Euler s theorem

From Euler s theorem, one can derive the following simple relation between the number and type of cycles n, (where the subscript / stands for the number of sides to the ring) necessary to close the hexagonal network of a graphene sheet ... 

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... 

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 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. 

Euler s theorem 209. 612 europium hydroxide, heat capacity 584-5 eutectic point in solid 4- liquid phase equilibria 421... 

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... 

Recall the elegant Euler s theorem that states that for a 2-d structure the number of nodes minus the number of edges (struts) plus the number of faces = 1, or — S +F = 1. This topology theorem is easy to prove. An examination of Table 3 shows some examples of this theorem and indicates how to prove it. 

Next, note that if we take a 2-dimensional faceted structure and fold it into 3-dimensions, we have added a face. Thus we have proven Euler s theorem in 3-dimensions N — S + F = 2. 

Combining this with Euler s theorem we obtain 3N—S 6, or S = 3N—6, our formula for determinate structures in 3-dimensions. 

Euler s Theorem Rotation is the general movement of a rigid body in space with a single point fixed. 

It is left as an exercise to the reader to show that etx = cosx + isinx, which is known as Euler s theorem. In the present notation... 

In the above MILP-optimization problem, Euler s theorem for the generation of stable and feasible molecular structures (fully connected graphs) needs to be added as a condition in order to ensure the generation of chemically feasible molecules. This condition is mathematically formulated as,... 

Most branches of theoretical science can be expounded at various levels of abstraction. The most elegant and formal approach to thermodynamics, that of Caratheodory [1], depends on a familiarity with a special type of differential equation (Pfaff equation) with which the usual student of chemistry is unacquainted. However, an introductory presentation of thermodynamics follows best along historical lines of development, for which only the elementary principles of calculus are necessary. We follow this approach here. Nevertheless, we also discuss exact differentials and Euler s theorem, because many concepts and derivations can be presented in a more satisfying and precise manner with their use. 

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 one example of the application of Euler s theorem, we refer again to the volume of a two-component system. Evidently the total volume is a function of the number of moles of each component ... 

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... 

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.]... 

An important geometrical growth parameter is the average number of sides per grain, (N), in the ensemble, which can be determined with Euler s theorem, which states that... 

The last member of Equation (2) shows that n, is the partial molar quantity associated with the Gibbs free energy, G. Euler s theorem gives then... 

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