Path function

Melissas V S, Truhlar D G and Garrett B C 1992 Optimized calculations of reaction paths and reaction-path functions for chemical reactions J. Chem. Phys. 96 5758... 

The properties that describe a system and its transformations can be grouped in two broad categories. Some properties depend only on the conditions that describe the system. These properties are called state functions. Other properties depend on how the change occurs. Properties that depend on how a change takes place are called path functions. 

An everyday example illustrates the difference between state functions and path functions. The Daltons, who live in San Francisco, decide to visit relatives in Denver. They take different routes, as Figure shows. Mr. Dalton takes a train directly from San Francisco to Denver, but Ms. Dalton goes to Dallas for a business meeting and then flies from Dallas to Denver. The Daltons daughter drives to Los Angeles, where she catches a flight to Denver. 

On arrival, each of the Daltons is asked two questions by their relatives How far is Denver from San Francisco and How far did you travel to get here Each answers 950 miles to the first question, because the distance between the two cities is a difference in values of a state function. Each answers differently to the second question, however, because each Dalton traveled a different distance to reach Denver. Distance traveled depends on the path and is a change in a path function. 

Energy is a state fianction, but heat and work are path functions. To illustrate this, Figure 6-11 describes two different paths for the combustion of 1 moi of methane. Path 1 represents what happens in an automobile fueled by natural gas As methane bums, the system does work on its surroundings by driving back the piston. At the same... 

A engine - famance In the engine, some of this energy accomplishes work and the rest is transferred as heat. In the furnace, all of this energy is transferred as heat. In other words, < engine fumance The heat transferred is different for the two paths, so q is a path function, not a state function. 

As with the distance each Dalton traveled in going from San Francisco to Denver (see Figure b-gi. q and w are path functions. As with the distance between San Francisco and Denver, A is a state function. The fact that heat transfer depends on the path while energy change is independent of path has important consequences in chemistry, as we describe later in this chapter. 

To determine A E using measured values of q, we also must know w. Because heat and work are path functions, however, we proceed differently for constant volume than for constant pressure. To distinguish between these different paths, we use a subscript v for constant-volume calorimetry and a subscript p for constant-pressure calorimetry. This gives different expressions for the two t q)es of calorimeters ... 

TdS = dH-VdP This equation relates only the properties of closed system. There is no path function term in the equation, and therefore it holds good for any... 

Step 2 Define the feedback path function. Let s presume that our measurement function is first order too. The measurement gain has been taken out and implemented in Step 1. 

Figure 4. The Data Reduction Process. The arrows depict the control path. Functions in the left column are "Mainstream" functions. The functions in the right hand column are used for custom Interactive data reduction.

The benefit of this course is that it provides all students taking the physical chemistry lecture course with the same mathematical foundation. In the physical chemistry lecture we can discuss the relationship between different thermodynamic functions without stopping to review partial derivatives. We can talk about the difference between work, heat, and energy without stopping to teach the difference between path functions and work functions. We can write... 

Calculate q, tv, AE, and AH for each step, and calculate overall values for each pathway. Explain how the overall values for the two pathways illustrate that AE and AH are state functions while q and tv are path functions. 

The isentrope, the path function that describes a continuity and not a jump, is different from the Hugoniot. Remember that a relief wave is a continuous... 

In the previous sections, we dealt in detail with the properties at the shock front, the jump process that takes material in front of the shock to the state behind the shock. We showed that this is indeed a discontinuous process, and that pressure disturbances cannot outrun the shock (in the strong shock region). We stated, but did not demonstrate, that the rarefaction wave (also called relief, or unloading wave they are all synonymous) is continuous, that it follows a path function, not a jump condition. Let us look into this statement now. 

Define or explain the following terms energy, system, closed system, nonflow system, open system, flow system, surroundings, property, extensive property, intensive property, state, heat, work, kinetic energy, potential energy, internal energy, enthalpy, initial state, final state, point (state) function, state variable, cyclical process, and path function. 

Note that unless the process (or path) under which work is carried out is specified from the initial to the final state of the system, you are not able to calculate the value of the work done. In other words, work done in going between the initial and final states can have any value, depending on the path taken. Work is therefore called a path function, and the value of W depends on the initial state, the path, and the final state of the system, as illustrated in the next example. 

The macroscopic state of any one-component fluid system in equilibrium can be described by just three properties, of which at least one is extensive. All other properties of the state of the same system are necessarily specified by the chosen three properties. For instance, if for a single component gas in equilibrium, pressure, temperature, and volume are known, all other properties which describe the state of that gas (such as number of moles, internal energy, enthalpy, entropy, and Gibbs energy) must have a specific single value. Since the state of a system can be described exactly by specific properties, it is not necessary to know how the state was formed or what reaction pathway brought a state into being. Such properties that describe the state of a system are called slate functions. Properties that do not describe the state of a system, but depend upon the pathway used to achieve any state, are called path functions. Work and heat are examples of path functions. 

The start point and end point of the expansion could also have been achieved by removing one mass, allowing the piston to rise, increasing the pressure to 2mg/A while holding the piston steady, and then replacing the second mass. This would have been a different pathway to achieve the same result. Since work is a path function, a different pathway results in a different amount of work in this case the work would have been mgh. 

A path function is a function that depends on the histoiy of the system. Examples of path functions include work and heat. 

The statement that the bond-path RDF is independent of conformational changes relies on the precision of the Cartesian coordinates of the atoms and the accuracy of calculation. In practice, the bond-path functions of two conformers are extremely similar but seldom coincidental. 

The function W (work) and the function Q (heat) which appear in one statement of the first law of thermodynamics (AU = Q — W) are path functions. These functions depend upon the path followed in going from one state to another. 

Notice that along each of the three paths considered (and, in fact, any other path between the initial and final states), the sum of Q and W. which is equal to A17, is 8163.7 J/mol, even though 0 and lU separately.-are different along the different paths. This illustrates that whereas the internal energy is a. sjate property and is path independent (i.e., its change in going from state 1 to state 2 depends only on these states and not on the path between them), the heat and work flows depend on the path and are therefore path functions. ... 

This example verifies, at least for the paths considered here, that the entropy is a state function. For reversible processes in dosed systems, the rate of change of entropy and the ratio Q/T are equal. Thus, for reversible changes. QIT s also a state function, even though the total heat flow O is a path function. B... 

The two experiments, immersion of the system in hot water and rotating a paddle in the same system, involved the same change in state but different heat and work effects. The quantities of heat and work that flow depend on the process and therefore on the path connecting the initial and final states. Heat and work are called path functions. 

In general, W y and Q y are not zero this is characteristic of path functions. 

Calculation of Reaction Paths and Reaction-Path Functions for Chemical Reactions. 

In order to calculate R(t, t ) efficiently, we introduce a path functional Hg x ) that is unity if at least one state along the trajectory x( ) lies in B and vanishes otherwise,... 

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