Adiabatic path

One may now consider how changes can be made in a system across an adiabatic wall. The first law of thermodynamics can now be stated as another generalization of experimental observation, but in an unfamiliar form the M/ork required to transform an adiabatic (thermally insulated) system, from a completely specified initial state to a completely specifiedfinal state is independent of the source of the work (mechanical, electrical, etc.) and independent of the nature of the adiabatic path. This is exactly what Joule observed the same amount of work, mechanical or electrical, was always required to bring an adiabatically enclosed volume of water from one temperature 0 to another 02. 

This can be illustrated by showing the net work involved in various adiabatic paths by which one mole of helium gas (4.00 g) is brought from an initial state in whichp = 1.000 atm, V= 24.62 1 [T= 300.0 K], to a final state in whichp = 1.200 atm, V= 30.7791 [T= 450.0 K]. Ideal-gas behaviour is assumed (actual experimental measurements on a slightly non-ideal real gas would be slightly different). Infomiation shown in brackets could be measured or calculated, but is not essential to the experimental verification of the first law. 

As we shall see, because of the limitations that the second law of thennodynamics imposes, it may be impossible to find any adiabatic paths from a particular state A to another state B because In this... 

It suffices to carry out one such experiment, such as the expansion or compression of a gas, to establish that there are states inaccessible by adiabatic reversible paths, indeed even by any adiabatic irreversible path. For example, if one takes one mole of N2 gas in a volume of 24 litres at a pressure of 1.00 atm (i.e. at 25 °C), there is no combination of adiabatic reversible paths that can bring the system to a final state with the same volume and a different temperature. A higher temperature (on the ideal-gas scale Oj ) can be reached by an adiabatic irreversible path, e.g. by doing electrical work on the system, but a state with the same volume and a lower temperature Oj is inaccessible by any adiabatic path. 

Martyna, G.J. Adiabatic path integral molecular dynamics methods. I. Theory. J. Chem. Phys. 104 (1996) 2018-2027. 

FIGURE 20.2 Adiabatic paths for bond dissociation in two different electronic states and the diabatic path. 

The work done by an ideal gas of constant specific heat in passing from one isotherm to another is the same for all adiabatic paths, is independent of the initial or final pressures or volumes, and is proportional to the difference of temperature between the isotherms. 

According to the Caratheodory theorem, the existence of an integrating denominator that creates an exact differential (state function) out of any inexact differential is tied to the existence of points (specified by the values of their x, s) that cannot be reached from a given point by an adiabatic path (a solution curve), Caratheodory showed that, based upon the earlier statements of the Second Law, such states exist for the flow of heat in a reversible process, so that the theorem becomes applicable to this physical process. This conclusion, which is still another way of stating the Second Law, is known as the Caratheodory principle. It can be stated as... 

Figure 2.11 Graph of empirical temperature 0 against. v, a state variable such as pressure, (a) Reversible adiabatic paths (solid lines 1 — 2 and 1— 2)...

For each state identified on the 9 isothermal as 1, 1, l",..., let us draw paths representing reversible adiabatic processes that intersect a second isotherm at 02. The intersections of the reversible adiabatic paths from states 1,1 and 1" on 9 with those on 02 are denoted by 2, 2 and 2", respectively. Along the three paths, 1-2, l -2, and l"-2", no heat is absorbed or liberated because the processes that connect these points are defined to be adiabatic. 

We wish to show that no points to the leftbb of 2 on the isotherm 62 are accessible from point 1 via any adiabatic path, reversible or irreversible. Suppose we assume that some adiabatic path does exist between 1 and 2. We represent this path as a dotted curve in Figure 2.11a. We then consider the cycle I —>2 —> 1 — 1. The net heat associated with this cycle would be that arising from the last step 1 — 1, since the other two steps are defined to be adiabatic. We have defined the direction 1 — 1 to correspond to an absorption of heat, which we will call qy. From the first law, the net work vv done in the cycle, is given by w = —q, since AU for the cycle is zero. Thus, for this process, iv is negative (and therefore performed by the system), since qy is positive, having been absorbed from the reservoir. The net effect of this cycle, then, is to completely convert heat absorbed at a high temperature reservoir into work. This is a phenomenon forbidden by the Kelvin-Planck statement of the Second Law. Hence, points to the left of 2 cannot be reached from point 1 by way of any adiabatic path. 

What about points to the right of 2 Can they be reached Consider an adiabatic path from point 1 to point 2a that is also located on the isothermal Qj. The cycle of interest is 1 — 2a —> 2 — 1. Again, two of the three steps are adiabatic. In this case, however, heat is evolved during the 2a —> 2 step from the conversion of work into heat. The complete conversion of work into heat is a well-known phenomenon and is not forbidden by the laws of thermodynamics. Thus, there are states to the right of 2 on the isotherm O2 that are accessible from 1 via an adiabatic path. 

