility of a change, and hence yields (i) a starting point to relate the economic implications of irreversibility in real processes, and (ii) an insight into the interference between two processes in a system. 

If we consider the change of local entropy of a system at steady state ds/dt = 0, the local entropy density must remain constant because external and internal parameters do not change with time. However, the divergence of entropy flow does not vanish div J, = . Therefore, the entropy produced at any point of a system must be removed or transferred by a flow of entropy taking place at that point. A steady state cannot be maintained in an adiabatic system, since the entropy produced by irreversible processes cannot be removed because no entropy flow is exchanged with the environment. For an adiabatic system, equilibrium state is the only time-invariant state. 

Here, <55 s 0 is the entropy change arising from irreversible processes occurring within a completely closed system. As Eq. (1.12.9a) shows, S can then only increase. As soon as these processes have ceased, 50 = 55 = 0, so that 5 has assumed an extremal value which is a maximum under the present constraints. For example, the entropy change in the free expansion of a gas can be determined by finding AS under quasistatic conditions, as specified later in Section 2.3. Since 5 is a function of state the same entropy change takes place in a free expansion under the same conditions. All this, of course, merely repeats what has been well established in earlier sections. 

There Is only one way for a system to be In equilibrium with its surroundings. There are many ways to be out-of-equilibrium. Therefore, the field of thermostatics is well-defined and mature, and the field of thermodynamics (or irreversible thermodynamics or whatever latest fashion calls it) is less well-defined and growing. By irreversible thermodynamics, we usually mean the study of processes in which spontaneous change is occurring and, therefore, thermostatic entropy is being created. Since, by definition the system is not in equilibrium, the central question is how to relate the entropy change to physical processes other than the definition in Equation 2, since that definition requires "reversible heat transfer."... 

This is known as the Clausius inequality and has important applications in irreversible processes. For example, dS > (dQ/T) for an irreversible chemical reaction or material exchange in a closed heterogeneous system, because of the extra disorder created in the system. In summary, when we consider a closed system and its surroundings together, if the process is reversible and if any entropy decrease takes place in either the system or in its surroundings, this decrease in entropy should be compensated by an entropy increase in the other part, and the total entropy change is thus zero. However, if the process is irreversible and thus spontaneous, we should apply Clausius inequality and can state that there is a net increase in total entropy. Total entropy change approaches zero when the process approaches reversibility. 

Head loss thus represents useable energy lost from the system and may be thought of as a measure of change in entropy. Likewise, the irreversible process of dissolving minerals in the aquifer system has an entropy change associated with it. One way in which the physical and chemical processes within such a system can be compared is through use of entropy concepts. 

In order to calculate the entropy changes in the two reservoirs after this irreversible process has occurred, we must devise a way of transferring the heat reversibly, since an entropy change can be calculated directly only for a reversible process. We can make use of an ideal gas to carry out the heat transfer process, as shown in Figure 5.56. The gas is contained in a cylinder with a piston. We first place it in the warm reservoir, at temperature Ti, and expand it reversibly and isothermally until it has taken up heat equal to q. The gas is then removed from the hot reservoir, placed in an insulated container, and allowed to expand reversibly and adiabatically until its temperature has fallen to T. Finally, the gas is placed in contact with the colder reservoir at Tc and compressed isothermally until it has given up heat equal to q. 

The first integral is equal to zero since the system was isolated during the irreversible process, so that any heat change in one part of the system is exactly compensated by an equal and opposite change in another part. The second integral is the entropy change when the process B A occurs, so that... 

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