Correction equations

FIGURE 14.2 Relative isotopic abundances of the naturally occurring elements, showing all the potential isobaric interferences. (From UIPAC Isotopic Composition of the Elements, Pure and Applied Chemistry 75[6], 683-799, 2003.) 

Practical Guide to ICP-MS A Tutorial for Beginners, Second Edition 

An interference correction for Cd would then be entered in the software as 

This is a relatively simple example, but explains the basic principles of the process. In practice, especially in spectrally complex samples, corrections often have to be made to the isotope being used for the correction in addition to the analyte mass, which makes the mathematical equation far more complex. 

This approach can also be used for some less severe polyatomic-type spectral interferences. For example, in the determination of V at mass 51 in diluted brine (typically 1000 ppm NaCl), there is a substantial spectral interference from at mass 51. By measuring the intensity of the C 10+ at mass 53, which is free of any interference, a correction can be applied in a similar way to the previous example. 

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For precise measurements, diere is a slight correction for the effect of the slightly different pressure on the chemical potentials of the solid or of the components of the solution. More important, corrections must be made for the non-ideality of the pure gas and of the gaseous mixture. With these corrections, equation (A2.1.60) can be verified within experimental error. 

There is some disagreement on the correct equation for describing the relationship between plate height and mobile-phase velocity. In addition to the van Deemter equation (equation 12.28), another equation is that proposed by Hawkes... 

For gas-absorption systems in which the inlet gas is concentrated, the correct equation is... 

The ambiguity of the total pressure effect can be seen by a comparison of the gas-phase- and liquid-phase-controlled cases when the gas phase controls, the liquid-phase resistance is negligible and Kg( = K npf is independent of the total pressure. For this case the coefficient K g< is inversely proportional to the total system pressure as shown in Eq. (14-66). On the other hand, when the liquid phase controls, the correct equation is... 

This equation, evaluated at = 1, is plotted for r = 2 and r = 0.5 in Fig. 16-20. Clearly, the solution for r= 0.5 is not physically correct. Equation (16-132), with dZi/d = Xi/, is apphed to this case to give the shock indicated by the dashed line. Alternatively, we could have obtained bed profiles by evaluating equations at Ti = const. 

The basically correct equation appears to be that of Giddings but, over the range of mobile phase velocities normally employed i.e., velocities in the neighborhood of the optimum velocity), the Van Deem ter equation is the simplest and most appropriate to use. 

The criterion for the validity of Equation 8-141 is Npg 1.0. A rough rule-of-thumb is Npg > 10. If this condition is not satisfied, the correct equation depends on the boundary conditions at the inlet and outlet. A procedure for determining dispersion coefficient Dg [ is as follows ... 

In order to use the data in systems handling liquids other than water correction equations and charts are used [66]. The charts are more convenient to use and are presented in Figures 9-46 A, B, C, D. First, determine the total or static hold-ups for water at 20°C second, determine separately the correction for viscosity, density, and surface tension third, multiply the water hold-up by each... 

The above corrections assume all of the molecules are in the ground electronic state so that the molecular parameters are the same for all molecules. The correction equations can give results in error when low-lying excited electronic states are present and significantly populated. An example is NO, which has an electronic state only 121.1 cm 1 above the ground state. (See Table 10.3 for this and other examples.)... 

Steady-state. An erroneous rate law is shown below for the reaction scheme believed to represent the reaction between Fe3+ and I-, in that an extraneous denominator term appears. In the scheme shown, I2 and Fel2+ obey the steady-state approximation. Show what the incorrect part of the expression is. Suggest a simple derivation of the correct equation that avoids extensive algebraic manipulations. 

An attempt to derive the isokinetic relationship still more generally considering a temperature-dependent a (2) is not quite correct. Equation (72), corresponding to eq. (4), p. 317, of (2), then has a solution... 

By solving the pressure and velocity correction equations, the corresponding correction terms and thus the new velocity field, which by construction fulfils the... 

In practice, the full versions of the pressure and velocity correction equation are rarely considered. In order to determine the velocity correction, the sum over neighboring nodes ... 

Chlorodiphenylphosphine 488 reacts with a-sulphinyl carbanions to give a-sulphinylphosphines 489 which undergo ready isomerization to a-sulphenylphosphine oxides 4W (equation 295). The report of Almog and Weissman that a-sulphinyl carbanions react with phosphorochloridates 491 to give a-phosphoryl sulphoxides 492 calls for correction (equation 296). Actually, the phosphorylation occurs at the oxygen atom of the ambident dimsyl anion, and is followed by the Pummerer-type reaction affording diethylphosphoric acid and tetraethyl pyrophosphate among other products . ... 

One has to be careful, however, in defining time constants. The first important step is to set up the correct equations appropriately. If the prime interest is not the accumulation of oxygen in the liquid as defined previously, but the depletion of oxygen from the gas bubbles, then the appropriate balance equation becomes... 

For the regression analysis of a mixture design of this type, the NOCONSTANT regression command in MINITAB was used. Because of the constraint that the sum of all components must equal unity, the resultant models are in the form of Scheffe polynomials(13), in which the constant term is included in the other coefficients. However, the calculation of correlation coefficients and F values given by MINITAB are not correct for this situation. Therefore, these values had to be calculated in a separate program. Again, the computer made these repetitive and Involved calculations easily. The correct equations are shown below (13) ... 

Conceded — Eq. (8.68) violates Babinet s theorem. Nevertheless, it is valid for vi < 0.5 and can easily be remembered, whereas die correct equation is somewhat more involved. 

We wish to develop an analytic expression for this situation. To do so, we will follow the same steps used in the standard development, but use the rigorously correct equation (i.e., equation 42-37 instead of the approximate equation previously used. The steps are the standard ones used for finding a minimum (or maximum) of a function take the derivative of equation 42-37, then set that derivative equal to zero. Since the derivative of interest is the derivative with respect to T, in preparation for this we reorganize equation 42-37 as follows we substitute equation 41-6 (reference [2], reorganized to Es = TEX (since Er is a constant) into equation 42-37 this enables us to eliminate Es from the equation ... 

Interestingly, in Voigtman s development of these equations, his expression corresponding to equation 48-103 is missing the l/Er term inside the radical, even though he arrived at the correct equation corresponding to equation 47-88, as we noted in Chapter 47 referenced as the paper [3],... 

First, in going from equation 54-3 to equation 54-4 [1], when we factored the constants from the derivative we should have taken out 1 /a1, whereas we factored out 1 /a. Therefore several equations from equation 54-4 on are off by a factor of [Pg.413]

Similarly, the correct equations for the second derivative of the Normal distribution are... 

Now that we have a correct equation, we want to solve this equation (or equation 69-3, which is essentially equivalent) for b. Now, if matrix A had the same number of rows and columns (a square matrix), we could form its inverse, and multiply both sides of equation 69-3 by A-1 ... 

The correct equation is (d). At first glance it may not appear that (d) produces a relative standard uncertainty, but look again. We can rewrite (d) as follows ... 

Because the pressure is unknown, Equation 12-4 or 12-5 is used by trial and error until the correct equation is identified. Equation 12-4 is applicable for pressures below 0.3855M. Because SM (from Table 12-3) is 85,000 psi, 0.385SM = 32,700 psi, and r = 3 ft = 36 in and tv = 0.5 in. By substituting into Equation 12-4 for cylindrical vessels, we obtain... 

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