State space reversible adiabat

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. 

Next let us eonsider the reversible adiabatic processes that are possible. To carry out a reversible adiabatic process, starting at an initial equilibrium state, we use an adiabatic boundary and slowly vary one or more of the work coordinates. A certain final temperature will result. It is helpful in visualizing this process to think of an A/-dimensional space in which each axis represents one of the N independent variables needed to describe an equilibrium state. A point in this space represents an equilibrium state, and the path of a reversible process can be represented as a curve in this space. 

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. 

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. 

To show that two reversible adiabats cannot cross for other systems we assume the opposite of what we want to prove and then show that this assumption leads to a contradiction with fact and therefore must be false. Assume that there are two different reversible adiabats in the state space of a closed simple system and that the curves coincide at state number 1, as depicted in Figure 3.6. We choose a state on each reversible adiabat, labeled state number 2 and state number 3 such that the reversible process leading from state 2 to state 3 has q > 0. Now consider a reversible cyclic process 1 2 3 1. Since steps 1 and 3 are adiabatic. 

Caratheodory devised a three-part proof that the mathematical statement of the second law follows from a physical statement of the second law. The first part is to establish that in the state space of the system only one reversible adiabat passes through any given point. This was shown in Chapter 3. The second part of the argument is to show that this fact implies that a function S exists whose differential vanishes along the reversible adiabat on which also vanishes. This implies that d rev possesses an integrating factor, which is a function y that produces an exact differential dS when it multiplies an inexact differential ... 

We will give only a nonrigorous outline of Caratheodory s prooC We represent the state of a simple closed system by a point in the state space with T on the vertical axis and y on the horizontal axis. The main idea is that if there is a single curve in this space along which dqtey vanishes there is also a differential of a function, dS, which vanishes on the same curve. Consider reversible adiabatic processes of a closed simple system starting from a particular initial state. Since no two adiabats can cross the reversible adiabat can be represented mathematically by a function... 

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