Euler’s theorem of homogeneous functions

Because the internal energy U [Eq. (22)] is a homogeneous function of degree 1 of the variables f , Euler s theorem of homogeneous functions yields... 

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,... 

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]... 

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 ... 

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 usefulness of the fact that thermodynamic functions are homogeneous in the first or zeroth degree is due mainly to Euler s theorem regarding homogeneous functions. 

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-... 

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. 

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) ... 

From (2) of the preceding section we see that the chemical potentials are homogeneous functions of zero degree with respect to the masses, hence from Euler s theorem ... 

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... 

The extensive thermodynamic variables are homogeneous functions of degree one in the number of moles, and Euler s theorem can be used to relate the composition derivatives of these variables. 

In connection with the development of the thermodynamic concept of partial molar quantities, it is desirable to be familiar with a mathematical relationship known as Euler s theorem. As this theorem is stated with reference to homogeneous functions, we will consider briefly the namre of these functions. 

Euler s Theorem. The statement of the theorem can be made as follows If fix, y) is a homogeneous function of degree n, then... 

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... 

Since AGM is a state function that is extensive in nu n2, and n3, i.e., a homogeneous function of the first degree in nu n2, and n3, Euler s theorem gives... 

As the Gibbs energy is a first-order homogenous function of the extensive variables A7 and n. the application of Euler s theorem yields... 

The entropy and volume are extensive properties, as are the number of moles of each component, but the temperature and pressure are not. Consequently, we may set H as a homogenous function of the first degree in the entropy and mole numbers if the pressure is kept constant. Then, by Euler s theorem,... 

Alternatively, equation 2.2-14 can be regarded as a result of Euler s theorem. A function f(xux2,...,xN) is said to be homogeneous of degree n if... 

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... 

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