Volume reversible/irreversible processes

We shall suppose the solute to be a mol of an ideal gas, occupying a volume v at the pressure o and the solvent a volume Y of Iig. 56. liquid just sufficient to dissolve all the gas under the pressure j)o- If the gas is brought directly into contact with the liquid, an irreversible process of solution occurs, but if it is first of all expanded to a very large volume, the dissolution may be made reversible, except for the first trace of gas entering the... 

For an irreversible adiabatic expansion in which some work is performed, the work performed is less in magnimde than that in the reversible process because the external pressure is less than the pressure of the gas by a finite amount. Thus, if the final volume is the same as in the reversible process, the final temperature will not be as low in the irreversible process, because, according to Equation (5.47), the temperature drop depends directly on the work performed by the expanding gas. Similarly, from Equations (5.42) and (5.44), AC7 and A//, respectively, also must be numerically smaller in the intermediate expansion than in the reversible expansion. In the adiabatic expansion, from a common set of initial conditions to the same final volume, the values of Af7 and A//, as well as the values of the work performed, seem to depend on the path (see summary in Table 5.2). At first glance, such behavior seems to contradict the assumption that U and H are state functions. Careful consideration shows that the difference occurs because the endpoints of the three paths are different. Even though the final volume can be made the same, the final temperature depends on whether the expansion is free, reversible, or intermediate (Table 5.2). 

Figure 13. Correlation of gas evolution on a graphite electrode in 1.0 M LiC104/PC/EC (50 50) with the irreversible process at 0.80 V during the first discharge. Note the level off of gas volume as soon as reversible lithium ion intercalation starts. (Reproduced with permission from ref 261 (Figure 2). Copyright 1993 The Electrochemical Soci-ety).

Following a similar approach, Shu et al. used an FC/PC mixture instead of neat PC as electrolyte solvent, and their analysis of propylene gas volume corroborates the observations of Arakawa and Ya-maki. Furthermore, because FC was present in their electrolyte, the reversible lithium intercalation could occur after a long plateau at 0.8 V (representing PC decomposition), therefore a correlation between the gas volume and this irreversible process was able to be established, as shown in Figure 13. Considering Aurbach s spectroscopic observations (to be discussed later), a modified mechanism (see Scheme 10) was proposed by Shu et al., wherein a competition exists between the surface reaction leading to radical anions and the formation of ternary... 

Consider a simple system where the only work is a volume expansion against an external pressure (Fig. 2.2). In this case, for either a reversible or irreversible process, it can be shown that... 

The maximum work the system can do occurs when dP -> 0. Wi ax " P0 y When the system does the maximum work, in other words, the system undergoes a reversible process, then from the first law of thermodynamics AU = q - w = qr - wmax or qr = AU + wmax q, is the maximum amount of heat which the system can absorb from the surroundings (heat reservoir) for the vaporisation of 1 mole of water. If the pressure drop, dP, is a finite amount, i.e., dP 0, in other words, the system undergoes an irreversible process, then the system does less work for the same volume expansion w = (Po-dP)P < hw Heat transferred from the surroundings to the system is q = AU + w... 

The varying P is substituted by nRT/V and the integration then performed over the changing volume. wrev is the work done by the gas in expanding reversibly from (Vf Pi) to (Vf, Pf). It can be equated to the area enclosed between the curve of P plotted versus V and the V axis (i.e. the abscissa) (see Frame 2). wKV is larger than the work done, Win, during the irreversible process of expansion and it also represents the maximum work obtainable from any expansion process which takes the gas from the state (Vi, Pi) to the state (Vf, Pf). 

For a vstem at constant pressure and temperature, we see that the Gibbs free energy is constant for a reversible process but decreases for an irreversible process, reaching aminimum value consistent with the pressure and temperature for the equilibrium state just as for a system at constant volume the Helmholtz free energy is constant for a reversible process but decreases for an irreversible process. As with A, we can get the equation of state and specific heat from the derivatives of <7, in equilibrium. We have... 

