Reversible Processes in Closed Systems

Thermodynamic analysis in the laboratory usually relates to reversible processes in closed systems that are at or close to equilibrium. The contrast with living systems is great These are open systems that exchange matter and energy with their surronndings, and they are not at equilibrium. 

Entropy Changes of Isothermal Reversible Processes in Closed Systems... 

Figure 2.8 Schematic plot of states accessible via adiabatic processes in closed systems. From the initial state 1 only states on the line can be reached by reversible adiabatic volume changes. States above the reversible adiabat can be reached only by processes that include irreversible adiabatic volume changes. States below the reversible adiabat cannot be reached by any adiabatic volume change.

Consider a closed system of one chemical component (e.g., a pure substance) in a single homogeneous phase. The only kind of work is expansion work, with V as the work variable. This kind of system has two independent variables (Sec. 2.4.3). During a reversible process in this system, the heat is = T dS, the work is dm = —pdV, and an infinitesimal internal energy change is given by... 

The differential equations above are valid for reversible processes taking place in closed systems in which there is no flow of matter among the various phases. 

The thermodynamic analysis of an actual Otto cycle is complicated. To simplify the analysis, we consider an ideal Otto cycle composed entirely of internally reversible processes. In the Otto cycle analysis, a closed piston-cylinder assembly is used as a control mass system. 

One particular pattern of behaviour which can be shown by systems far from equilibrium and with which we will be much concerned is that of oscillations. Some preliminary comments about the thermodynamics of oscillatory processes can be made and are particularly important. In closed systems, the only concentrations which vary in an oscillatory way are those of the intermediates there is generally a monotonic decrease in reactant concentrations and a monotonic, but not necessarily smooth, increase in those of the products. The free energy even of oscillatory systems decreases continuously during the course of the reaction AG does not oscillate. Nor are there specific individual reactions which proceed forwards at some stages and backwards at others in fact our simplest models will comprise reactions in which the reverse reactions are neglected completely. 

Reversible processes are those processes that take place under conditions of equilibrium that is, the forces operating within the system are balanced. Therefore, the thermodynamics associated with reversible processes are closely related to equilibrium conditions. In this chapter we investigate those conditions that must be satisfied when a system is in equilibrium. In particular, we are interested in the relations that must exist between the various thermodynamic functions for both phase and chemical equilibrium. We are also interested in the conditions that must be satisfied when a system is stable. 

This equation, combining the first and second laws, is derived for the special case of a reversible process. However, it contains only properties of the system. Properties depend on state alone, and not on the kind of process that produces the state. Therefore, Eq. (6.1) is not restricted in application to reversible processes. However, the restrictions placed on the nature of the system cannot be relaxed. Thus Eq. (6.1) applies to any process in a system of constant mass that results in a differential change from one equilibrium state to another. The system may consist of a single phase (a homogeneous system), or it may be made up of several phases (a heterogeneous system) it may be chemically inert, or it may undergo chemical reaction. The only requirements are that the system be closed and that the change occur between equilibrium states. 

For an ideal gas, we can obtain the change of entropy in terms of volume (or pressure), temperature, and the number of moles. For a unit mass of fluid undergoing a mechanically reversible process in a closed system, the first law yields... 

The auraferraborane clusters of System 7 show a marked dependence on the identity of the MPhj ligand. The As complex is easier to reduce than the corresponding P complex and the reduction product is more stable (tj/j 0.1 s for the As complex, and tj/2 0.01 s for the P complex ). The major difference, however, is found for the oxidations while the P complex shows two well-defined electrochemically and chemically quasi-reversible processes in CV, the As complex only gives one well-defined oxidation peak at a potential ca 0.3 V higher than F ,i for the P complex (i.e. close to F ,2 for the P complex). A difference of ca 0.3 V in the redox potential upon a change from PPhj to AsPhj seems very unlikely based on the other data in Table 18, and the explanation of the observation is probably that a small pre-peak in the CV of the As complex (at a potential close to F ,i for the P complex) is in fact the first oxidation of the As complex. The unexpected small size may be due to a reorganization process prior to electron transfer, which is much slower for the As than for the P complex. 

The two preceding equations for an ideal gas undergoing a reversible process in a closed system take several forms through elimination of one of the variables P,V,oiT hy Eq. (3.13). Thus, with P = RT/V they become ... 

Moreover, Q = AH for mechanically reversible, constant-pressure,closed-system processes [Eq. (2.23)] and for the transfer of heat in steady-flow exchangers where AEp and AEj are negligible and Wg = 0. hr eitlrer case. 

