Energy reversible adiabatic processes

Theorem. From any state of a system, the maximum energy that can be extracted adiabatically in a CCP process is the work done in a reversible adiabatic process that ends in a stable equilibrium state. Moreover, the energy change of a system starting from a given state and ending at a stable equilibrium state is the same for all reversible adiabatic CCP processes. We call this energy the adiabatic availability. 

Solution (a) Using the steam tables. From the steam tables we find Sa = 7.8356 kJ/kg K. The reversible adiabatic process is isentropic therefore, Sb = Sa = 7.8356 kJ/kg K. In the final state we know pressure (20 bar) and entropy therefore, the state is fully specified. The temperature and internal energy are obtained by interpolation at Pb = 20 bar, Sb = 7.8356 kJ/kg K and the results are summarized in the table below ... 

If there is work along any infinitesimal path element of the irreversible adiabatic process (dw 0), we know from experience that this work would be different if the work coordinate or coordinates were changing at a different rate, because energy dissipation from internal friction would be different. In the limit of infinite slowness, with the same change of work coordinates and the process remaining adiabatic, the internal friction would vanish, the process would become reversible, and the net work and final internal energy would differ from those of the irreversible process. Because the final state of the reversible adiabatic process is different from B, there is no reversible adiabatic path between states A and B. 

Consider now a reversible adiabatic process across the boundaries of a system. Since in the present case energy is an analytic function of Fand of the n we may use Eq. (1.7.4) to write (the subscript S here denotes an adiabatic process)... 

A spontaneous change occurs naturally (by itself) under specified conditions, without an ongoing input of energy from outside the system. In particular, a chemical reaction proceeding toward equilibrium is an example of a spontaneous change. For a thermodynamically reversible adiabatic process a quantitative statement of the second law can be formulated as [9] ... 

In a reversible, adiabatic process SQrev = TdS = 0. This condition, therefore, corresponds to a process with constant S ("isentropic ). Thus, the differential we are looking for is ( )g. In the Maxwell relations derived in eqn. (p), it is seen that the differential can be expressed by the natural variables (5, V) of the internal energy U... 

Again, for a reversible, adiabatic process the entropy of the system remains constant. Likewise, the entropy changes of the surroundings and universe are zero. We see that the results for reversible, adiabatic compression are identical to the results presented in Table 3.1 for reversible, adiabatic expansion. An energy balance gives ... 

Flere the subscripts and/refer to the initial and final states of the system and the work is defined as the work perfomied on the system (the opposite sign convention—with as work done by the system on the surroundings—is also in connnon use). Note that a cyclic process (one in which the system is returned to its initial state) is not introduced as will be seen later, a cyclic adiabatic process is possible only if every step is reversible. Equation (A2.1.9), i.e. the mtroduction of t/ as a state fiinction, is an expression of the law of conservation of energy. 

Theorem.—A process yields the maximum amount of available energy when it is conducted reversibly.—Proof. If the change is isothermal, this is a consequence of Moutier s theorem, for the system could be brought back to the initial state by a reversible process, and, by the second law, no work must be obtained in the whole cycle. If it is non-isothermal, we may suppose it to be constructed of a very large number of very small isothermal and adiabatic processes, which may be combined with another corresponding set of perfectlyJ reversible isothermal and adiabatic processes, so that a complete cycle is formed out of a very large number of infinitesimal Carnot s cycles (Fig 11). 

The adiabatic expansion of a gas is an example of (b). In the reversible adiabatic expansion of one mole of an ideal monatomic gas, initially at 298.15 K, from a volume of 25 dm3 to a final volume of 50 dm3, 2343 J of energy are added into the surroundings from the work done in the expansion. Since no heat can be exchanged (in an adiabatic process, q = 0), the internal energy of the gas must decrease by 2343 J. As a result, the temperature of the gas falls to 188 K. 

In Figure 4.4 for example, the direct reaction from R to P would be a non-adiabatic process. Although there is no simple and general answer to this question, most primary photochemical reactions can be considered to be adiabatic when the primary photoproduct (PPP) retains a large part of the excitation energy. In some cases this is fairly obvious, when the photoproduct is formed in an excited state for instance in a reversible proton transfer reaction (see section 4.3). 

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. 

In this equation, P represents, strictly speaking, the external pressure. However, if the adiabatic process is carried out reversibly, the actual pressure of the system is virtually identical with the external pressure ( 8a), so that under these conditions P is the pressure of the system. For an idecd gas, dE may be replaced by CydT, as shown by equation (9.22), since the energy content is independent of the volume, so that... 

Assume that the system used to carry out the Carnot cycle is an amount of ideal gas contained in a cylinder fitted with a frictionless piston. The concept of an ideal gas is introduced in Section 1.2. Of consequence at this point is the premise that for an ideal gas the internal energy, U, is a function of temperature only. The Carnot cycle consists of reversible isothermal and adiabatic processes. An isothermal process is one in which the system temperature is kept constant. An adiabatic process requires that no heat be transferred between the system and its surroundings. The steps are as follows ... 

Phase changes under isothermal conditions and those in flowing fluids show a fundamental difference. In the first case the latent heat of evaporation has to be transferred between the system and the environment. Many flow processes, however, are adiabatic and some of them are almost reversible. In such adiabatic flows the latent heat must be provided from the internal energy of the fluid. The ratio of internal energy and latent heat which depends mostly on the molar specific heat of the substance characterizes the extent of phase change attainable in adiabatic processes. Obviously, adiabatic phase changes can take place much faster than isothermal phase changes. 

To reduce the energy required for a distillation process with almost pure water as overhead vapor, a heat-pump arrangement is proposed. The column is operating at atmospheric pressure. The reboiler needs a vapor condensing at 150 °C. The overhead water vapor should be compressed by a reversible adiabatic compression process such that it can be tbe condensing vapor in tbe reboiler. Determine the heat delivered by the condensing vapor per pound of the vapor calculate the ratio of the heat delivered in the reboiler to the work required in the reversible compressor. Use tbe steam tables from Smith and Van Ness (1975) or any other suitable source. (Ans. 1020 Btu/fb (567 cal/g) 7.254.)... 

More recently, D. Emin [24] developed an alternative analysis of activated hopping by introducing the concept of coincidence. The tunneling of an electron from one site to the next occurs when the energy state of the second site coincides with that of the first one. Such a coincidence is insured by the thermal deformations of the lattice. By comparing the lifetime of such a coincidence and the electron transit time, one can identify two classes of hopping processes. If the coincidence lime is much laigcr than the transit lime, the jump is adiabatic the electron has lime to follow the lattice deformations. In the reverse case, the jump is non-adia-batic. 

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