Functions homogeneous

Functions are said to be homogeneous and of degree n if we can multiply every variable in the function by a constant, then factor out the constant. For example, f(x, y, z) is homogeneous in degree n if 

Clearly, homogeneity is concerned with the exponents of the variables in a function. Homogeneity is not always quite so obvious, however. For example 

Some functions are homogeneous in only some of the variables and not in others, which is the usual case in thermodynamics. Thus 

V being the total volume, n the number of moles of gas, R the gas constant, T the absolute temperature, and F tbe pressure. Clearly V is homogeneous first degree in (n, T, P), homogeneous first degree in n only or in T only, and of degree — 1 in P only. The derivative 

Intensive properties were defined as those that do not depend on the mass of the system considered. They are thus homogeneous in the zeroth degree in the masses of the components. For example, for the density 

A homogeneous function F of the first order in any number of the variables x, y, z. is defined by  

2 Thermodynamic Properties from Diiferentiation of Fundamental Equations 

Avoiding integration is often advantageous in theoretical applications of thermodynamics because there are no constants of integration. On the other hand, very often the derivatives needed involve variables that are difficult to measure experimentally. Even with the Gibbs function surface, which is closely linked to the convenient experimental variables of temperature, pressure and 

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. 

As the net result of multiplying each independent variable by the parameter X merely has been to multiply the function by X, the function is called homogeneous. Because the exponent of X in the result is 2, the function is of the second degree. 

Now we turn to an example of experimental significance. If we mix certain quantities of benzene and toluene, which form an ideal solution, the total volume V will be given by the expression 

The volume function then is homogeneous of the first degree, because the parameter X, which factors out, occurs to the first power. Although an ideal solution has been used in this illustration. Equation (2.31) is true of all solutions. However, for nonideal solutions, the partial molar volume must be used instead of molar volumes of the pure components (see Chapter 9). 

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 

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The concept of homogeneity naturally extends to functions of more than one variable. For example, a generalized homogeneous function of two variables, f(x,y), can be written in the form... 

Since NK must be a linear and homogeneous function of the numbers of cavities of different types, it follows directly from the previous equation that... 

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... 

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. 

Like the entropy expression the fundamental relation as a function of U is also a first-order homogeneous function, such that for constant A,... 

Much of the early work which would lead to the identification of proteins as defined chemical entities started from observations on enzymes, either those involved in fermentation or on the characterization of components in gastric secretions which powerfully catalyzed the hydrolysis of different foodstuffs. As well as the digestive enzymes, a number of relatively pure proteins could be isolated from natural sources where they made up the major component (Table 1). Because of the importance and difficulty of isolating pure proteins and demonstrating their homogeneity, functionally active and relatively abundant... 

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... 

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 ... 

In principle, eqns. (67)—(71) can be solved and the concentrations of each of the species can be determined as functions of time. Although —rg can be obtained as a homogeneous function of C-, it takes the form... 

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-... 

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.]... 

Consider a crystal which is in equilibrium having n chemical components (k = 1,2,..., ). We can define (at any given P and T) a Gibbs function, G, as a homogeneous function that is first order in the amount of components... 

It is always convenient to use intensive thermodynamic variables for the formulation of changes in energetic state functions such as the Gibbs energy G. Since G is a first order homogeneous function in the extensive variables V, S, and rtk, it follows that [H. Schmalzried, A.D. Pelton (1973)]... 

By expanding the Helmholtz free energy F at constant T in an arithmetic series in terms of ujk, we see that the linear terms vanish in view of the equilibrium condition (Euler relation for homogeneous functions of second order, F is given as... 

So far no approximations have been made. In scaling theory the assumption is now made that f(u) may be a homogeneous function, i.e. 

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