Hyperbolic cotangent

Expressions for the odd moments in terms of even moments, and for the even moments in terms of the odd ones, have been reported [177]. We expand the hyperbolic cotangent function according to... 

Despite its utility at room temperature, simple Marcus theory cannot explain the DeVault and Chance experiment. All Marcus reactions have a conspicuous temperature dependence except in the region close to where AG = —A. Marcus theory does not predict that a temperature-dependent reaction will shift to a temperature-independent reaction as the temperature is lowered. Hopfield proposed a quantum enhancement of Marcus theory that would permit the behavior seen in the experiment [11]. He introduced a characteristic frequency of vibration hco) that is coupled to electron transfer, in other words, a vibration that distorts the nuclei of the reactant to resemble the product state. This quantum expression includes a hyperbolic cotangent (Coth) term that resembles the Marcus expression at higher temperatures, but becomes essentially temperature independent at lower temperatures. Other quantized expressions, such as a full quantum mechanical simple harmonic oscillator behavior [12] and that of Jortner [13], give analogous temperature behavior. 

We substitute this result into Eq. (A2.64), using the hyperbolic cotangent, and obtain. 

A more general theoretical model [29], which results in Equation 11.38, provides a better description of an elastomeric stress-strain curve all the way up to fracture, by including the finite extensibilities of the chains. In this equation, 1 is the inverse Langevin function. " is a transcendental function which is defined by Equation 11.39. Coth is the hyperbolic cotangent function. The superscript of -1 represents the functional inversion (and not merely simple reciprocal) of the function in square brackets. 

The other hyperbolic trigonometric functions are the hyperbolic tangent, denoted by tanh(x) the hyperbolic cotangent, denoted by coth(x) the hyperbolic secant, denoted by sech(x) and the hyperbolic cosecant, denoted by csch(x). These functions are given by the equations... 

Mean-field theories of the surface tension of polymer solutions have been developed using the Cahn square gradient approach for interfacial properties of solutions and mixtures both for attractive and for repulsive air/liquid interfaces (Cahn and Hilliard 1958), in a way analogous to the treatment of surface segregation in polymer blends given in section 5.1. For situations in which a surface excess was formed, the volume fraction profile was a hyperbolic cotangent, whereas repulsive profiles were described by hyperbolic tangent functions. Values of the surface tension of semi-dilute solutions of polyst)n ene in toluene (a depletion layer) and polydimethyl siloxane in toluene (an attractive interface, a surface excess formed) were well described by this theory. 

Now, for small values of the argument, the hyperbolic cotangent is given by the expansion... 

Results in Eq. 8.114 are identical to those derived by Markham [68] for an analogous Hamiltonian worked out to describe spectra of electron traps in crystals. In the classical limit, jS ti(Ok, we can take a first-order expansion of the hyperbolic cotangent function coth (phcbk/l) IksT/fimk, thus obtaining the expressions... 

Second, when the chemical reaction is fast, ki is large, the hyperbolic cotangent equals one and... 

At low flow, the hyperbohc cotangent is a constant, and thus k is also constant. At high flow, the hyperbolic cotangent approaches the reciprocal of its argument, and k varies with the square root of flow. This behavior is shown in Fig. 17.1-5. 

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