Euler’s integral theorem

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) ... 

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. 

EULER S INTEGRAL THEOREM FOR HOMOGENEOUS EUNCTIONS OF ORDER m... 

If one equates (29-27) and (29-28) and lets A approach unity and y y, then Euler s integral theorem provides the prescription to calculate any homogeneous function in terms of its extensive variables and the associated slopes ... 

One of the most important consequences of Euler s integral theorem, as applied to stability criteria and phase separation, is the expansion of the extensive Gibbs free energy of mixing for a multicomponent mixture in terms of partial molar properties. This result is employed to analyze chemical stability of a binary mixture. 

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 ... 

Intercepts and Common Tangents to Agmixing ts. Composition in Binary Mixtures. Euler s integral theorem and the Gibbs-Duhem equation provide the tools to obtain expressions for Agmixing and (9 Agmixing/9y2)r,/) in binary mixtures. This information allows one to evaluate the tangent at any mixture composition via the point-slope formula. For example, if i i = and p,2 = M2 when the mole fraction of component 2 is y, then equations (29-73) and (29-76) yield ... 

Chemical Stability. Condition (29-123c) requires that (gmixture)22 > 0. Euler s integral theorem for a multicomponent mixture yields (i.e., see equation 29-69) ... 

By either a direct integration in which Z is held constant, or by using Euler s theorem, we have accomplished the integration of equation (5.16), and are now prepared to understand the physical significance of the partial molar property. For a one-component system, Z = nZ, , where Zm is the molar property. Thus, Zm is the contribution to Z for a mole of substance, and the total Z is the molar Zm multiplied by the number of moles. For a two-component system, equation (5.17) gives... 

Euler s theorem 612 exact differentials 604-5 extensive variables 598 graphical integrations 613-15 Simpson s rule 614-15 trapezoidal rule 613-14 inexact differentials 604-5 intensive variables 598 line integrals 605-8... 

These expressions are formally exact and the first equality in Eq. (123) comes from Euler s theorem stating that the AT potential u3(rn, r23) is a homogeneous function of order -9 of the variables r12, r13, and r23. Note that Eq. (123) is very convenient to realize the thermodynamic consistency of the integral equation, which is based on the equality between both expressions of the isothermal compressibility stemmed, respectively, from the virial pressure, It = 2 (dp/dE).,., and from the long-wavelength limit S 0) of the structure factor, %T = p[.S (0)/p]. The integral in Eq. (123) explicitly contains the tripledipole interaction and the triplet correlation function g (r12, r13, r23) that is unknown and, according to Kirkwood [86], has to be approximated by the superposition approximation, with the result... 

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 ... 

Many extensive thermodynamic properties are homogeneous in the mole numbers. This fact can be exploited through Euler s theorem to immediately integrate total differentials of such extensive properties. A function f x, y) is said to be homogeneous of order n if... 

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