Reversible adiabatic surface

There is, however, the additional dimension of temperature in the A-dimensional space. Do the paths for possible reversible adiabatic processes, starting from a common initial point, lie in a volume in the A-dimensional space Or do they fall on a surface described by r as a function of the work coordinates If the paths he in a volume, then every point in a volume element surrounding the initial point must be accessible from the initial point by a reversible adiabatic path. This accessibility is precisely what Caratheodory s principle of adiabatic inaccessibility denies. Therefore, the paths for all possible reversible adiabatic processes with a common initial state must lie on a unique surface. This is an (A — 1)-dimensional hypersurface in the A-dimensional space, or a curve if N is 2. One of these surfaces or curves will be referred to as a reversible adiabatic surface. 

Now consider the initial and final states of a reversible process with one-way heat (i.e., each nonzero infinitesimal quantity of heat has the same sign). Since we have seen that it is impossible for there to be a reversible adiabatic path between these states, the points for these states must lie on different reversible adiabatic surfaces that do not intersect anywhere in the A-dimensional space. Consequently, there is an infinite number of nonintersecting reversible adiabatic surfaces filling the A-dimensional space. (To visualize this for A = 3, think of a flexed stack of printer paper each sheet represents a different adiabatic surface in three-dimensional space.) A reversible, nonadiabatic process with one-way heat is represented by a path beginning at a point on one reversible adiabatic surface and ending at a point on a different surface. If q is positive, the final surface lies on one side of the initial surface, and if q is negative, the final surface is on the opposite side. 

The existenee of reversible adiabatic surfaces is the justification for defining a new state function S, the entropy. S is specified to have the same value everywhere on one of these surfaces, and a different, unique value on each different surface. In other words, the reversible adiabatic surfaces are surfaees of constant entropy in the A-dimensional space. The fact that the surfaces fill this spaee without intersecting ensures that 5 is a state function for equilibrium states, because any point in this space represents an equilibrium state and also lies on a single reversible adiabatic surface with a definite value of S. 

We know the entropy function must exist, because the reversible adiabatic surfaces exist. For instance. Fig. 4.9 on the next page shows a family of these surfaces for a closed system of a pure substance in a single phase. In this system, A is equal to 2, and the surfaces... 

Figure 4.9 A family of reversible adiabatic curves (two-dimensional reversible adiabatic surfaces) for an ideal gas with V and T as independent variables. A reversible adiabatic process moves the state of the system along a curve, whereas a reversible process with positive heat moves the state from one curve to another above and to the right. The curves are calculated for n = 1 mol and Cv,m - (3/2)/ . Adjacent curves differ in entropy by 1J K. ...

How can we assign a unique value of S to each reversible adiabatic surface We can order the values by letting a reversible process with positive one-way heat, which moves the point for the state to a new surface, correspond to an increase in the value of S. Negative one-way heat will then correspond to decreasing S. We can assign an arbitrary value to the entropy on one particular reversible adiabatic surface. (The third law of thermodynamics is used for this purpose—see Sec. 6.1.) Then all that is needed to assign a value of S to each equilibrium state is a formula for evaluating the difference in the entropies of any two surfaces. 

Suppose the same experimental system undergoes a second reversible process, not necessarily with one-way heat, along a different path connecting the same pair of reversible adiabatic surfaces. This could be path C D in Fig. 4.10(a). The net heat entering the... 

Figure 2.12 A set of parallel, isentropic surfaces ordered so that S, > S2 > S3. The solid curve marked 6 rev = 0 represents a reversible adiabatic path that connects two states that lie on the entropy surface. Si. The dashed curves marked 6qm = 0 are irreversible paths that connect states on different entropy surfaces. Only one of these two paths will be allowed the other will be forbidden.

Presumably all points on the same surface can be connected by some solution curve (reversible adiabatic process). Flowever, states on surface S2, for example, cannot be connected to states on either Si or S3 by any reversible adiabatic path. Rather, if they can be connected, it must be through irreversible adiabatic paths for which dS 0. We represent two such paths in Figure 2.12 by dashed lines. 

If the states of the system are functions of t and more than one other parameter, we note that the conclusions drawn from Fig. 4-1 hold for any plane in the space of the variables characterizing the system that is parallel to the t axis. Thus the region around 1 that is inaccessible adiabatically from 1 may be rotated about 1, thereby sweeping out a finite volume inaccessible adiabatically from 1. This volume includes points that are arbitrarily close to 1. We note that reversible adiabatic paths through 1 are restricted to lying on a surface through 1. 

While the importance of the breakdown of the BOA in thermal chemistry is still controversial, the time-reversed process of creating chemistry from hot electrons is well established. Because experiments are generally performed under conditions where there is no adiabatic chemistry, hot electron induced chemistry is easily identified and studied, even when the cross-section for the chemistry is very small. Typical scenarios involve photochemistry, femtochemistry and single molecule chemistry on surfaces. A few well-studied examples are discussed briefly in Section 4.8. Because a detailed discussion of these active fields would take this chapter far from its original purpose, they are only treated briefly to illustrate the relationship to other aspects of bond making/breaking at surfaces. 

Now let all these adiabatics and isothermals be produced until they cut the neighbouring ispthermals and adiabatics, as in the figure. In this way we have divided the surface of the diagram into a number of quadrilateral figures each bounded by an isothermal and an adiabatic. Each of these quadrilaterals represents a simple reversible cycle, to which the equation... 

Here uG — is the difference in the partial molar internal energy of the gas G and the adsorbed layer 1, and —p (iV/tym)s is the adiabatic differential heat of compression. Reversible adsorption can only occur at temperatures such that diffusion of the adsorbed gas is fast enough to establish an equilibrium distribution at the surface during the time of the measurements. This criterion holds for chemisorption as well, but implies that observations are made at correspondingly higher temperatures. 

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