The exponential integral

Methods using tabulated values of the exponential integral Putting U = E/RT, eqn. (23) becomes... 

Methods using a simple approximation for the exponential integral... 

Alternative simple approaches to approximations for the exponential integral exploit the linearity of the log P(t/) against U relation over short ranges [533,542,543]... 

For the integral being evaluated here, b = 2/a0 and n = 3, so that the exponential integral can be written as... 

This integral is related to the exponential integral (see Table 14.1). It cannot be solved in closed analytical form, but it can be evaluated numerically using the E-Z Solve software the upper limit may be set equal to 10f. 

Closely related to the gamma function are the exponential-integral ei(a) defined by the equation... 

Equation (41) involves the exponential integral which can be evaluated numerically or else is available as a standard tabulated function (see, for instance, ref. 33). Therefore... 

The exponential integral in eqn. (56) is a standard tabulated function [33], so predictions of conversion can be made and are plainly a function of Tk alone. Such calculations have been performed by various authors [43—46]. Hilder [47], when repeating these calculations, found eqn. (57) to be a simple and adequate approximation to eqn. (56). 

A better estimate of the time ArDA, and hence of the period ATp, can be obtained using an integration of eqn (5.78). This involves the exponential integral, Ei(x ) where xt = (M/K) — 0h a function freely available in standard tabulations. We then find... 

Many methods have been devised for the application of Eqs. (2-16) to (2-20) to thermo-analytical data which involve various approximations of the exponential integral, p(y). Notable examples are the methods of Doyle 24,25), Horowitz and Metzger i6), Coats and Redfern 27), and Ozawa 28-30>. The method of Ozawa is frequently used. By taking Doyle s approximation for p(y) in Eqs. (2-20) and (2-21) Ozawa obtained the approximate relationship... 

If a = a, an integer, the last integral is Ea-j(ablT), E being the exponential integral... 

The inequalities for the exponential integral stated above, equation (12), then give... 

To test whether the reaction is first order, we simply fit the data to the exponential integrated first-order rate equation (Table 3.1) using a non-linear optimisation procedure and the result is shown in Fig. 3.3. The excellent fit shows that the reaction follows the mathematical model and, therefore, that the process is first order with respect to [N2O5], i.e. the rate law is r = A bsI Os]. The rate constant is also obtained in the fitting procedure, k0bs = (6.10 0.06) x 10 4 s 1. We see that, even with such a low number of experimental points, the statistical error is lower than 1%, which shows that many data points are not needed if... 

The integrals in Eqs. (B) and (C) must be evaluated through the exponential integral, E(x), a special function whose values are tabulated in handbooks and are also found from such software packages as MAPLE . The necessary equations, as found from MAPLE , are ... 

The infinite continued fraction, Eq. (281), is very convenient for the purpose of calculations so that the complex dielectric susceptibility, Eq. (282), can be readily evaluated for all values of the model parameters r, p/, and a. For a = 1, the anomalous rotational diffusion solution, Eq. (281), coincides with that of Sack [40] for normal rotational diffusion. Moreover, in a few particular cases, Eqs. (281) and (282) can be considerably simplified. In the free rotation limit (( = 0), which corresponds to the continued fraction [Eq. (281)] evaluated at x = 0, that fraction can be expressed (just as for normal rotational diffusion [40]) in terms of the exponential integral function E z) [51] so that the normalized complex susceptibility is... 

Exercise 5.6.4. Consider A —> B —> C when the first reaction is of the second order, and express the concentrations in terms of the commonly occurring functions and the exponential integral Ei y) = Jtw dtjt. (See Appendix.)... 

It is assumed that the exponential and logarithmic functions are thoroughly familiar to the reader as also are their relatives the circular and hyperbolic functions. However, the Bessel functions are introduced to the student much later and have less claim to familiarity. The exponential integral is also a function which occurs in several places and is worthy of some explanation. This appendix is therefore intended to provide a little background on the applications of these functions. 

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