State functions, definition

We may then re-define the Gibbs free energy function in terms of instead of n. Hence, the modified state function definition yields ... 

The are many ways to define the rate of a chemical reaction. The most general definition uses the rate of change of a themiodynamic state function. Following the second law of themiodynamics, for example, the change of entropy S with time t would be an appropriate definition under reaction conditions at constant energy U and volume V ... 

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). 

Volume is an extensive property. Usually, we will be working with Vm, the molar volume. In solution, we will work with the partial molar volume V, which is the contribution per mole of component i in the mixture to the total volume. We will give the mathematical definition of partial molar quantities later when we describe how to measure them and use them. Volume is a property of the state of the system, and hence is a state function.1 That is... 

Equation (A 1.16) shows that the integral evaluated over any two paths that connect states 1 and 2 must be equal. The value of the integral, AZ, cannot depend upon the path but must be associated with the choice of states so that AZ Zi — Z. This is consistent with our earlier definition of a state function. 

The yield of a gas-solid heterogeneous reaction depends not on the total time that molecules spend in the reactor but on the time that they spend on the catalyst surface. The contact time distribution provides a standardized measure of times spent in the absorbed state. A functional definition is provided by the following equation applicable to a first-order, heterogeneous reaction in an isothermal reactor ... 

Following from Equation (3.3), we say that internal energy is a state function. A more formal definition of state function is, A thermodynamic property (such as internal energy) that depends only on the present state of the system, and is independent of its previous history . In other words, a state function depends only on those variables that define the current state of the system, such as how much material is present, whether it is a solid, liquid or gas, etc. 

Equations (3.11)-(3.13) (repeated here), together with the statement that t/ is a state function, constitute a complete statement of the first law. In effect. Equations (3.11) and (3.12) are operational definitions of At/ and Q, respectively, whereas Equation (3.13) is an empirical statement of the relationship among the quantities... 

The definition of entropy requires that information about a reversible path be available to calculate an entropy change. To obtain the change of entropy in an irreversible process, it is necessary to discover a reversible path between the same initial and final states. As S is a state function, AS is the same for the irreversible as for the reversible process. 

The second law of thermodynamics can be expressed in terms of another state function, the entropy (S). The thermodynamics definition considers the change in entropy dS that occurs as a result of a physical or chemical change, and is based on the expression... 

The first satisfactory definition of entropy, which is quite recent, is that of Kittel (1989) entropy is the natural logarithm of the quantum states accessible to a system. As we will see, this definition is easily understood in light of Boltzmann s relation between configurational entropy and permutability. The definition is clearly nonoperative (because the number of quantum states accessible to a system cannot be calculated). Nevertheless, the entropy of a phase may be experimentally measured with good precision (with a calorimeter, for instance), and we do not need any operative definition. Kittel s definition has the merit to having put an end to all sorts of nebulous definitions that confused causes with effects. The fundamental P-V-T relation between state functions in a closed system is represented by the exact differential (cf appendix 2)... 

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... 

As one can see, the operator has a property of the wave operator (it transforms the projection of the exact wave function into the exact wave function), however, it should be stressed that the operator converts just one projected wave function into the corresponding exact wave function so we will denote it as a state-specific wave operator in contrast to the so-called Bloch wave operator [46] that transforms all d projections into corresponding exact states. From definition (11) it is iimnediately seen that the state-specific wave operators obey the following system of equations for a = 1,..., d... 

The right-hand side of (3.48b) has the appearance of a difference in a state function (U + PV) between the two states 1, 2. This state function is identified as the enthalpy (symbolized by H), with fundamental definition... 

Most of the formalism to be developed in the coming sections of these lecture notes will be independent of the specific definition of the configurational basis, in which we expand the wave function. We therefore do not have to be very explicit about the exact nature of the basis states hn>. They can be either Slater determinants or spin-adapted Configuration State Functions (CSF s). For a long time it was assumed that CSF s were to be preferred for MCSCF calculations, since it gives a much shorter Cl expansion. Efficient methods like GUGA had also been developed for the solution of the Cl problem. Recent... 

A functional definition of / V I insoluble is that some of the compound remains in the solid state when about 10-3 mol of the compound is placed in contact with one liter of water. It does not mean that none of the compound dissolves. 

A PWC function is defined as an (N — l)-electron bound parent state (atom or ion) with well-defined spin, parity, angular momentum and energy Ia = (Sa, na, La, Ea), coupled first to the spin-angular part of a single-particle state for the Nth electron, with definite orbital angular momentum la, to form a state with definite parity n, spin S, angular momentum L, and their projections L and M (for brevity, we indicate these global quantum numbers with the collective index T)... 

Fig. 1. Double interacting chains of orthonormal states, with definite values ofy, representing the Renner-Teller system for orbital p-like functions. N represents the total number of phonons.

A noteworthy point is that in certain cases, it is convenient to speak in terms of function. For example, the main function of the plastic film is to keep the food fresh for a specific duration of time. In cases where a comparative study is commissioned, and alternatives are required, the function definition is important to clearly state what the product is meant to do, so alternatives can be sought. 

Assume now, that the basis states a) are the states with definite spin quantum number aa. It means, that only one spin component of the wave function, namely tp0.(da) is nonzero, and tpa(aa) = 0. In this case the only nonzero matrix elements are those with aa = a1 and <77 = as, they are... 

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