The Variables of State

The SI is the logical extension of the metric system of units that has been in use for about 200 years. It was adopted by the General Conference on Weights and Measures (CGPM) in 1960. 

It is a coherent system of seven base units, one for each of the seven dimensionally independent quantities  

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One must now examine the integrability of the differentials hi equation (A2.1.121 and equation (A2.1.13), which are examples of what matliematicians call Pfoff differential equations. If the equation is integrable, one can find an integrating denominator k, a fiiiictioii of the variables of state, such that = d( ) where d( ) is... 

In phase diagrams for two-component systems the composition is plotted vs. one of the variables of state (pressure or temperature), the other one having a constant value. Most common are plots of the composition vs. temperature at ambient pressure. Such phase diagrams differ depending on whether the components form solid solutions with each other or not or whether they combine to form compounds. 

Since Jri - is dependent only on the values of the variables of state at the beginning and end of the process, it follows that the work done by the system is the same for all reversible and isothermal processes, which bring the system from state I to state 2. 

In the equilibrium thermodynamics, the physical properties of the system are fully identified by the fundamental thermodynamic potential / = /(oq,. .., xn) as a real-valued function of n real variables, which are called the variables of state. The macroscopic state of the system is fixed by the set of independent variables of state. x=(oq,. .., xn). Each variable of state x(, which is related to the certain thermodynamic quantity, describes some individual property of the system. The first and the second partial derivatives of the thermodynamic potential with respect to the variables of state define the thermodynamic quantities (observables) of the system, which describe other individual properties of this system. The first differential and the first partial derivatives of the fundamental thermodynamic potential with respect to the variables of state can be written as... 

Solving this system of equations, we obtain m functions of the variables of state,... 

Then the partial derivatives of the first thermodynamic potential (Eq. (39)) with respect to the variables of state can be written as... 

The thermodynamic potential of the canonical ensemble, the Helmholtz free energy, is the first thermodynamic potential g=F, which is a function of the variables of state u 1 = T, x2=V, x3=N, and x4=z. It is obtained from the fundamental thermodynamic potential / =E (the energy) by the Legendre transform (Eq. (7)), exchanging the variable of state x1 =S of the fundamental thermodynamic potential with its conjugate variable u 1 = / . In the canonical ensemble, the first partial derivatives (Eq. (1)) of the fundamental thermodynamic potential are defined asu2=-p, u3=p, and u 4 = - S. The entropy (Eq. (46)) for the Tsallis and Boltzmann-Gibbs statistics in the canonical ensemble can be rewritten as... 

In conclusion, let us summarize the main principles of the equilibrium statistical mechanics based on the generalized statistical entropy. The basic idea is that in the thermodynamic equilibrium, there exists a universal function called thermodynamic potential that completely describes the properties and states of the thermodynamic system. The fundamental thermodynamic potential, its arguments (variables of state), and its first partial derivatives with respect to the variables of state determine the complete set of physical quantities characterizing the properties of the thermodynamic system. The physical system can be prepared in many ways given by the different sets of the variables of state and their appropriate thermodynamic potentials. The first thermodynamic potential is obtained from the fundamental thermodynamic potential by the Legendre transform. The second thermodynamic potential is obtained by the substitution of one variable of state with the fundamental thermodynamic potential. Then the complete set of physical quantities and the appropriate thermodynamic potential determine the physical properties of the given system and their dependences. In the equilibrium thermodynamics, the thermodynamic potential of the physical system is given a priori, and it is a multivariate function of several variables of state. However, in the equilibrium... 

There undoubtedly exists a best (and as yet undiscovered) cubic equation of state, but best only in a coarse statistical sense. Such an equation, constructed so as to avoid manifestly unreasonable predictions of all possible thermodynamic properties of interest, probably would not produce really adequate estimates of any single property (except perhaps over very limited ranges of the variables of state). The search for this equation, if successful, would yield an expression of limited usefulness as a cubic equation. [It could, of course, provide the basis for more precise, noncubic expressions, obtained e.g. by coupling the cubic equation with deviation functions (see, e.g., Gray et al. (17) and Redlich (18)).]... 

