Adiabatic processes reversible expansion

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. 

It is useful to compare the reversible adiabatic and reversible isothermal expansions of the ideal gas. For an isothermal process, the ideal gas equation can be written... 

In general the conditions under which a change in state of a gas takes place are neither isothermal nor adiabatic and the relation between pressure and volume is approximately of the form Pvk = constant for a reversible process, where k is a numerical quantity whose value depends on the heat transfer between the gas and its surroundings, k usually lies between 1 and y though it may, under certain circumstances, lie outside these limits it will have the same value for a reversible compression as for a reversible expansion under similar conditions. Under these conditions therefore, equation 2.70 becomes ... 

So far we have not specified whether the adiabatic expansion under consideration is reversible. Equations (5.40), (5.42), and (5.44) for the calculation of the thermodynamic changes in this process apply to the reversible expansion, the free expansion, or the intermediate expansion, so long as we are dealing with an ideal gas. However, the niunerical values of W, AU, and AH will not be the same for each of the three types of adiabatic expansion because T2, the final temperature of the gas, will depend on the type of expansion, even though the initial temperature is identical in aU cases. 

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). 

CARNOT CYCLE. An ideal cycle or four reversible changes in the physical condition of a substance, useful in thermodynamic theory. Starting with specified values of die variable temperature, specific volume, and pressure, the substance undergoes, in succession, an isothermal (constant temperature) expansion, an adiabatic expansion (see also Adiabatic Process), and an isothermal compression to such a point that a further adiabatic compression will return the substance to its original condition. These changes are represented on the volume-pressure diagram respectively by ub. he. ctl. and da in Fig. I. Or the cycle may he reversed ad c h a. 

Consider now adiabatic processes wherein no heat transfer occurs. We represent on the PV diagram of Fig. 5.6 an irreversible, adiabatic expansion of a fluid from an initial equilibrium state at point A to a final equilibrium state at point B. Now suppose the fluid is restored to its initial state by a reversible process. If the initial process results in an entropy change of the fluid, then there must be heat transfer during the reversible restoration process such that... 

A certain gas obeys the equation of state P(V -nb) - nRT and has a constant volume heat capacity, Cv, which is independent of temperature. The parameter b is a constant. For 1 mol, find W, AE, Q, and AH for the following processes (a) Isothermal reversible expansion. (b) Isobaric reversible expansion. (c) Isochoric reversible process, (d) Adiabatic reversible expansion in terms of Tlf Vlt V2, Cp, and Cv subscripts of 1 and 2 denote initial and final states, respectively. (c) Adiabatic irreversible expansion against a constant external pressure P2, in terms of Plf P2, Tj, and 7 = (Cp/Cy). 

The ratio, y, of the principal specific heats of a gas is of importance in expansion and compression processes, since it may be shown that a reversible, adiabatic (isen-tropic) expansion or compression will obey the law ... 

Equation (2.4-21) can be used for a reversible adiabatic compression as well as for an expansion. It is an example of an important fact that holds for any system, not just an ideal gas For a reversible adiabatic process in a simple system the final temperature is a function of the final volume for a given initial state. All of the possible final state points for reversible adiabatic processes starting at a given initial state lie on a single curve in the state space, called a reversible adiabat. This fact will be important in our discussion of the second law of thermodynamics in Chapter 3. 

The Carnot engine operates on a two-stroke cycle that is called the Carnot cycle. We begin the cycle with the piston at top dead center and with the hot reservoir in contact with the cylinder. We break the expansion stroke into two steps. The first step is an isothermal reversible expansion of the system at the temperature of the hot reservoir. The final volume of the first step is chosen so that the second step, which is an adiabatic reversible expansion, ends with the system at the temperature of the cold reservoir and with the piston at bottom dead center. The compression stroke is also broken into two steps. The third step of the cyclic process is a reversible isothermal compression with... 

Consider when the same ideal gas undergoes an adiabatic, reversible expansion (as opposed to isothermal). We will assume that the heat capacity of this gas does not change with temperature, that is, constant beat capaci. This process is illustrated in Figure 2.16. 

An ideal gas undergoes an adiabatic, reversible expansion process in a closed system (a) If Cj, is constant, show that ... 

To illustrate how entropy tells us about the directionahty of nature, we first pick a set of four mechanical processes similar to those described in Figure 3.2a (1) reversible expansion, (2) irreversible expansion, (3) reversible compression, and (4) irreversible compression. In this case, however, we choose adiabatic rather than isothermal processes. The adiabatic process represents the limit of no heat transfer between the system and the... 

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 ... 

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