Reversible Processes in Ideal Gases

This section demonstrates calculations of changes in macroscopic properties caused during several specific reversible processes in ideal gases. These will serve as auxiliary calculation pathways for evaluating changes in state functions during irreversible processes. We use this procedure extensively in Chapter 13 on spontaneous processes and the second law of thermodynamics. 

Recall from Section 12.1 that a true reversible process is an idealization it is a process in which the system proceeds with infinitesimal speed through a series of equilibrium states. The external pressure therefore, can never differ by more than an infinitesimal amount from the pressure, P, of the gas itself. The heat, work, energy, and enthalpy changes for ideal gases at constant volume (called isochoric processes) and at constant pressure (isobaric processes) have already been considered. This section examines isothermal (constant temperature) and adiabatic (q = 0) processes. 

An isothermal process is one conducted at constant temperature. This is accomplished by placing the system in a large reservoir (bath) at fixed temperature and allowing heat to be transferred as required between system and reservoir. The reservoir is large enough that its temperature is almost unchanged by this heat transfer. In Section 12.4, U for an ideal gas was shown to depend only on temperature therefore, AU = 0 for any isothermal ideal gas process. From the first law it follows that 

In a reversible process, Pe t = Pgas = but the relation w = -Pext from Section 12.2 cannot be used to calculate the work, because that expression applies only if the external pressure remains constant as the volume changes. In the reversible isothermal expansion of an ideal gas. 

The work in the complete process is the sum of the areas of the rectangles in the figure. As the step size dV is made smaller, it approaches the infinitesimal 

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The equations developed in this section have been derived for mechanically reversible nonflow processes involving ideal gases. However, those equations which relate state functions only are valid for ideal gases regardless of the process and apply equally to reversible and irreversible flow and nonflow processes, because changes in state functions depend only on the initial and Anal states of the system. On the other hand, an equation for Q or W is specific to the case considered in its derivation. 

Figure 11.3 Reversible mixing process for ideal gases A and B confined in a cylinder. Piston 1 is permeable to A but not B piston 2 is permeable to B but not A.

The enthalpy change can be found from the following two fundamental thermodynamic relationships which, in the case of ideal gases, are valid for irreversible processes as well as reversible ones ... 

Further insight can be gained from the idealized T - S diagram for the cycle. Figure 9-14. The compression of the air and fuel streams is represented here as a single adiabatic reversible (constant S) process in which the temperature of the gases rises above ambient. The heating of... 

In complete equilibrium, the ratio of the population of an atomic or molecular species in an excited electronic state to the population in the groun d state is given by Boltzmann factor e — and the statistical weight term. Under these equilibrium conditions the process of electronic excitation by absorption of radiation will be in balance with electronic deactivation by emission of radiation, and collision activation will be balanced by collision deactivation excitation by chemical reaction will be balanced by the reverse reaction in which the electronically excited species supplies the excitation energy. However, this perfect equilibrium is attained only in a constant-temperature inclosure such as the ideal black-body furnace, and the radiation must then give -a continuous spectrum with unit emissivity. In practice we are more familiar with hot gases emitting dis-... 

In this appendix, we shall discuss a system of ideal gases. We shall examine the suitability of the terms mixing entropy and mixing Gibbs energy, and the validity of the statement that the mixing process is essentially reversible , see, for example, Denbigh (1966). 

Reversible processes are but one example of a host of concepts of a similarly idealized nature in chemistry and physics— for example, ideal gases and solutions, absolute zero temperature, infinitely dilute solutions, perfect black-body radiation, isolated systems, perfect insulators, and so on. In every case, the adoption of the idealized case simplifies or makes possible the application of mathematics to physical... 

This is a special process in which the system exchanges reversible work with the surroundings but no heat. Experimentally, this can be accomplished by thermally insulating the system and conducting the process in a quasi-static manner. Such system can either be compressed or expanded. The conditions reversible and adiabatic fully specify the path, as we will see below. The process is not specific to ideal-gases, but the calculation of this path for a general fluid will be delayed until Chapter d. 

Isothermal-isobaric mixing. Consider Nj moles of pure ideal gas 1 and N2 moles of pure ideal gas 2 initially in separate containers at the same T and P. We mix these two gases in such a way that the mixture remains at the same T and P note this is the reverse of the process shown in Figure 4.1. We want to determine whether the change in entropy is positive, negative, or zero. The entropy change is given by... 

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