Since the only constraint we have placed on Qi is that it be less than 9, the second isotherm can be as arbitrarily close to the first as we wish. The conclusion that states exist on this second isotherm that cannot be reached from a point on the first isotherm by any adiabatic path is therefore general. Thus, we can argue that there are states located in the plane defined by 6 and xi that are inaccessible from state 1. 

In addition, we have established that there is a sense of direction to the location of the inaccessible states. State 2, the state reached from 1 by a reversible adiabatic path, represents the division between the states on the second isotherm that are accessible and inaccessible from state 1. We represent this schematically in Figure 2.1 lb, where the reversible adiabatic path separates states that are accessible from state 1 from those that are inaccessible. The observation that the reversible path serves as the boundary between the two sets of states will be useful later when we show the direction of allowed processes in terms of the sign of A5(universe). 

Now we can consider the effect of variations of 9 with a second variable, x2. Since we have been general about the nature of the variable, xj, we can expect to obtain similar behavior for the variable X2. We construct isotherms using the same B and 62, but this time in the direction of x2. Our initial point will be the same state 1 as earlier. The value of x2 in state 1 will fix the location of this state on the isotherm in the new direction. A reversible adiabatic path can be constructed that connects state 1 with a state on the second isothermal in the x2 direction. Irreversible states located on one side of this point will be inaccessible from state 1 by adiabatic paths, while states located on the other side of that point will be accessible. Thus, there exist states located on the plane defined by 9 and X2 that are inaccessible from point 1. Similar conclusions can be drawn by considering isotherms localized on the planes formed by 9 and each of the x,. 

Thus, we can conclude that, within the neighborhood of every state in this thermodynamic system, there are states that cannot be reached via adiabatic paths. Given the existence of these states, then, the existence of an integrating denominator for the differential element of reversible heat, Sqrev, is guaranteed from Caratheodory s theorem. Our next task is to identify this integrating denominator. 

The Caratheodory theorem establishes the existence of an integrating denominator for systems in which the Caratheodory principle identifies appropriate conditions — the existence of states inaccessible from one another by way of adiabatic paths. The uniqueness of such an integrating denominator is not established, however. In fact, one can show (but we will not) that an infinite number of such denominators exist, each leading to the existence of a different state function, and that these denominators differ by arbitrary factors of . Thus, we can make the assignment that A F (E ) = = KF(E) = 1. 

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. 

In the analysis in Section 2.2b, we showed that for any given initial state there are states that are accessible via irreversible adiabatic paths from the initial state, as well as states that are inaccessible from that initial state by way of irreversible adiabatic paths. Figure 2.11b showed that a reversible adiabatic path containing the initial state marked the division between the states that were accessible or inaccessible from that state, with all accessible states lying on one side of the reversible adiabatic path, and all inaccessible states lying on the other side of it. 

Figure 6.12. Possible adiabatic paths on a temperature-volume coordinate system.

Note that the reversible isothermal path produces maximum useful work, while the reversible adiabatic path produces maximum cooling. 

Figure 34. Two possible mechanisms for true associative ionization (a) adiabatic path and ionization by dynamic coupling (b) diabatic path and ionization by electronic coupling.

Fig. 7.7 (a) Field ionization data for Na nd states of n = 30, 32, 34, and 36. (b) Light lines extreme members of m = 0 Stark manifolds (fourth order perturbation theory) dotted lines adiabatic paths to ionization for n = 30, 32, 34, and 36 dark lines diabatic paths to ionization for lowest members of m = 2 manifolds for n = 30, 32, 34, and 36. The lines indicating the classical ionization fields are calculated on the basis of Ref. 5 (from ref. 4). 

Fig. 7.9 Adiabatic and diabatic paths to ionization for n = 15 states in the center and on the edges of the Stark manifold. The diabatic paths are shown by solid bold lines and the adiabatic paths by broken bold lines. In both cases ionization occurs at the large black dots. The diabatic paths are identical to hydrogenic behavior. The adiabatic n = 15 paths are trapped between the adiabatic n = 14 and n = 16 levels. Adiabatic ionization always occurs at lower fields than diabatic ionization.

In passing we note that this analysis can be generalized to an arbitrary (but adiabatic) path B(t), this enables one to see that the correction to the Berry phase is geometric, but that its geometric nature is very different from the Berry phase of an isolated spin-half [20]. 

The Hamiltonian along the adiabatic path, as developed by Gorling and Levy [11], has the following form... 

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