Path-dependent functions

Types of Thermodynamic Function Path Dependent Function, q and w... 

The lone remaining aspect of this topic that requires additional discussion is the fact that the mechanical threshold stress evolution is path-dependent. The fact that (df/dy)o in (7.41) is a function of y means that computations of material behavior must follow the actual high-rate deformational path to obtain the material strength f. This becomes a practical problem only in dealing with shock-wave compression. 

Students often ask, What is enthalpy The answer is simple. Enthalpy is a mathematical function defined in terms of fundamental thermodynamic properties as H = U+pV. This combination occurs frequently in thermodynamic equations and it is convenient to write it as a single symbol. We will show later that it does have the useful property that in a constant pressure process in which only pressure-volume work is involved, the change in enthalpy AH is equal to the heat q that flows in or out of a system during a thermodynamic process. This equality is convenient since it provides a way to calculate q. Heat flow is not a state function and is often not easy to calculate. In the next chapter, we will make calculations that demonstrate this path dependence. On the other hand, since H is a function of extensive state variables it must also be an extensive state variable, and dH = 0. As a result, AH is the same regardless of the path or series of steps followed in getting from the initial to final state and... 

In earlier days, A was called the work function because it equals the work performed on or by a system in a reversible process conducted at constant temperature. In the next chapter we will quantitatively define work, describe the reversible process and prove this equality. The name free energy for A results from this equality. That is, A A is the energy free or available to do work. Work is not a state function and depends upon the path and hence, is often not easy to calculate. Under the conditions of reversibility and constant temperature, however, calculation of A A provides a useful procedure for calculating u ... 

As we have seen before, exact differentials correspond to the total differential of a state function, while inexact differentials are associated with quantities that are not state functions, but are path-dependent. Caratheodory proved a purely mathematical theorem, with no reference to physical systems, that establishes the condition for the existence of an integrating denominator for differential expressions of the form of equation (2.44). Called the Caratheodory theorem, it asserts that an integrating denominator exists for Pfaffian differentials, Sq, when there exist final states specified by ( V, ... x )j that are inaccessible from some initial state (.vj,.... v )in by a path for which Sq = 0. Such paths are called solution curves of the differential expression The connection from the purely mathematical realm to thermodynamic systems is established by recognizing that we can express the differential expressions for heat transfer during a reversible thermodynamic process, 6qrey as Pfaffian differentials of the form given by equation (2.44). Then, solution curves (for which Sqrev = 0) correspond to reversible adiabatic processes in which no heat is absorbed or released. 

If a laser beam produces in the outer atmosphere a spectrum spanning from the ultraviolet to at least the red, then the return light will follow different optical paths depending on the wavelength (Fig. 19). The air refraction index is a function of air temperature T and pressure P ... 

Equation (28) is still exact. To introduce the classical-path approximation, we assume that the nuclear dynamics of the system can be described by classical trajectories that is, the position operator x is approximated by its mean value, namely, the trajectory x t). As a consequence, the quantum-mechanical operators of the nuclear dynamics (e.g., Eh (x)) become classical functions that depend parametrically on x t). In the same way, the nuclear wave functions dk x,t) become complex-valued coefficients dk x t),t). As the electronic dynamics is evaluated along the classical path of the nuclei, the approximation thus accounts for the reaction of the quantum DoE to the dynamics of the classical DoF. 

It can be stated as a generalisation that historic or innovation cultural and institutional aspects play an important role (and supposedly also one that is gaining in importance) for comprehending innovation events . This conclusion is reached, for example, if we examine by way of summary the stated distinctions of the main obstacle system inertia (such as path dependency, investment cycles, complexity of the iimovation system and level of iimovation) as well as the distinctive features of the main driver competition (such as competition or market type and especially the role of the public). Case studies doubtless have an important function for comprehending these elements and developments. On the basis of such studies the innovation system model can also be improved and developed further (cf Figure 20). 

Again it may be desirable to vary the chronological order for the introduction of the solvent, soluble organic reagents, organometallic precursor, ligand, and promoter. Dramatic variations in catalytic activity may occur, going from very active systems to systems that possess more-or-less no activity and are therefore fully deactivated. In at least one case, the variability of catalytic activity as a function of start-up procedure has been studied by in situ FTIR. The term path-dependent catalysis has been used to emphasize the importance of the start-up procedure [60]. 

This relationship identifies the surface energy as the increment of the Gibbs free energy per unit change in area at constant temperature, pressure, and number of moles. The path-dependent variable dWs in Eq. (2.60) has been replaced by a state variable, namely, the Gibbs free energy. The energy interpretation of y has been carried to the point where it has been identified with a specific thermodynamic function. As a result, many of the relationships that apply to G also apply to y ... 

In other words, each generation is weighted after differentiation with a function that depends on the path length. One easily verifies Eq. (C.84) for the random polycondensates when a - Gaussian statistics) is assumed. 

We have seen here that internal energy (U) differs from heat (q) and work (w) in that it depends only on the state of the system, not on the path it takes. Functions which depend only on the initial and final states and not on path are called slate functions. 

Olvera de la Cruz and Sanchez [76] were first to report theoretical calculations concerning the phase stability of graft and miktoarm AnBn star copolymers with equal numbers of A and B branches. The static structure factor S(q) was calculated for the disordered phase (melt) by expanding the free energy, in terms of the Fourier transform of the order parameter. They applied path integral methods which are equivalent to the random phase approximation method used by Leibler. For the copolymers considered S(q) had the functional form S(q) 1 = (Q(q)/N)-2% where N is the total number of units of the copolymer chain, % the Flory interaction parameter and Q a function that depends specifically on the copolymer type. S(q) has a maximum at q which is determined by the equation dQ/dQ=0. 

Figure 1.1 Comparison of path dependent functions (q and w) and path independent change (AU) during the expansion of a gas from (V, T,) to (Vf, 7)) via two different states (V, T() (path 1) and (Vf, T) (path 2). 1/ and T represent the volume and temperature of the gas. q and w represent the heat absorbed by the gas and the work done by the gas on the surroundings in expanding against the external pressure, P. Wf and W2 are both negative (using the convention discussed in Frame 7) since work of expansion is expended by the gas and lost from the (gas) system.

We note here, as was found in the example of the calculation of the work done during the irreversible adiabatic of expansion of a gas (equation 9.4 Frame 9), that under specific conditions a path dependent function can become identically equal to the change in a state function. 

Path-dependent properties are a function of the sequencing of the transition steps from an initial to a final state, and it is only when each event is enacted in an identical way that the integrals of path-dependent functions are themselves identical. 

The difficulty with this is that it makes C, like Q, a path-dependent quantity rather than a state function. However, it does suggest the possibility that more than one heat capacity might be usefully defined. 

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