Euler theorem equilibrium

The formalism of the statistical mechanics agrees with the requirements of the equilibrium thermodynamics if the thermodynamic potential, which contains all information about the physical system, in the thermodynamic limit is a homogeneous function of the first order with respect to the extensive variables of state of the system [14, 6-7]. It was proved that for the Tsallis and Boltzmann-Gibbs statistics [6, 7], the Renyi statistics [10], and the incomplete nonextensive statistics [12], this property of thermodynamic potential provides the zeroth law of thermodynamics, the principle of additivity, the Euler theorem, and the Gibbs-Duhem relation if the entropic index z is an extensive variable of state. The scaling properties of the entropic index z and its relation to the thermodynamic limit for the Tsallis statistics were first discussed in the papers [16,17],... 

Let Nb and Nb be the equilibrium numbers of L- and -cules, respectively, and Nyy be the total number of water molecules in the system, Njy = Nb + Nb> Viewing the system as a mixture of two components, we write the volume of the system, using the Euler theorem, as... 

The substitution moment Isy differs from the equilibrium moment ley by first order terms of the expansion. Since g is a homogeneous function of degree one-half of the atomic masses [24], the second term on the right-hand side of Eq. 91a is, by Euler s theorem ... 

Systems at equilibrium have measurable properties. A property of a system is any quantity that has a fixed and invariable value in a system at equilibrium. If the system changes from one equilibrium state to another, the properties therefore have changes that depend only on the two states chosen, and not on the manner in which the system changed from one to the other. This dependence of properties on equilibrium states and not on processes is reflected in the alternative name for them, state variables. Recall from the discussion of Euler s theorem in Chapter 2 that extensive variables are proportional to the quantity of matter being considered—for example, volume and (total) heat capacity. Intensive variables are independent of quantity, and include con-... 

Stability criteria are discussed within the framework of equilibrium thermodynamics. Preliminary information about state functions, Legendre transformations, natural variables for the appropriate thermodynamic potentials, Euler s integral theorem for homogeneous functions, the Gibbs-Duhem equation, and the method of Jacobians is required to make this chapter self-contained. Thermal, mechanical, and chemical stability constitute complete thermodynamic stability. Each type of stability is discussed empirically in terms of a unique thermodynamic state function. The rigorous approach to stability, which invokes energy minimization, confirms the empirical results and reveals that r - -1 conditions must be satisfied if an r-component mixture is homogeneous and does not separate into more than one phase. 

But we don t have to use Euler s theorem. We can simply expand our definition of G, which so far is restricted to closed (constant composition) systems. If we exclude chemical work, which means we deal only with systems at complete stable equilibrium, we know from Equation (4.65)... 

P and JJL in Equation 2.92 are the equilibrium pressure and chemical potentials, respectively. Applying Euler s theorem for homogeneous functions, we obtain from Equation 2.92... 

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