Irreversible Adiabatic Volume Changes

There are an infinite number of irreversible adiabatic volume change paths that might be considered, but to keep the book to a reasonable size we will restrict ourselves to three. Each is instructive in its own way. 

To investigate the variation of temperature during an isenthalpic change of pressure, one is naturally interested in the derivative (dT/dP)H, called the Joule-Thompson coefficient, / jj. An easy way to derive an expression for this quantity is to use Table 2.1. From Table 2.1, 3T)h = —V + T dV/dT)p, and (3P)h = Cp. Then 

As an example of how the Joule-Thompson coefficient might be used, consider some hot spring fluids (approximated by pure water) rising vertically in the crust. When boiling begins, the pressure is 165 bars and the temperature is 350°C. At this point, VhjO = 31.35 cm mol = 0.7493 calbar mol , a = 0.01037 and C° = 43.60 calK moP, and 

Whatever the means of cooling the fluid to the point of phase separation (boiling). 

The systematics of adiabatic expansions that we have presented can be seen as simply an exercise in manipulating thermodynamic concepts, but in fact the extent to which fluids circulate at elevated temperatures and pressures in the Earth s crust means that volume changes, both adiabatic and non-adiabatic, are often important in constructing models explaining fluid behavior. Applications of isenthalpic expansion to minerals and rock masses, discussed by Waldbaum (1971), are possible, but to date none have been documented convincingly. 

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Figure 2.8 Schematic plot of states accessible via adiabatic processes in closed systems. From the initial state 1 only states on the line can be reached by reversible adiabatic volume changes. States above the reversible adiabat can be reached only by processes that include irreversible adiabatic volume changes. States below the reversible adiabat cannot be reached by any adiabatic volume change.

Solid curves for irreversible adiabatic volume changes at finite rates in the... 

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). 

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 ... 

Now we repeat the experiment using a different adiabatic process B. The system is still closed, and the initial and final states are still [T, V ] and [T , V ], but we use a sequence of steps with various weights, so the volume changes in a different way hence, the degree of irreversibility differs from that in process A. In general, to achieve the required final state [T , y ] we may have to use some combination of compressions and expansions. The work required for this second process is... 

Suppose the system is a solid body whose temperature initially is nonuniform. Provided there are no internal adiabatic partitions, the initial state is a nonequilibrium state lacking internal thermal equilibrium. If the system is surrounded by thermal insulation, and volume changes are negligible, this is an isolated system. There will be a spontaneous, irreversible internal redistribution of thermal energy that eventually brings the system to a final equilibrium state of uniform temperature. 

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. 

Consider the free expansion of a gas shown in Fig. 3.8 on page 79. The system is the gas. Assume that the vessel walls are rigid and adiabatic, so that the system is isolated. When the stopcock between the two vessels is opened, the gas expands irreversibly into the vacuum without heat or work and at constant internal energy. To carry out the same change of state reversibly, we confine the gas at its initial volume and temperature in a cylinder-and-piston device and use the piston to expand the gas adiabatically with negative work. Positive heat is then needed to return the internal energy reversibly to its initial value. Because the reversible path has positive heat, the entropy change is positive. 

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