Equations, mathematical rearrangement

Many mathematical problems that are encountered in chemistry fall into the following algebraic forms. Solutions to these problems are simplified by first isolating the desired term on one side of the equation. This rearrangement is accomplished by treating both sides of the equation in an identical manner until the desired term is... 

To conform to the dimensional homogeneity, the dimensions of the variable on the left-hand side of the equation must be equal to those on the right-hand side. With some simple mathematical rearrangements, Eq. 2.1 can be transformed into an... 

The Rabinowitsch equation allows the calculation of the shear rate at the pipe wall from three measurable quantities R, Q, and AP. Since the flow properties of a fluid are independent of the pipe structure, the curve of the fluid is the same as the r-y curve. Therefore, by using Equation 8.18 and the Rabinowitsch equation, the flow curve, i.e, t-y or log r-log y, can be obtained. However, the use of the Rabinowitsch equation sometimes is tedious, and hence it often is simplified by using the following mathematical rearrangements. 

When rearranged, this equation is a cubic equation (an equation in x3), which can be solved with a graphing calculator or mathematical software. However, because K is very small, we suppose that x will turn out to be so small that we can use the approximation procedure ... 

In attempting to determine if a given set of experimental data is of the same mathematical form as equation 7.3.29, there are three routes that permit the graphical determination of the parameters Vmax and K. The most frequently used plot is known as a Lineweaver-Burk or reciprocal plot. It is based on rearrangement of equation 7.3.29 into the following form. 

By rearranging equation 9-1, we can also express it as follows, wherein the fact that it is a mathematical identity becomes apparent ... 

Here it has been shown that the conclusion about is related to the mathematical approximation used in interpreting the data by way of Method I it is easy to show that the second and third conclusions are also dependent on the initial assumptions in Method I. Rearrangement of Equation 28 and substitution of the constant capacitance constraint yields a relationship between i/>q and In a + ... 

In a first order reaction, the change in concentration of the reactants or products with time is exponential (see Figure 2.8), and this can be illustrated mathematically by integrating and rearranging Equation 2.6 to give ... 

This is called a steady-state approximation and is expressed mathematically by setting the rate of ES formation equal to the rate of ES consumption (Equations 4.6 and 4.7). After a number of rearrangements, Equation 4.7 can be solved for [ES] (Equation 4.8). The collection of three rate constants is replaced with a single term, Km, the Michaelis constant. 

The mathematical equations used, for the release of drug for other geometries, are shown in Table 6.4. With the exception of Equation (6.61d) for short times, one should use the nonlinear parameter estimation methods to determine the diffusivity of a drug. As shown in Table 6.4, it is recommended to use the approximation equations for long times and rearrange them into ... 

The Gran plot (Fig. 13.2c2). This method consists of the mathematical transformation of the titration curve into straight lines via rearranged Nernst equations. Using a selective electrode that responds only to a... 

Sometimes an equation that is not in standard form can be changed to the form y = mx + b by rearrangement or mathematical manipulation. An example is the equation k = Ae Ea/RT, where A, a and R are constants, k is the dependent variable, and 1/T is the independent variable. This equation can be changed to standard form by taking the natural logarithm of both sides,... 

The mathematical analysis of the dynamics of systems interacting with encompassing reservoirs, whose detailed dynamics is of no direct interest, is facilitated by the use of projection operators or projectors. A simple example is provided by the use of such projectors to rearrange a system of linear equations. Let... 

The basic PK parameters for a one-compartment IV infusion model can also be determined from infusion period plasma concentration data. This analysis requires an estimate of the steady-state concentration (Qi), however, so plasma concentration measurements must be made for infusion times up to about 7 ty eUm in order to perform this analysis. The mathematical basis for this analysis comes from Equation (10.127), which can be rearranged to become... 

In the last equation the term containing jr has been dropped because x will be small near equilibrium and the x term will be negligibly small. The equation may now be rearranged and integrated using the following integration, which is found in standard mathematical handbooks. 

Considering first the summation part in this equation, one obtains by rearrangement and simplification (using standard mathematical relations) ... 

It can be seen from the form of Equation 18.23 that the derivative approach can be used to determine [C]o. The mathematical relationships for the integral methods can be obtained by rearranging and integrating. 

Equation (5.9) can be rearranged into several new forms that yield straight lines when one new variable is plotted against the other. The advantages of this mathematical manipulation are that (1) and K can be determined readily by fitting a straight line to the transformed data (2) departures of the data from a straight line are more easily detected than nonconformity to a hyperbola (these departures may indicate an... 

One reason chemistry can be challenging for beginning students is that they often do not possess the required mathematical skills. Thus we have paid careful attention to such fundamental mathematical skills as using scientific notation, rounding off to the correct number of significant figures, and rearranging equations to solve for a particular quantity. And we have meticulously followed the rules we have set down, so as not to confuse students. 

The hyperbolas encountered in mathematics always have two limbs, whereas the plot of Vo against A appears to have only one. However, if we rearrange the Michaelis-Menten equation, we shall obtain... 

A most satisfactory treatment of kinetic data is the direct linear plot of Eisenthal and Cornish Bowden (1974). Axes are drawn with -S on the abscissa and V on the ordinate, but instead of making the usual hyperbolic plot of S/v (see Enzyme kinetics), corresponding points (each reading of -S and its related v value) are joined by straight lines. The point of intersection of this family of lines gives the values of V and K . Mathematically, this plot corresponds to a rearrangement of the general equation = v + vK ,/... 

Where T and Tei are the kinetic energy operators for the nuclei and electrons, and Vmi, Vne and Vee are the electrostatic potential energies arising from intemuclear, nucleus-electron, and interelectronic interactions. At this point, the Born-Oppenheimer approximation is invoked, i.e., the assumption that the movement of electrons is much faster than that of the nuclei and therefore the two can be decoupled from one another. This is mathematically represented by a separation of variables in the wavefunction, Y(R,r)= (r(R)) %(R), where <1> and % are the electronic and nuclear wavefunctions, respectively, and O is a function of r parameterized by R. Thus, with some rearrangement, Equation (21) becomes... 

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