Adiabatic expansion reversible change

In an adiabatic expansion or compression, the system is thermally isolated from the surroundings so that q = 0. If the change is reversible, we can derive a general relationship between p, V, and T, that can then be applied to a fluid (such as an ideal gas) by knowing the equation of state relating p, V, and T. 

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. 

A reversible adiabatic expansion of an ideal gas has a zero entropy change, and an irreversible adiabatic expansion of the same gas from the same initial state to the same final volume has a positive entropy change. This statement may seem to be inconsistent with the statement that 5 is a thermodynamic property. The resolution of the discrepancy is that the two changes do not constitute the same change of state the final temperature of the reversible adiabatic expansion is lower than the final temperature of the irreversible adiabatic expansion (as in path 2 in Fig. 6.7). 

In the absence of viscosity, rarefaction waves are thermodynamically reversible phenomena, that is, no change in entropy is involved and the ordinary laws of adiabatic expansion can be applied. Shock waves, on the other hand, ate irreversible there is a continuous dissipation of energy into heat (Ref 1)... 

The fact that a gas can be cooled (/xJT > 0) or warmed (/zJT < 0) by merely expanding under adiabatic (adiabatic conditions, A U =w, so the work performed by the gas in reversible adiabatic expansion must be compensated by the change AU in internal energy, that is, by a temperature change (since heat capacity is nonzero). 

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... 

For the reversible adiabatic expansion of a perfect gas, the change in energy content is related to the change in volume by... 

In order that the heat exchanges between the gas and the reservoirs take place reversibly it is essential that the temperature of the gas is made equal to that of the particular reservoir before they are brought into thermal contact. This change in temperature must be effected reversibly and without recourse to any other heat reservoir. Such a change can be effected by adiabatic and reversible compression or expansion of the gas. 

We can use these equations to calculate the changes in various properties of an ideal gas undergoing a reversible, adiabatic expansion or compression. This is illustrated in Example 10.16. 

Notice for the reversible adiabatic expansion considered in Example 10.16 that the temperature of sample changed from 298 K to 118 K—a very dramatic temperature decrease. Because of this significant temperature decrease, the final volume of the gas is much smaller than if the expansion were carried out isothermally, where the temperature would remain at 298 K. For a reversible isothermal expansion at 298 K from P] = 10.0 atm and Vj = 12.2 L to P2 = 1.00 atm, the final volume is... 

As expected from the much greater volume change, the work delivered to the surroundings in the reversible, isothermal expansion is much greater than for the reversible adiabatic expansion. The two types of expansions starting at Pj = 10.0 atm and Vj = 12.2 L are compared in Fig. 10.19. Note that for reversible, isothermal expansion... 

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. 

Problem Calculate the work of expansion in ergs when the pressure of 1 mole of an ideal gas at 25 C is changed adiabatically and reversibly from 1.0 atm. to 5.0 atm. The molar heat capacities may be taken as equal to those of air. (Compare the problem in 8b, which is for an isothermal expansion between the same pressure limits.)... 

Let ns next consider a particular type of a non-isothermal reversible cycle consisting of an isothermal expansion of a system (solid, liquid, or gas), followed by an adiabatic expansion, this in turn being followed by an isothermal compression, and this by an adiabatic compression, thereby bringing the system back to its original state Such a cycle, consisting of two isothermal volume changes and two adiabatic volume T.,. changes, is called a Car-... 

By measuring the temperature change accompanying a differential volume change in a free expansion across a valve and separately in a reversible adiabatic expansion, the two derivatives cT/dV)H and [cT/dV)s can be experimentally evaluated. 

In an adiabatic expansion of a gas, mechanical work is done by the gas as its volume increases and the gas temperature falls. For an ideal gas undergoing a reversible adiabatic change it can be shown that pvy=Ki V p -r=K2... 

Section 3.4.1 explained that during a rapid spontaneous expansion of the gas in the cylinder shown in Fig. 3.4, the pressure pb exerted by the gas at the moving piston is less than the pressure at the stationary wall. Consequently the work given by u = — f Ph dE is less negative for a spontaneous adiabatic expansion than for a reversible adiabatic expansion with the same initial state and the same volume change. 

Figure 3.20 Adiabatic expansion work with internal friction for a fixed magnitnde of AF, as a function of the average rate of volume change. The open circles indicate the reversible limits.

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