Euler s theorem on homogeneous functions

Since the dipole term is homogeneous of degree 1 and the quadrupole term is homogeneous of degree 2 in the r, and R, from Euler s theorem on homogeneous functions we have [9a, 41]... 

Euler s theorem on homogeneous functions states that, if/(xi,..., Xy) is homogeneous of degree n, then... 

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

The equation is an example of the result of applying Euler s theorem on homogeneous functions to V treated as a function of n and / b-... 

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) ... 

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. 

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 ... 

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... 

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 ... 

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. 

Here T denotes a homogeneous function of the wth degree of the qk s, which may, moreover, depend on the qk s. By Euler s 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) ... 

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