For a reversible process, where the system is always infinitesmally close to equilibrium, the equality in Eq. (1.3) is satisfied. The resulting equation is known as the fundamental equation of thermodynamics... 

The familiar notion [37,38] of a close to equilibrium process differs from our concept of a nearly reversible process. For close to equilibrium processes, the initial and final equilibrium states E and Ep must have nearly identical A values. Thus, for such processes ot is always small, and Eq. (A.36) is guaranteed to be valid. In contrast for nearly reversible processes, the A values for states T and Tp need not be close. Instead, for such processes the actual system states Tp- (r) during the process must differ only slightly from the equilibrium states Tp- with the same A values. Thus for nearly reversible processes a does not have to be small, and Eq. (A.36) may or may not be valid.)... 

Since the differential entropy change S and the heat flow Q for a reversible process in a closed system are related as... 

The functions A and G are called potentials because allow the calculation of the maximum work only from Initial and final states. Consider a reversible process in a closed system. The maximum work can be produced only by a reversible transformation. Eqs (5.15) and (5.26), can be combined to give the following relation ... 

The first two partial derivatives in Eq. (6-39) can be determined from the equations for closed systems, since the composition is held constant. For an infinitesimal reversible process in which no mass is transferred, we obtain from Eq. (6-18)... 

It was also found with non-ionic surfactants that flocculation of the system occurred either at or very close to the cloud point (the lower consolute boundary) of the surfactant. Moreover, this was found to be a reversible process in that on cooling below the cloud point the particles redispersed provided that the temperature was not taken too far above the cloud point (c. 5-10°C) [109]. 

For isothermal processes on closed systems, the reversible work is given by the change in Helmholtz energy, as already noted in (3.2.15). 

Steam undergoes the following reversible process in a closed system from initial conditions 10 bar, 400 °C, to 550 °C under constant pressure, then to 8 bar under constant volume. Determine the energy balances. How would the energy balances change if steam from the same initial state were first cooled at constant volume to 8 bar, then heated at constant pressure to the same final state ... 

Imagine an ideal gas confined to a 1-L flask at 1 atm pressure, as shown in Figure 19.5 . The flask is connected by a closed stopcock to another 1-L flask, which is evacuated. Now suppose foe stopcock is opened while keeping foe system at a constant temperature. The gas will spontaneously expand into foe second flask until the pressure is 0.5 atm in both flasks. During this constant-temperature (isothermal) expansion into a vacuum, foe gas does no work (w = 0). Furthermore, because foe energy of an ideal gas depends only on temperature, which is constant during the process, AE = 0 for foe expansion. Nevertheless, foe process is spontaneous. The reverse process, in which the gas that is evenly distributed between the two flasks spontaneously moves entirely into one of foe flasks, is inconceivable. This reverse process, however, would also involve no heat... 

A ffien a spontaneous process with a reversible limit is proceeding slowly enough for its states to closely approximate those of the reversible process, a small change in forces exerted on the system by the surroundings or in the external temperature at the boundary can change the process to one whose states approximate the sequence of states of the reverse process. In other words, it takes only a small change in external conditions at the boundary, or in an external field, to reverse the direction of the process. 

It should now be apparent that a satisfactory formula for defining the entropy change of a reversible process in a closed system is... 

A real process approaches a reversible process in the limit of infinite slowness. For all practical purposes, therefore, we may apply Eq. 5.2.2 to a process obeying the conditions of validity and taking place so slowly that the temperature and pressure remain essentially uniform—that is, for a process in which the system stays very close to thermal and mechanical equilibrium. 

Combination of these equations using the fact that T cannot be negative gives an important relation for dV for reversible processes in a simple closed system ... 

It is found experimentally that the stretching of a mbber object approximately obeys three properties (1) the volume remains constant (2) the tension force is proportional to the absolute temperature and (3) the energy is independent of the length at constant temperature. An ideal rubber exactly conforms to these three properties. Since the volume is constant, the first term on the right-hand side of Eq. (28.9-1) vanishes for an ideal rubber. For reversible processes in a closed system made of ideal mbber, the first and second laws of thermodynamics give the relation ... 

While the above statements are intuitively satisfying, for calculational purposes we need a mathematical statement. Clausius provided that by defining a new quantity, which he named entropy. It is defined as a property of some mass of matter that, for reversible processes in a closed system, obeys Eq. 2.14 ... 

Notice that the system should be closed because in an open systan entropy can behave in any way, i.e., it can decrease, increase or remain constant In closed systems entropy lanains constant only in reversible processes. In irreversible processes entropy always increases. 

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