Quadratic equation in terms

Equation 6-153 can be expressed in the form of a quadratic equation in terms of the fractional conversion XA as... 

This is a quadratic equation in terms of the ozone concentration. When faced with this level of complexity, it is often convenient to guess the answer and to see if one of the three terms can be dropped because it is small relative to the others. We know that the correct ozone concentration is quite a bit lower than 1013 cm-3 let us guess that it is 1012 cirT3. Using this value, the three terms in the quadratic equation become... 

This is a quadratic equation in terms of the variable of interest, S—OM, the amount of metal adsorbed, as is demonstrated by rearranging equation 4.26 ... 

Substituting Eq. (8) into Eq. (7) and solving for [free ligand] yields a quadratic equation in terms of Ki, [total binding sites], and [totd ligand] all of which it may be possible to measure or estimate. The solution to the equation is... 

E21 + E 7 E 18 E 17/(2 E 8 2 E 3] 1 First coefficient of quadratic equation in terms of filtrate volume, when filtering at constant pressure Second coefficient - which does not change during iterati[Pg.519]

This is a quadratic equation in cAi, the solution of which can be written in terms of cAi t... 

The derivation of Equation (5.73) is dependent on the second law of thermodynamics and will be performed in Section 10.4.) Using Figure 5.8, we can see that Equation (5.73) (a quadratic equation in Tj) should have two distinct real roots for Tj at low pressures, two identical real roots at Pmax. and two imaginary roots above Pmax- At low pressure and high temperamre, which are conditions that correspond to the upper inversion temperature, the second term in Equation (5.73) can be neglected and the result is... 

A quadratic equation has a squared term in it. For instance,x2 + 2x-3 = 0isa quadratic equation. An interesting feature of these equations is that many quadratic equations have two completely different solutions. In the equation x2 + 2x - 3 = 0, x is equal to either +1 or -3. Either number works. Number problems using quadratic equations in their solutions may or may not use both answers. After solving the problem, you go back and look at the original question to see if both answers make sense. 

A similar treatment for the other transformations leads, in every case, to quadratic equations in a f term, and one of the roots is obviously to be discarded. For both the AF and the OAS transformations, the solution is... 

After expanding the cubic and quadratic binomial, we have the linearized equation in terms of perturbation x and the stationary solution Xs... 

Equation (2.8.8) is a quadratic equation in 9t which may be solved to find Of in its dependence on x2. However, in view of all of the above approximations it is generally appropriate to neglect the term in (0f/Tf)2. Equation (2.8.8) may then be inverted to read... 

In the answer to Worked Problem 2.1, we obtained the required roots of the quadratic equation in the form of a sum of a real number (-1) and an imaginary number (2i or -2i). Such numbers are termed complex numbers, and have the general form ... 

These approximations convert the cubic into a quadratic equation in one unknown. These quadratic equations must be used in situations where the pH is less than 4 (Equation 5.61). or greater than 10 (Equation 5.62). In Equation (5.61) the [H30 ]actuai is likely to be comparable to both [HAJtotai and [A ]totai and the term in [HsO Jactuai in the right hand side of Equation (5.61) must not be dropped. Similarly, for pH values greater than 10 the term in [OH lactuai will become comparable to both [HAJtotai and [A Jtotai and must not be dropped,... 

A common application of a quadratic equation in elementary chemistry is the calculation of the hydrogen ion concentration in a solution of a weak acid. If activity coefficients are assumed to equal unity, the equilibrium expression in terms of molar concentrations is... 

We may rewrite Eqn. 34 in terms of these quantities (Eqn. 35) assuming w wP and AGl are constant in a series they may be determined from the experimental data from the coefficients of the quadratic equation in AG. ... 

The other quadratic factors can also be ivritten down at once and will not be given here. It should be noted, however, that a check on the G elements arises from the fact that the last term wiU always involve /iaMc and never or This can be seen if the factored secular equation in terms of external symmetry coordinates (combinations of cartesians) is visualized. In terms of these the kinetic energy is diagonal, with the appropriate atomic masses as diagonal coefficients. The Aig factor, according to Table 10-1, would involve one H mass and one C mass = 1, = 1, = n > — 0) so that the product of the roots... 

In this form of the energy equation the dissipation is represented as the work of the filtered variables through the corresponding displacement and velocity increments. In order to obtain an energy equation in terms of positive definite quadratic terms, the increments Au and Av are eliminated in the damping terms by use of the filter equations (27). These two relations are now used to eliminate the coupled damping terms from the energy balance equation (28) that then takes the form... 

The first and second terms in Eq. (12) do not depend on the dynamical variable R. Here e determines the energy of a pure electronic excitation and it depends on an electrostatic interaction of a chrcmiophore with other molecules. A distribution over values of e leads to the appearance of an inhomogeneous broadening. The linear and quadratic equations in variaUe g terms describe the linear and quadratic electron-phonon interaction whidi transforms 5-like optical lines into optical bands. Let us consider first the effect of the linear interaction. 

This is now a linear differential equation in terms of the lower case correction function. Assuming this linear equation can now be solved, the correction function (lower case u) ean then be added to the approximation function (upper ease U ) and an improved solution obtained. The procedure can then be repeated as many times as necessary to achieve a desired degree of aeeuracy in the solution. It ean be seen that if a valid solution is obtained, the F funetion in Eq. (11.40) ap-proaehes zero so the correction term will approach zero. As with other Newton like methods, the solution is expected to converge rapidly as the exaet solution is approaehed. This technique is frequently referred to as quasilinerization and it has been shown that quadratic convergence occurs if the procedure eonverges. 

The quadratic angle bending term in MM+ is identical to that of equation (12) on page 175, apart from a factor 1/2. Three Gq values... 

To correlate to the acentric factor a quadratic Taylor series in terms of the compressibility factor was formulated. This equation is represented as... 

Equation (5-69) describes rate-equilibrium relationships in terms of a single parameter, the intrinsic barrier AGo, which therefore assumes great importance in interpretations of such data. It is usually assumed that AGo is essentially constant within the reaction series then it can be estimated from a plot of AG vs. AG° as the value of AG when AG = 0. Another method is to fit the data to a quadratic in AG and to find AGq from the coefficient of the quadratic term. ... 

This section describes a number of finite difference approximations useful for solving second-order partial differential equations that is, equations containing terms such as d f jd d. The basic idea is to approximate f 2 z. polynomial in x and then to differentiate the polynomial to obtain estimates for derivatives such as df jdx and d f jdx -. The polynomial approximation is a local one that applies to some region of space centered about point x. When the point changes, the polynomial approximation will change as well. We begin by fitting a quadratic to the three points shown below. 

The third term, Uqt, in Eq. (27) is due to the partial electron transfer between an ion and solvents in its immediate vicinity. The model Hamiltonian approach [33], described in Section V, has shown that Uqt (= AW in Ref. 33) per primary solvent molecule, for an ion such as the polyanion, can also be expressed as a function of E, approximately a quadratic equation ... 

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