Adiabatic Volume Changes

Thus far we have developed just about all the thermodynamic concepts required by Earth scientists with the exception of those needed to deal with solutions. Since all naturally occurring substances are solutions of one kind or another (although some can usefully be treated as pure substances), this is quite an important limitation, and we will proceed to discuss the treatment of solutions in Chapter 10. However, a great deal can be done with the thermodynamics of pure systems, and in this chapter we discuss a couple of applications of the concepts so far developed which are of particular interest to Earth scientists—the thermal effects associated with adiabatic volume changes, and the T-P phase diagrams of pure minerals. 

From the First Law, AU = q + w, we see that adiabatic processes (q = 0) are those for which AU = w. Thus the final state achieved after such a change will depend entirely on the work done during the change, or conversely, for any two states 

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Fig. 8.2. Schematic cartoons illustrating various types of adiabatic volume changes. In each case the points on the PV diagram correspond to those in Figure 8.1. (a) Reversible, Isentropic. (b) Joule Expansion, (c) Joule-Thompson Expansion.

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. 

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.

From state 1 any reversible adiabatic volume change leaves the system somewhere on the curve shown in Figure 2.8. Further, each reversible adiabat is unique that is, reversible adiabats do not intersect. If they did, then it would be possible to find two different values of the adiabatic work for the same change of state this would violate the first law. 

To find the work during the adiabatic volume change, we can use the relation... 

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

Adiabatic elasticity, Q, measured under such conditions that no heat is allowed to enter, or escape from, the fluid during the volume change. 

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... 

Consider a small adiabatic change. The volume changes by an amount dV and the temperature by an amount dT. Now U depends only on temperature for an ideal gas, so... 

The work required for the adiabatic process is seen to be somewhat less numerically than for the isothermal expansion, because of the smaller volume change. 

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-... 

Thermal effects are often the key concern in reactor scaleup. The generation of heat is proportional to the volume of the reactor. Note the factor of V in Equation 5.31. For a scaleup that maintains geomedic similarity, the surface area increases only as Sooner or later, temperature can no longer be controlled by external heat transfer, and the reactor will approach adiabatic operation. There are relatively few reactions where the full adiabatic temperature change can be tolerated. Endothermic reactions will have poor yields. Exothermic reactions will have thermal runaways giving undesired byproducts. It is the reactor designer s job to avoid limitations of scale or at least to understand them so that a desired product will result. There are many options. The best process and the best equipment at the laboratory scale are rarely the best for scaleup. Put another way, a process that is less than perfect at a small scale may be better for scaleup precisely because it is scaleable. 

By measuring the temperature change and the specific volume change accompanying a small pressure change in a reversible adiabatic process, one can evaluate the derivative... 

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. 

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. 

The first two of these equations relate to changes in state at constant entropy, that is, adiabatic, reversible changes in state. The derivative (dT/dV)s represents the rate of change of temperature with volume in a reversible adiabatic transformation. We shall not be much concerned with Eqs. (10.23) and (10.24). 

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... 

Consider expansion of a against p, so > 0. Then, we must have P > PP to make the Uis of (7.2.18) positive. Similarly, when phase a is compressed, then < q and (7.2.18) requires P < PP. That is, for both expansions and compressions of phase a, the pressure difference (P - PP) drives an adiabatic, constant-mass change of volume, and the work associated with such volume changes "flows" from regions of high pressure to regions of low pressure. Similar statements apply for other work modes. 

We emphasize that states in mechanical equilibrium have both the driving force (P -PP) and the volume change (dU ) equal to zero. Moreover, neither isothermal nor adiabatic constant-mass changes in volume can occur without a mechanical driving force that is, we cannot have 0 with P - pP = 0. 

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