Negative exponent

Prony series is the name given to a series of exponential terms, usually with the variable in the exponent negative and diminishing in absolute value. Reference is often made to R. Prony s paper in J. Ecole Polytechnique of 1795, but the derivation of the series that bears his name from his original paper is doubtful See note by Z. Rigbi, BulL Soc. Rheol., 23 (1), 1980,14... 

The intercept on the adsorption axis, and also the value of c, diminishes as the amount of retained nonane increases (Table 4.7). The very high value of c (>10 ) for the starting material could in principle be explained by adsorption either in micropores or on active sites such as exposed Ti cations produced by dehydration but, as shown in earlier work, the latter kind of adsorption would result in isotherms of quite different shape, and can be ruled out. The negative intercept obtained with the 25°C-outgassed sample (Fig. 4.14 curve (D)) is a mathematical consequence of the reduced adsorption at low relative pressure which in expressed in the low c-value (c = 13). It is most probably accounted for by the presence of adsorbed nonane on the external surface which was not removed at 25°C but only at I50°C. (The Frenkel-Halsey-Hill exponent (p. 90) for the multilayer region of the 25°C-outgassed sample was only 1 -9 as compared with 2-61 for the standard rutile, and 2-38 for the 150°C-outgassed sample). 

A Boltzmann factor in which the energy of crystallization appears in a negative exponent. According to Eq. (4.11), this energy increases-hence the exponential decreases-with increasing r. 

The rate of a process is expressed by the derivative of a concentration (square brackets) with respect to time, d[ ]/dt. If the concentration of a reaction product is used, this quantity is positive if a reactant is used, it is negative and a minus sign must be included. Also, each derivative d[ ]/dt should be divided by the coefficient of that component in the chemical equation which describes the reaction so that a single rate is described, whichever component in the reaction is used to monitor it. A rate law describes the rate of a reaction as the product of a constant k, called the rate constant, and various concentrations, each raised to specific powers. The power of an individual concentration term in a rate law is called the order with respect to that component, and the sum of the exponents of all concentration terms gives the overall order of the reaction. Thus in the rate law Rate = k[X] [Y], the reaction is first order in X, second order in Y, and third order overall. 

The positive exponent is associated with the same species as identifies the r (i.e., for rj, Ml Pi-), while the negative exponent is associated with the other species (for ri, M2 P2 )-... 

Related to the preceding is the classification with respect to oidei. In the power law rate equation / = /cC C, the exponent to which any particular reactant concentration is raised is called the order p or q with respect to that substance, and the sum of the exponents p + q is the order of the reaction. At times the order is identical with the molecularity, but there are many reactions with experimental orders of zero or fractions or negative numbers. Complex reactions may not conform to any power law. Thus, there are reactions of ... 

Rules. Eliminate temperature terms in the denominator. (Terms with negative exponents in the power law model are considered to belong to the denominator, in the hyperbolic model. Author.)... 

The exponents a and p may be integers or fraetions and may be both positive and negative values, as well as the value of zero. In some eases, die exponents are independent of temperature. Where die... 

To avoid the cumbersome use of negative exponents to express concentrations that range over 14 orders of magnitude, S0rensen, a Danish biochemist, devised... 

If the characteristic exponents of (6-42) have negative real parts, the identically zero solution is asymptotically stable. 

Equation (1.20) is frequently used to correlate data from complex reactions. Complex reactions can give rise to rate expressions that have the form of Equation (1.20), but with fractional or even negative exponents. Complex reactions with observed orders of 1/2 or 3/2 can be explained theoretically based on mechanisms discussed in Chapter 2. Negative orders arise when a compound retards a reaction—say, by competing for active sites in a heterogeneously catalyzed reaction—or when the reaction is reversible. Observed reaction orders above 3 are occasionally reported. An example is the reaction of styrene with nitric acid, where an overall order of 4 has been observed. The likely explanation is that the acid serves both as a catalyst and as a reactant. The reaction is far from elementary. 

In this case, as follows from Equation (3.106) both roots are negative and the function. s(t) is a sum of two exponents ... 

The exponent (or power) to which the base number (10) has to be raised is the logarithm. To find the pH of a substance, the negative of the logarithm of the hydrogen ion concentration must be taken. 

Particular solutions to Eq. (131) are R(p) = e p/2, where only the negative exponent yields an acceptable function at infinity. This result suggests the substitution Rip) = e pf2Sip), which results in the differential equation... 

The meaning of 10 to a negative exponent is that the coefficient is divided by that number of 10s (multiplied together). 

Self-Similar Relaxation with Negative Exponent Value. 223... 

The liquid-solid transition for these systems seems to have the same features as for chemical gelation, namely divergence of the longest relaxation time and power law spectrum with negative exponent. 

A self-similar relaxation spectrum with a negative exponent (-n) has the property that tan S is independent of frequency. This is convenient for detecting the instant of gelation. However, it is not evident that the claim can be reversed. There might be other functions which result in a constant tan S. This will be... 

Power law relaxation is no guarantee for a gel point. It should be noted that, besides materials near LST, there exist materials which show the very simple power law relaxation behavior over quite extended time windows. Such behavior has been termed self-similar or scale invariant since it is the same at any time scale of observation (within the given time window). Self-similar relaxation has been associated with self-similar structures on the molecular and super-molecular level and, for suspensions and emulsions, on particulate level. Such self-similar relaxation is only found over a finite range of relaxation times, i.e. between a lower and an upper cut-off, and 2U. The exponent may adopt negative or positive values, however, with different consequences and... 

For negative exponent values, the symbol — n with n > 0 will be used. The self-similar spectrum has the form... 

The self-similar behavior is most obvious when it occurs in this form, i.e. if the exponent is negative and the self-similar region is extensive. G(t), G (co), G"(co), and H(X) all have power law format and they have been used interchangeably in the literature. Less obvious is the self-similar behavior for positive exponent values. 

A wide variety of polymeric materials exhibit self-similar relaxation behavior with positive or negative relaxation exponents. Positive exponents are only found with highly entangled chains if the chains are linear, flexible, and of uniform length [61] the power law spectrum here describes the relaxation behavior in the entanglement and flow region. 

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