A Simple Calculation

For illustration we show the calculation of the C, (electronic spin) term for the simplest case, the hydrogen ground state. Since this is a light element, Eq. (11) is a very good approximation. 

In order to calculate the mixing fraction cpNc, one resorts to first-order perturbation theory. We will proceed with a first-order calculation for the ground (Is) state of hydrogen. Although all p states are mixed into the ground state by //pnc we will calculate only the contribution of the nearest opposite parity state, the 2p. First-order perturbation theory then takes the form 

We substitute VpNc from Eq. (11), and integrate out the delta functions, leaving 

The values for the wave functions at the origin are known exactly for hydrogen and can be substituted in. The 2p state can have total angular momentum J = 3/2 and J = 1/2. Only the J = 1/2 term contributes, as can be seen by evaluating directly the matrix elements of VpNc- This, of course, is expected because our potential is a (pseudo)-scalar. 

Note the factor of i in this expression. It will always appear in matrix elements of //pnc when the interaction is of the form of Eq. (2) or its special cases, Eq. (8) or Eq. (11), and when the conventional choice of wave function phase is made as was done here. The relative phase of matrix elements is independent of phase convention, and plays an important role in the PNC interference effects to be discussed later. This point was illustrated in the qualitative discussion in Section 1.3. 

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Surface Area. The most important features influencing the performance of carbon blacks are aggregate size and surface area. Surface area is measured by gas- and Hquid-phase adsorption techniques, and depends on the amount of adsorbate required to form a surface monolayer. If the area occupied by a single-adsorbate molecule is known, a simple calculation will yield the surface area. A low temperature nitrogen absorption method, based on the original method of Bmnauer, Emmett, and Teller (BET) (30), has been adopted by ASTM as standard method D3037-86 (2). 

This is a simple calculation to determine the maximum symmetrical fault level of a system, to select the type of equipment, devices and bus system etc. But to decide on a realistic protective scheme, the asymmetrical value of the fault current must be estimated by including all the likely impedances of the circuit. 

In the last chapter we examined data for the yield strengths exhibited by materials. But what would we expect From our understanding of the structure of solids and the stiffness of the bonds between the atoms, can we estimate what the yield strength should be A simple calculation (given in the next section) overestimates it grossly. This is because real crystals contain defects, dislocations, which move easily. When they move, the crystal deforms the stress needed to move them is the yield strength. Dislocations are the carriers of deformation, much as electrons are the carriers of charge. 

An example that shows that the cohesive strength of a material is less than that of the adhesional strength of the interface is that of the nominal 50,000 mile steel belted radial tire. It is a simple calculation to show that, on average, a tire leaves a monolayer of rubber particles on the road every time it makes a rotation. In essence, the strength of the adhesional bonding between the road and the tire is greater than that of the rubber within the tire. 

From a human reliability perspective, a number of interesting points arise from this example. A simple calculation shows that the frequency of a major release (3.2 x lO"" per year) is dominated by human errors. The major contribution to this frequency is the frequency of a spill during truck unloading (3 X10" per year). An examination of the fault tree for this event shows that this frequency is dominated by event B15 Insufficient volume in tank to imload truck, and B16 Failure of, or ignoring LIA-1. Of these events, B15 could be due to a prior human error, and B16 would be a combination of instrument failure and human error. (Note however, that we are not necessarily assigning the causes of the errors solely to the operator. The role of management influences on error will be discussed later.) Apart from the dominant sequence discussed above, human-caused failures are likely to occur throughout the fault tree. It is usually the case that human error dominates a risk assessment, if it is properly considered in the analysis. This is illustrated in Bellamy et al. (1986) with an example from the analysis of an offshore lifeboat system. 

In the case of a simple calculation shows that the dissociation energy... 

The complexity of the plant and the type and number of sensitivity tests to be carried out will determine whether a formal model or a simple calculation is needed. 

A simple calculation based on the solubility product of ferrous hydroxide and assuming an interfacial pH of 9 (due to the alkalisation of the cathodic surface by reaction ) shows that, according to the Nernst equation, at -0-85 V (vs. CU/CUSO4) the ferrous ion concentration then present is sufficient to permit deposition hydroxide ion. It appears that the ferrous hydroxide formed may be protective and that the practical protection potential ( —0-85 V), as opposed to the theoretical protection potential (E, = -0-93 V), is governed by the thermodynamics of precipitation and not those of dissolution. 

Before we can decide whether there is a large or a small difference between the unitary terms of the Cl" ions and the 1 ion, the difference in the cratic term must be calculated. The question is whether the difference of 41.0 e.u. between 116.8 and 75.8 merely represents the difference between the cratic terms, or whether it includes as well a difference between the uuitary terms of the I" ion and the Cl" ion. For ions provided by any two such sparingly soluble substances this question can be settled at once by a simple calculation. 

To conclude Chapter 6, let us connect it with Chapter 4 through a simple calculation involving the intensity of cobalt Ka as measured by counting 1 square centimeter over a 1-second interval. The measured intensity for massive cobalt corresponds to about 500,000 counts. By means of Equations 6-7 and 6-9, we calculate from this number a value of 23 counts for AI (Equation 6-9) for a monolayer of cobalt atoms. The measured value from Table 4-4 is 15 counts. The good agreement shows that absorption effects may be calculated with confidence. 

Another important result states that the characteristic function of a sum of statistically independent random variables is the product of the characteristic functions of the individual summands. The reader should compare this statement with the deceptively similar sounding one made on page 154, and carefully note the difference between the two. The proof of this statement is a simple calculation... 