Pressure P and temperature of the pore solution T are used as the variables of state for the system studied. A local thermal equilibrium of the system is assumed, i.e., equality of the pore solution temperature and that of rocks at each point of the continuum studied. 

The system under consideration is such that all its parts are at the same temperature. We assume that no phase transitions occur during the processes discussed in order that we may make assumptions about the continuity of the heat absorbed or liberated during these processes as a function of the variables of state. This restriction is easily removed. 

The doctrine of molecular chaos, leading to the interpretation of entropy as probability, is in a somewhat different case again. It is based, though not upon direct experiment, upon the primary hypothesis of aU chemistry, that of the existence of molecules, and upon the assumption, common to most of physics, that these particles are in motion. It is related very closely to such facts of common observation as diffusion and evaporation, and it takes its place among the major theories about the nature of things. In scope and significance it is of a different order from rather colourless assertions about the geometry of lines and surfaces constructed with the variables of state. 

Combining Eqs. (4.1) and (4.2) yields the final equation of state. It is useful to scale the variables of state and the resulting thermodynamic functions, such as the internal energy. With the explicit expressions at hand, for sufficiently long chains this lends itself to an effective principle of corresponding states, with the scaling parameters defined as follows for an 5-mer ... 

Thermal analysis is the analytical technique that establishes the experimental data for the variables of state. Details about the definition of thermal analysis are given in Fig. 2.4. The six most basic thermal analysis techniques which allow the determina-... 

The variables of state for thermomechanical analysis are deformation (strain) and stress. The SI units of deformation are based on length (meter, m), volume, (cubic meter, m ) and angle (radian, rad, or degree) as listed in Fig. 4.143 (see also Fig. 2.3). Stress is defined as force per unit area with the SI unit newton m", also called by its own name, pascal. Pa. Since these units are not quite as frequently used, some conversion factors are listed below. The stress is always defined as force per area. 

Infinitely small changes in the variables of state entail infinitely small changes in properties P ... 

In Iiq. (3) the partial derivatives indicate the changes in P with one of the variables of state, the others being unchanged dP is an exact differential as a consequence of... 

The internal variable of chemical reactions is the extent of reaction f, defined by the variables of state rii (amount of substance Y ) and supplementary conditions, which are given by the stoichiometry of the chemical reaction investigated,... 

In order that the deviation from ideality expressed as the activity coefficient may be significant, the selection of a reference for systems should be consistent. The activity coefficient depends, besides the variables of state associated with it, on the concentration units employed (mole fraction, molality etc.). In the case of polymer solutions volume fractions segment fractions and weight fractions Wt are the most convenient [15]. 

The above property of 17, which is the essential content of the first law, may also be expressed by the statement dU is an exact differential in the variables of state. The word exact merely means that the integral is independent of path.f... 

Consider a transformation at thermodynamic equilibrium for given values of the variables of state. Its affinity is null. If we vary one or more of the variables of state by an infinitesimal amount, the affinity takes on a new value + d. In order for that new state to also be a state of equilibrium of... 

Real-gas thermodynamic properties may be expressed as functions of the variables of state, according to relations which may be developed from first principles. A comprehensive list of such relations has been given by Beattie and Stockmayer. For example, the molar enthalpy H, the molar entropy S, and the molar Gibbs energy G of a real gas can be written in terms of p, T, and p (the amount density) as follows ... 

Since one cannot expect in all cases valid outlet variables of state for the elements connected in series to be already available at a specific time for determining the variables of state of a specific circuit element at a given point of time, the simulation algorithm is so often cyclically repeated for all elements until all calculated variables of state in two successive simulation runs are identical (virtual simulation). Under specific circumstances, for example in case of a feedback of elements with the delay times "zero", several truth values alternate irrespective of the number of simulation runs. In such cases the simulation is terminated after N runs (N = empirically determined value for the maximum number of virtual simulation runs) and the value X = undefined is assigned to the corresponding variables of state. 

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