Euler’s theorem, for homogeneous functions

Equation (6.27) merely says that if the independent extensive arguments of U are multiplied by A [cf. (6.25b-d)], then U itself must be multiplied by the same factor [cf. (6.25a)]. [Mathematically, the property (6.27) identifies the internal energy function (6.26) as a homogeneous function of first order, and the consequence to be derived is merely a special case of what is called Euler s theorem for homogeneous functions in your college algebra textbook.]... 

A property of great usefulness possessed by partial molar quantities derives from Euler s theorem for homogeneous functions, which states that, for a homogeneous function f n-[..Dj,...) of degree 1,... 

This relationship is known as Euler s Theorem for Homogeneous Functions of Degree One. However, in addition to the dependence on the x,- the function F may also display a dependence on parameters such as pressure P or temperature T that, of course, remains unaffected by the above manipulations. 

This will seem like a reasonable conclusion to anyone who recalls our discussion of Euler s Theorem for homogeneous functions in Chapter 2, since V is homogeneous in the first degree in the masses (or mole numbers) of the components NaCl and H2O. It is, in other words, an extensive state variable. 

Extensive thermodynamic state functions such as the internal energy U depend linearly on mass or mole numbers of each component. This claim is consistent with Euler s theorem for homogeneous functions of the first degree with respect to molar mass. If a mixture contains r components and exists as a single phase, then U exhibits r - - 2 degrees of freedom and depends on the following natural variables, all of which are extensive ... 

Mathematically minded peopie simpiy invoke Euler s theorem for homogeneous functions ( C.2.3). In plain language, this says that for any extensive... 

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... 

This relationship is known as Euler s theorem for homogeneous functions of degree one. This is a theorem of great importance in thermodynamics, as will become evident later. 

Generalization of Euler s theorem on homogeneous functions to functionals [24, 32] allows one to write for the extensive quantity z... 

The internal energy f/ is a first-power homogeneous function. According to Euler s theorem for such functions it follows from Liquations 1 and 6 that... 

Incorporating such a departure in a total differential of the internal energy of the surface phase, and using Euler s theorem on homogeneous functions, the following relation can be shown to be valid for the surface phase (Guggenheim, 1967) ... 

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. 

The same result can be obtained from an application of Euler s theorem, explained in more detail in Appendix 1. The thermodynamic quantities, Z, are homogeneous functions of degree one with respect to mole numbers.c At constant T and p, one can use Euler s theorem to write an expression for Z in terms of the mole numbers and the derivatives of Z with respect to the mole numbers. The result isd... 

Extensive thermodynamic properties at constant temperature and pressure are homogeneous functions of degree 1 of the mole numbers. From Euler s theorem [Equation (2.33)] for a homogeneous function of degree n... 

At an elementary level, one of the dogmas taught to almost every chemist is that in thermodynamics only differences bctwmi thermodjmamic potentials at various state points matter. This is essentiallj a consequence of the discussion in Section 1.3 where we emphasized that exact differentials exist for thermodynamic potentials such as 14, S, T, Q, or fl. These potentials therefore satisfy Eq. (1.18). However, one is frequently confronted with the problem of calculating absolute values of thennodynamic potentials theoretically. An example is the determination of phase equilibria, which is one of the key issues in this book cliapter. In this context a theorem associated with the Swiss mathematician Leonhard Euler is quite useful. We elaborate on Euler s theorem in Appendix A.3 where we also introduce the notion of homogeneous functions of degree k. 

Nonideal solutions may be represented by a formula which is symmetrical to Eq. (14-23). This formula is obtained by use of Euler s theorem (Appendix A, Theorem A-6) for homogeneous functions. These extensive thermodynamic... 

Since there are N ideal gas molecules, Euler s integral theorem for homogeneous thermodynamic state functions reveals that the chemical potential of a pure material is equivalent to the Gibbs free energy G T, p, N) on a per molecule basis (see equation 29-30(7) ... 

All extensive thermodynamic state functions are homogeneous to the first degree with respect to system mass. Hence, m = 1 for U, H, A, and G. Integration via Euler s theorem yields the following results, where the extensive natural variables are highlighted in bold ... 

Step 2. Use Euler s integral theorem to construct an expression for the extensive thermodynamic state function, which is homogeneous to the first degree with respect to its extensive independent variables. Since all natural variables of U are extensive, the restricted sum in Euler s theorem includes all the variables ... 

Euler s Theorem states that, for a homogeneous function/, when, A = 1, we have... 

In thermodynamics, extensive thermodynamic functions of interest to us are homogeneous functions of degree 1. The arbitrary multiplier A will be equal to the mass or moles of the system N (or N). For applying Euler s theorem in thermodynamics, consider the internal eneigy U = U S, V,N). Internal energy U is first order (m = 1) in mass and S,V,N are all proportional to mass, then we have... 

Note 3.8 (Homogeneousfunction and Euler s theorem). A function (x) is said to be the homogeneous function of degree n if, for any scalar k, we have... 

Determine which (if any) of the following functions are homogeneous with respect to all three independent variables x, y, and z. Find the degree of each homogeneous function. All letters except /, x, y, and z denote constants. For the expressions that are homogeneous, verify that they conform to Euler s theorem. 

The above parts show the minimum principle for vector processes in the frame of the generalized Onsager constitutive theory by the directions of Onsager s last dissip>ation of energy principle. We had seen above that in case of source-free balances, this principle is equivalent with the principle of minimal entropy production. The equivalence of the two theorems in the frame of the linear constitutive theory was proven by Gyarmati [2] first. Furthermore, we showed that in case when the principle of minimal entropy production is used for the determination of the possible forms of constitutive equations, the results are similar to the linear theory in the frame of the Onsager s constitutive theory, where the dissipation potentials are homogeneous Euler s functions. 

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