Exponential functions Excel

This is an indication of the collective nature of the effect. Although collisions between hard spheres are instantaneous the model itself is not binary. Very careful analysis of the free-path distribution has been undertaken in an excellent old work [74], It showed quite definite although small deviations from Poissonian statistics not only in solids, but also in a liquid hard-sphere system. The mean free-path X is used as a scaling length to make a dimensionless free-path distribution, Xp, as a function of a free-path length r/X. In the zero-density limit this is an ideal exponential function (Ap)o- In a one-dimensional system this is an exact result, i.e., Xp/(Xp)0 = 1 at any density. In two dimensions the dense-fluid scaled free-path distributions agree quite well with each other, but not so well with the zero-density scaled distribution, which is represented by a horizontal line (Fig. 1.21(a)). The maximum deviation is about... 

Subsequent work by Johansson and Lofroth [183] compared this result with those obtained from Brownian dynamics simulation of hard-sphere diffusion in polymer networks of wormlike chains. They concluded that their theory gave excellent agreement for small particles. For larger particles, the theory predicted a faster diffusion than was observed. They have also compared the diffusion coefficients from Eq. (73) to the experimental values [182] for diffusion of poly(ethylene glycol) in k-carrageenan gels and solutions. It was found that their theory can successfully predict the diffusion of solutes in both flexible and stiff polymer systems. Equation (73) is an example of the so-called stretched exponential function discussed further later. 

It has been known for 40 years, since the initial PGSE experiment of Stejskal and Tanner [20], that NMR is an excellent tool for measuring the statistical displacements of molecules with time, usually called diffusion. The exponential attenuation a of the PGSE signal as a function of time between the PGSE pulses is given by [21]... 

A second important application of CMD has been to study the dynamics of the hydrated proton. This study involved extensive CMD simulations to determine the proton transport rate in on our Multi-State Empirical Valence Bond (MS-EVB) model for the hydrated proton. = Shown in Fig. 4 are results for the population correlation function, (n(t)n(O)), for the Eigen cation, HsO, in liquid water. Also shown is the correlation function for D3O+ in heavy water. It should be noted that the population correlation function is expected to decay exponentially at long times, the rate of which reflects the excess proton transport rate. The straight line fits (dotted lines) to the semi-log plots of the correlation functions give this rate. For the normal water case, the CMD simulation using the MS-EVB model yields excellent agreement with the experimental proton hopping... 

Figure 2.4 Graph created with Microsoft Excel showing the concentrations of DDT in trout from Lake Michigan (see the above table) as a function of time and showing a fitted exponential line (using the TrendLine feature). The negative exponent is the rate constant.

The parameter C is often considered independent of temperature and it has been shown to take a universal value of 0.0894 for long-chain hydrocarbons (Cutler et al., 1958). Nanda and Simha (1964) obtained excellent correlations for a number of polymers using this value of C. This value was therefore retained in the Handbook. The parameter B has the dimensions of pressure, but it is a function only of temperature. Different forms have been used in the literature to describe this temperature dependence, but the exponential form has proven to be a reliable representation. Thus B(T) is given by... 

Kinetic traces for reactions with L-Met gave excellent fits to a double-exponential function. The obtained constants, kobsi and kobs2. were plotted against the concentration of the entering L-Met molecule. In the case of kobsi, a linear... 

Here the term 1 — X, the fraction of feed propane decomposed, is a measure of the gross amount of product which has been formed. A least squares fit of the data of Table I gave A<> = 2.761 X 1017 sec 1 B = 40.886 E = 77.910 kcal/mol H = 2170 with R2 0.99. This gives X as a function of t to about 0.013, and must be considered an excellent fit. It should be noted that Saito and co-workers (8-13) have used essentially the same equation to fit their rate data for a number of paraffins. Saito s equation does not contain an exponential in the product inhibition term of the denominator. However, it is doubtful whether any available data justify calculating such a term (assumption that H = 0 makes the fit trivially worse). 

Choose the Type tab and then choose the appropriate fitting function from the gallery of functional forms. (Depending on the data in the series, the exponential, power or logarithmic choices may not be available.) If you choose the polynomial form you can select the order of the polynomial by using the spinner. If you choose 3, for example. Excel will fit a polynomial of order three... 

Sokhan and Quirke434 devise a method of computing interfacial friction and the Maxwell slip coefficients (a) by equilibrium MD, in which a is computed from the relaxation time, which itself can be estimated by an exponential fit to the collective velocity autocorrelation function. Their equilibrium method is compared to an NEMD method devised by the authors previously and excellent agreement is found. They study the density dependence of the slip... 

The fluorescence decays of the locally excited state can be analysed as a two exponential decay function in the temperature domain between 298 K and 233 K. Below this temperature the decays are one exponential. Furthermore excellent agreement between the decay parameters measured in the locally excited and excimer region of the emission spectrum is obtained. 

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