Homogeneous function, degree

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 ... 

If all the n s are increased in the same ratio, U and V are also increased in the same ratio, and are therefore homogeneous functions of the first degree in those variables. Ujn0 and V/ o... 

A classification of dispersed systems on this basis has been worked out by Pawlow (30) (1910), who introduces a new variable called the concentration of the dispersed phase, i.e., the ratio of the masses of the two constituents of an emulsion, etc. When the dispersed phase is finely divided the thermodynamic potential is a homogeneous function of zero degree in respect of this concentration. 

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... 

Euler s theorem states that if a function/(.y. r. r.) is homogeneous of degree n, then... 

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. 

For the special case for which n = 2, it can be shown that the linearization method defined above becomes identical to the Newton-Raphson method. The result may be generalized to apply to any homogeneous function of degree n. 

Proceeding to a general dehnition, we can say that a function, /(x, y, z,. .) is homogeneous of degree n if, upon replacement of each independent variable by an arbitrary parameter X times the variable, the function is multiphed by X", that is, if... 

Examine the following functions for homogeneity and degree of homogeneity ... 

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... 

Although the function / is a homogeneous function of the mole numbers of degree 1, the partial molar quantities, and are homogeneous functions of degree 0 that is, the partial molar quantities are intensive variables. This statement can be proved by the following procedure. Let us differentiate both sides of Equation (2.32) with respect to x ... 

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... 

Using Eq. (31) in Eq. (35) and using Euler theorem again, since is a homogeneous function of the first degree in Nx and N2, gives... 

Exercise 1.9 Find a homogeneous function of degree /2 on pind a ho-... 

It is of the utmost importance to recognize that balances can only be made on extensive variables or quantities. If you double the system, an extensive measure is doubled, whereas an intensive measure remains the same. Thus the mass of two identical bricks is twice the mass of one, but the density, or mass per unit volume, remains the same in the duplicated system because we have doubled both the mass and the volume. The first (mass) behaves like a homogeneous function of degree one, the second (volume) of degree zero. Thus in the simple example used in Example 1, we did not make our balance on the concentration, moles per unit volume = c, but on the amount, moles - Vc. 

But hk is an intensive variable, i.e., a homogeneous function of degree zero, and this implies (by a well-known result attributed to Euler, quoted previously) that... 

The energy of a single-phase system is a homogenous function of the first degree in the entropy, volume, and the number of moles of each component. Thus, by Euler s theorem2... 

The extensive properties are homogenous functions of the first degree in the mole numbers at constant temperature and pressure. Then, the partial molar quantities are homogenous functions of zeroth degree in the mole numbers at constant temperature and pressure that is, they are functions of the composition. We use the mole fractions here, but we could use the molality, mole ratio, or any other composition variable that is zeroth degree in the mole numbers. For mole fractions, Xk = Xk(T, P, x) and the differential of Xk may be written as... 

We have already said that the extensive quantities are homogenous functions of the first degree in the mole numbers, and consequently the partial molar quantities are homogenous functions of zeroth degree in the mole... 

This equation is consistent with the concept that the Gibbs energy is a homogenous function of the first degree in the mole numbers. Comparison of this equation with Equation (7.49) together with the relation that Pt = Py( shows that... 

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... 

The internal energy is homogeneous of degree 1 in terms of extensive thermodynamic properties, and so equation 2.2-8 leads to equation 2.2-14. All extensive variables are homogeneous functions of the first degree of other extensive properties. All intensive properties are homogeneous functions of the zeroth degree of the extensive properties. 

Among the four principal thermodynamic energy functions, U, H, F, and G, the free enthalpy G (Gibbs energy) associated with the intensive variables T and p is a homogeneous function of the first degree with respect to the extensive independent variable of the number of moles n. of the constituent substances present in the system considered, so that it can be expressed as the sum of the chemical potentials of all constituent substances at constant temperature and pressure. 

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