Degrees of polymerization can be calculated from quantitative 13C NMR data by considering the number of substituted (reacted) relative to unsubstituted (not yet reacted) ortho and para phenolic carbons where [5] is the sum of substituted ortho and para carbons and [5] + [f/ is the total ortho and para carbons. The fraction of reacted ortho and para sites is denoted by fs [Eq. (7.2)]. Thus, the number-average number of phenol units per chain Or) can be calculated using Eq. (7.3). This leads to a simple calculation of Mn = x x 106 — 14 ... 

Theories neglect that catalysts usually have limited turnover numbers due to destructive side reactions. This may not be so obvious in analytical experiments but it has severe consequences for large scale applications. A simple calculation can illustrate this problem if a redox polymer with a monomer molecular weight of 400 Da and a density of 1 g cm " is considered with all redox centers addressable from the electrode and accessible to the substrate with a turnover number of 1000, then, to react 1 nunol of substrate at a 1 cm electrode surface, at least 5 pmol of active catalyst centers corresponding to 2 mg of polymer, or a dry film thickness of 20 pm are required. This is 20 times more than the calculated optimum film thickness for rather favorable conditions... 

We will illustrate the necessity of including solute from CCN by a simple calculation, recalling that pH = 5.6 is the supposed equilibrium value for water in contact with 300 ppm of CO2. (That calculation will appear later.) In clean, marine air, the concentration of submicrometer aerosol particles (by far the most numerous) is small, say 0.25 pg m . It is known from measurements that the molecular form is often NH4HSO4, and we assume it is all dissolved in 0.125 g/m of liquid water in a cloud - which is typical for fair-weather marine clouds. Thus the average concentration of sulfate ion [SO4 ], mol/L, is... 

Statistical and algebraic methods, too, can be classed as either rugged or not they are rugged when algorithms are chosen that on repetition of the experiment do not get derailed by the random analytical error inherent in every measurement,i° 433 is, when similar coefficients are found for the mathematical model, and equivalent conclusions are drawn. Obviously, the choice of the fitted model plays a pivotal role. If a model is to be fitted by means of an iterative algorithm, the initial guess for the coefficients should not be too critical. In a simple calculation a combination of numbers and truncation errors might lead to a division by zero and crash the computer. If the data evaluation scheme is such that errors of this type could occur, the validation plan must make provisions to test this aspect. 

The addition polymerization of carbon-carbon double bonds is driven by the thermodynamic favorability of forming two a bonds in the polymer from a 0+71 bond in the monomer. A simple calculation based upon theoretically... 

In order to do this, a simple calculator may be used to aid in the mathematical manipulations. A desktop computer into which the data is entered may be used to generate a standard curve automatically along with the unknown values which are automatically read from the curve. If a paper tape print out... 

It has been found that a plot of pg / Vg [1-pgl vs pg is hnear for the pressure range of 0.05 to 0.4, with aslope of (C - 1) / (Vmono C ) and intercept of 1/ (Vmono C ). Let us now do a simple calculation using BEH data obtained. Suppose we have a 20 gm. sample having a density of 2.0. We measure the surface area as 6 m. From the area of a sphere. A = r d2, and the volume of a sphere, V = 4/37tD3., we find the total volume of n spheres to be 10 cc, i.e.- n 4/3 r D = 10. The surface area of n 7rD2 spheres is 6 m3. The total number of spheres present, n, is the same in both formulas. Therefore, by substitution, we find D= 10 p. If we obtain a particle diameter by some other method and find that it is mueh smaller than that of the BET method, we infer that the peatieles are porous. We thus speak of the porosity and need to correct for the pore surface area if we are to meike a reasonable estimate of the true diameter by the BET method. 

The current amplitude cannot be increased at free will, because the clamp voltage will usually not exceed + 100 mV. A simple calculation reveals that, given the limitations above, channels need to have a conductance of 10 pS, and time constants in the millisecond range to be detected. Any channel smaller or faster than this will escape detection in this analysis. 

Arrhenius s theory contains the idea that the effects of ions on coUigative properties are additive (i.e., that interactions between the ions are absent). However, a simple calculation shows that at distances of less than 10 to 20 nm between ions, marked... 

Goodness-of-fit tests may be a simple calculation of the sum of squared residuals for each organ in the model [26] or calculation of a log likelihood function [60], In the former case,... 

Where pathways have their origin in the same medium or have a common point of intersection, a simple calculation is used to adjust the concentration at the origin or the intersection such that SPPPLVs taken together provide the target organism, usually humans, with an exposure providing exactly Up. Thus, for SPPPLVs via three pathways from the same source ... 

Once the FE, or percent dispersion, is known, a simple calculation converts dispersion to average particle size. However, this calculation is based on the... 

As an attractive statement of such a two-sided approach, consider cutting off the integral of (9.18) at a value e = e above which the determination of (s) is not sufficiently accurate. For e > e, consider exploiting a s) and (9.17) to characterize ,y(e) in that high-interaction-energy regime. A simple calculation then shows that... 

Thus, considering equation 4-2, we note that the matrix expression looks like a simple algebraic expression relating the product of two variables to a third variable, even though in this case the variables in question are entire matrices. In equation 4-2, the matrix /f represents the unknown quantities in the original simultaneous equations. If equation 4-2 were a simple algebraic equation, clearly the solution would be to divide both sides of this equation by A, which would result in the equation B = C/A. Since A and C both represent known quantities, a simple calculation would give the solution for the unknown B. 

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