Euler’s number

AH= adsorption enthalpy (negative) R = universal gas constant e = Euler s number (2.71828...)... 

By contrast, m represents the base in logarithmic and exponential relations. The irrational Euler s number e = 2.7182... is often used as the base. When this is done, we speak of natural logarithms, abbreviated to In, or (natural) exponential functions, which we call e-functions because of their relation to the number e. We then obtain ... 

Distance between two particles or surfaces Nemst potential electric field Euler s number, base of natural logarithms Elementary charge (electron charge absolute value)... 

The temperature follows an exponential function whose time constant r depends on the magnitude of the heat leakage and the apparent heat capacity of the measuring system, but it can be easily determined experimentally. If at t=0, a short heat pulse is given (e.g., by means of electric current in a resistor), we obtain in ideal conditions the temperature function shown in Figure 6.8. The time constant can be determined from the width of the function at the ordinate Tp + AT/e (e Euler s number). 

According to Figure 5-2 the relationship between logistics customer service and revenues is described by an S-shaped curve (Ballou, 2003). It is described mathematically by the sigmoid function f(p) shown in Eq. (5-6). Within this function e denotes Euler s number, s refers to the logistics customer service which is weighted to... 

There are exceptions to this simple equation that occur infrequentiy but nevertheless must be considered. A more complete relationship for the number of exchangers, E, in a network is obtained by applying Euler s network relation from graph theory (6) ... 

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 ... 

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... 

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. 

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. 

A relationship, known as Euler s formula, exists between a complex number [x + jy] (x is the real part, y is the imaginary part of the complex number (j = P )) and a sine and cosine function. Many authors and textbooks prefer the complex number notation for its compactness and convenience. By substituting the Euler equations cos(r) = d + e -")/2 and sin(r) = (d - e t )l2j in eq. (40.1), a compact complex number notation for the Fourier transform is obtained as follows ... 

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... 

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... 

With the exceptions of cavities containing square faces, all hydrate cavities (as well as clathrasil and Buckyball family cavities) follow Euler s theorem (Lyusternik, 1963) for convex polyhedra, stated as (F + V = E + 2). The number of faces (F) plus the vertices (V) equals the edges (E) plus 2. Euler s theorem is easily fulfilled in cavities having exactly 12 pentagonal faces and any number of hexagonal faces except one. 

The 512 cavity geometry (faces, F = 12 vertices, V = 20 edges, E = 30) follows Euler s theorem of F + V = E + 2. Chen (1980, p. 109) suggested that the 512 cavity seems geometrically favored by nature because it maximizes the number of bonds (30) to molecules (20) along the surface, when compared to similar cavities. Holland and Castleman (1980) studied a number of clusters and determined that the 512 cluster had a hydrogen bond advantage over ice, and that it was less strained than other clathrate clusters. 

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