Thermodynamic state functions

In the broadest sense, thermodynamics is concerned with mathematical relationships that describe equiUbrium conditions as well as transformations of energy from one form to another. Many chemical properties and parameters of engineering significance have origins in the mathematical expressions of the first and second laws and accompanying definitions. Particularly important are those fundamental equations which connect thermodynamic state functions to real-world, measurable properties such as pressure, volume, temperature, and heat capacity (1 3) (see also Thermodynamic properties). 

First, we shall explore a conceptual relation between kinetics and thermodynamics that allows one to draw certain conclusions about the kinetics of the reverse reaction, even when it has itself not been studied. Second, we shall show how the thermodynamic state functions for the transition state can be defined from kinetic data. These are the previously mentioned activation parameters. If their values for the reaction in one direction have been determined, then the values in the other can be calculated from them as well as the standard thermodynamic functions. The implications of this calculation will be explored. Third, we shall consider a fundamental principle that requires that the... 

A change in any thermodynamic state function is independent of the path used to accomplish that change. This feature of state functions tells us that the energy change in a chemical reaction is independent of the manner in which the reaction takes place. In the real world, chemical reactions often follow very complicated paths. Even a relatively simple overall reaction such as the combustion of CH4 and O2 can be very complicated at the... 

Enthalpy is a thermodynamic state function that describes heat flow at constant pressure. 

Why do some reactions go virtually to completion, whereas others reach equilibrium when hardly any of the starting materials have been consumed At the molecular level, bond energies and molecular organization are the determining factors. These features correlate with the thermodynamic state functions of enthalpy and entropy. As discussed In Chapter 14, free energy (G) is the state function that combines these properties. This section establishes the connection between thermodynamics and equilibrium. 

Clearly, the differential obtained, namely, d S = SqJT is exact and S, the entropy, is a thermodynamic state function, that is, it is independent of the path of integration. While Eq. (88) was obtained with the assumption of an ideal gas, the result is general if reversible conditions are applied. 

The enthalpy of formation of a compound is a so-called thermodynamic state function, which means that the value depends only on the initial and final states of the system. When the formation of crystalline NaCl from the elements is considered, it is possible to consider the process as if it occurred in a series of steps that can be summarized in a thermochemical cycle known as a Born-Haber cycle. In this cycle, the overall heat change is the same regardless of the pathway that is followed between the initial and final states. Although the rate of a reaction depends on the pathway, the enthalpy change is a function of initial and final states only, not the pathway between them. The Born-Haber cycle for the formation of sodium chloride is shown as follows ... 

When the stretched rubber band is relaxed, the signs of the thermodynamic state functions change ... 

The concept of a state function can be quite difficult, so let us consider a simple example from outside chemistry. Geographical position has analogies to a thermodynamic state function, insofar as it does not matter whether we have travelled from London to New York via Athens or flew direct. The net difference in position is identical in either case. Figure 3.4 shows this truth diagrammatically. In a similar way, the value of A U for the process A C is the same as the overall change for the process A -> B -> C. We shall look further at the consequences of U being a state function on p. 98. 

A fundamental thermodynamic state function (symbolized by S), and as such, not dependent on the path by which a particular state is reached. For a reversible process, the differential change in entropy, dS, is equal to the amount of energy absorbed by the system, dq, divided by the absolute temperature, T. Thus,... 

Thermodynamic state functions change with temperature this will be true if values of the heat capacity of any component is nonzero (which is almost always true). Whenever the heat capacity is not a constant, the various thermodynamic state functions will show nonlinear dependencies on temperature. 

See specific constant of interest Thermodynamic state function,... 

The first law of thermodynamics, which can be stated in various ways, enuciates the principle of the conservation of energy. In the present context, its most important application is in the calculation of the heat evolved or absorbed when a given chemical reaction takes place. Certain thermodynamic properties known as state functions are used to define equilibrium states and these properties depend only on the present state of the system and not on its history, that is the route by which it reached that state. The definition of a sufficient number of thermodynamic state functions serves to fix the state of a system for example, the state of a given mass of a pure gas is defined if the pressure and temperature are fixed. When a system undergoes some change from state 1 to state 2 in which a quantity of heat, Q, is absorbed and an amount of work, W, is done on the system, the first law may be written... 