The present volume involves several alterations in the presentation of thermodynamic topics covered in the previous editions. Obviously, it is not a trivial exercise to present in a novel fashion any material that covers a period of more than 160 years. However, as best as I can determine the treatment of irreversible phenomena in Sections 1.13, 1.14, and 1.20 appears not to be widely known. Following much indecision, and with encouragement by the editors, I have dropped the various exercises requiring numerical evaluation of formulae developed in the text. After much thought I have also relegated the Caratheodory formulation of the Second Law of Thermodynamics (and a derivation of the Debye-Hiickel equation) as a separate chapter to the end of the book. This permitted me to concentrate on a simpler exposition that directly links entropy to the reversible transfer of heat. It also provides a neat parallelism with the First Law that directly connects energy to work performance in an adiabatic process. A more careful discussion of the basic mechanism that forces electrochemical phenomena has been provided. I have also added material on the effects of curved interfaces and self assembly, and presented a more systematic formulation of the basics of irreversible processes. A discussion of critical phenomena is now included as a separate chapter. Lastly, the treatment of binary solutions has been expanded to deal with asymmetric properties of such systems. 

Consider now an irreversible process in a closed system wherein no heat transfer occurs. Such a process is represented on the P V diagram of Fig. 5.6, which shows an irreversible, adiabatic expansion of 1 mol of fluid from an initial equilibrium state at point A to a final equilibrium state at pointB. Now suppose the fluid is restored to its initial state by a reversible process consisting of two steps first, the reversible, adiabatic (constant-entropy) compression of tile fluid to tile initial pressure, and second, a reversible, constant-pressure step that restores tile initial volume. If tlie initial process results in an entropy change of tlie fluid, tlien tliere must be heat transfer during tlie reversible, constant-P second step such tliat ... 

The final term, representing the rate of entropy generation Sq, reflects the second-law requirement that it be positive for irreversible processes. There are two sources of irreversibility (a) those within the control volume, i.e., internal irreversibilities,and (b) those resultingfrom heat transfer across finite temperature differences between system and surroundings, i.e., external thermal irreversibilities. In the limiting case where Sq = 0, the process must be completely reversible, implying ... 

The change in volume of a gas again illustrates the difference between reversible and irreversible processes. The adiabatic compression of a gas (see p. 91) is reversible, as the initial state may be re-estabhshed completely by an adiabatic expansion. In practice, however, it is impossible to construct vessels absolutely impermeable to heat. No actual compression is therefore strictly adiabatic, as some of the heat produced is always lost by conduction or radiation to the surroundings. The less the permeability of the walls of the vessel, the smaller this loss in heat will be, and the more nearly will the change in volume approximate to a reversible process. 

We have already noted that work done is not a state function this is also true of the mechanical work of expansion. The derivation above has shown that the work is related to the process carried out rather than to the initial and final states. We can consider the reversible expansion of a gas from volume Vy to volume V2, and can also consider an irreversible process, in which case less work will be done by the system. 

Clearly, the second process just described is a reversible process, while the first is irreversible. There is another important characteristic of reversible and irreversible processes. In the irreversible process just described, a single mass is placed on the piston, the stops are released, and the piston shoots up and settles in the final position. As this occurs the internal equilibrium of the gas is completely upset, convection currents are set up, and the temperature fluctuates. A finite length of time is required for the gas to equilibrate under the new set of conditions. A similar situation prevails in the irreversible compression. This behavior contrasts with the reversible expansion in which at each stage the opposing pressure differs only infinitesimally from the equilibrium pressure in the system, and the volume increases only infinitesimally. In the reversible process the internal equilibrium of the gas is disturbed only infinitesimally and in the limit not at all. Therefore, at any stage in a reversible transformation, the system does not depart from equilibrium by more than an infinitesimal amount. 

Important is that equilibrium states passed during such reversible processes are stable. Such a reversible process must be slow because the time scale (for volume) in the usual (irreversible) processes is much less than the relaxation time— practically finite time of achieving an equilibrium state during a change of V from the old (perturbed) value to the new equilibrium value during the process in B. These results... 

Equation (3.2.18) is the first form of the fundamental equation for open systems. In the case of a reversible change, each term in (3.2.18) has a simple physical interpretation (T dS) is the heat crossing system boundaries (-PdV) is the work that alters the system volume and (dL//dN,)dN, is related to the work that causes component i to diffuse across system boundaries. For irreversible processes no such simple interpretations apply nevertheless, since the Ihs is an exact differential, (3.2.18) is valid regardless of whether a change of state is reversible. In a similar fashion we can extend each of the other forms of the fundamental equation to open systems. The results are... 

The rate of entropy production cannot be negative however, the changes in entropy of the system may be positive, negative, or zero. For a reversible process, the entropy production becomes zero when the process is internally reversible as well as the heat transfer between the control volume and its surroundings are reversible. The entropy of an isolated system during an irreversible process always increases, which is called the increase of entropy principle. The energy (power) dissipated because of irreversibility would be... 

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