Binomial expansions

Irude model only considers the dipole-dipole interaction if higher-order terms, due to e-quadrupole, quadrupole-quadrupole, etc., interactions are included as well as other i in the binomial expansion, then the energy of the Drude model is more properly an as a series expansion ... 

The intensities correspond to the coefficients of a binomial expansion (Pascal s triangle)... 

Compounds that contain chlorine, bromine, sulfur, or silicon are usually apparent from prominent peaks at masses 2, 4, 6, and so on, units larger than the nominal mass of the parent or fragment ion. Eor example, when one chlorine atom is present, the P + 2 mass peak will be about one-third the intensity of the parent peak. When one bromine atom is present, the P + 2 mass peak will be about the same intensity as the parent peak. The abundance of heavy isotopes is treated in terms of the binomial expansion (a -I- h) , where a is the relative abundance of the light isotope, b is the relative abundance of the heavy isotope, and m is the number of atoms of the particular element present in the molecule. If two bromine atoms are present, the binomial expansion is... 

The binomial expansion, Eq. (37), is particularly useful in numerical applications. For example, if a =, ... 

Ooeffidents in binomial expansion, PASCAL S TRIANGLE Ooenzyme A-dependent enzymes,... 

Equation (82) can be used in eqns. (74)—(76) to obtain expressions for P and. The summations which appear in these expressions can be written as series which are recognisable as some form of a binomial expansion. So that... 

The values of the equilibrium constants for the reactions shown in Equation (1) calculated by classical theory correspond to combinations of terms in the appropriate binomial expansion and all the equilibrium constants are given by the general equation... 

The prefactor outside the curly brackets gives the number of bonds that are strengthened by the absence of the atom at the vacancy site. The contribution inside the curly brackets gives the change in the bond energy due to the change in coordination from to ( — 1). For close-packed lattices 1 so that using the binomial expansion... 

Hence, using the binomial expansion and substituting eqn (6.6) into (6.4), we have... 

Expanding the square roots by the binomial expansion (see Appendix A) and retaining no terms higher than second order yields... 

We use the binomial expansion to find the coefficients of the Legendre polynomial of degree . For convenience, we multiply through by 2 ... 

In general, n equivalent nuclei of spin I will produce an EPR multiplet consisting of (2nl + 1) equally spaced lines. For spins of I = Vi, the intensity ratios are given by the coefficients of the binomial expansion (a + b)n but for I > /2, the formula for intensities is much more complicated. For modest values of n it is easy to find the intensity ratios by sketching a branching diagram thus... 

If an assumption of K/R S> 1 is made, the preceding equation can be rewritten by first taking two terms of a binomial expansion for it ... 

Let us assume that we have at our disposal a sample containing N-particles to be measured. Let p denote the fraction (expressed decimally) of particles falling below a stipulated size and q the fraction exceeding this size then Np denotes the number of particles less than the stipulated size and Nq those which are greater. Let this process be repeated tt-times. Then since each event is independent of the previous one, the frequency of 0, 1, 2,. . . particles being less than the stated size must be given by the binomial expansion... 

The general binomial expansion, Eq (23-1), can be fitted to most size distributions encountered in micromeritics. The special expansion Eq (23-2) can be simplified somewhat. Provided neither p nor q is small, it may be shown that when n becomes very great the frequency of particles less than a stipulated diameter d is... 

Asymmetrical Distributions—These are included in our binomial expansion Eq (23-1). However, it is more convenient to use another equation similar to Eq (23-8) for such distributions by merely changing variables. Many frequency distribution data which plot asymmetrically on arithmetic grid become symmetric if the independent variable is plotted logarithmically. When a normal distribution results by this method we may apply Eqs (23-5) and (23-6) by taking the logarithms of the variables, thus ... 

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