It is also convenient to define another thermodynamic state function, enthalpy, H. 

It should be emphasized that the criterion for macroscopic character is based on independent properties only. (The importance of properly enumerating the number of independent intensive properties will become apparent in the discussion of the Gibbs phase rule, Section 5.1). For example, from two independent extensive variables such as mass m and volume V, one can obviously form the ratio m/V (density p), which is neither extensive nor intensive, nor independent of m and V. (That density cannot fulfill the uniform value throughout criterion for intensive character will be apparent from consideration of any 2-phase system, where p certainly varies from one phase region to another.) Of course, for many thermodynamic purposes, we are free to choose a different set of independent properties (perhaps including, for example, p or other ratio-type properties), rather than the base set of intensive and extensive properties that are used to assess macroscopic character. But considerable conceptual and formal simplifications result from choosing properties of pure intensive (R() or extensive QQ character as independent arguments of thermodynamic state functions, and it is important to realize that this pure choice is always possible if (and only if) the system is macroscopic. 

Similar transfer functions can be defined for other thermodynamic state functions, e.g. H. However, if these functions are to be combined to yield other state functions, e.g. S, then care must be exercised to ensure that, as in the use of equation (15), the same standard state is always used. 

The thermodynamic state functions obtained by non-linear regression in both experiments are identical within the error limits and are considered reliable contrary to the parameters of higher complexation that suffer from dramatically increased errors and their cross correlation. 

The pseudothermodynamic equilibrium constant of the activated complex (A q) is related to the thermodynamic state functions by... 

For book-keeping purposes the production of entropy during chemical change is considered as reducing the useful energy of the system by disorderly dispersion. In many cases this waste can be calculated statistically from the increase in disorder. To be in line with other thermodynamic state functions, any system is considered to be in some state of disorder at all temperatures above absolute zero, where entropy vanishes. 

With these Equations (4-33) to (4-35) the first goal is reached of describing partition by using parameters resulting from thermodynamic state functions. 

Thus, the deposition temperature and the thermodynamic state function of the adsorption are combined and they can easily be determined from each other. The retention time for a short-lived radioactive species is calculated as the radioactive lifetime of the nuclide ... 

Equation (1.60) illustrates a very important thermodynamic property in which the quantity dqrev / T becomes zero when the cyclic process is completed regardless of the paths taken from the initial to final states. Such a property is known as a state function, as are P, V, T, E, and H. Clausius suggested defining a new thermodynamic state function, called entropy and denoted as S, where dqrev / T = dS, so that... 

A, E, G, H, T, and S are thermodynamic state functions, independent of the path of a process. Changes in A and G from one state to another at constant temperature can be written by ... 

In Section 2.1, we remarked that classical thermodynamics does not offer us a means of determining absolute values of thermodynamic state functions. Fortunately, first-principles (FP), or ab initio, methods based on the density-functional theory (DFT) provide a way of calculating thermodynamic properties at 0 K, where one can normally neglect zero-point vibrations. At finite temperatures, vibrational contributions must be added to the zero-kelvin DFT results. To understand how ab initio thermodynamics (not to be confused with the term computational thermochemistry used in Section 2.1) is possible, we first need to discuss the statistical mechanical interpretation of absolute internal energy, so that we can relate it to concepts from ab initio methods. 

Each ensemble has "natural" thermodynamic variables, but the usual relationships between macroscopic thermodynamic functions will allow us to obtain the "other" thermodynamic state functions. 

Helmholtz energy (sometimes also called Helmholtz free energy, or Helmholtz function) is the thermodynamic state function equal to the maximum possible nonexpansion work output, which can be done by a closed system in an isothermal isochoric process (T = const, V = const). In terms of the -> internal energy and -> entropy... 

The thermodynamic state function of a system that indicates the amount of energy available for the system to do useful work at constant T and P. 

It turns out that the water pot example is a nice analog for a general process system such as a reactor. In the process system we characterize the energy storage with the thermodynamic state function exergy B, instead of heat Qf. Heat is of course not a general thermodynamic state function but it plays the role of one in the water pot example. In the... 

Entropy in this sense is different from the thermodynamic state function 5, which has a large reversible component, for instance as defined at phase transitions. An entropy change in the system is compensated for by an almost equal, but opposite change in the surroundings. 

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