Two equations

The multicomponent form of the Underwood equation can be used to calculate the vapor flow at minimum reflux in each column of the sequence. The minimum vapor rate in a single column is obtained by alternate use of two equations ... 

Lane improved on these tables with accurate polynomial fits to numerical solutions of Eq. 11-17 [16]. Two equations result the first is applicable when rja 2... 

The gradient model has been combined with two equations of state to successfully model the temperature dependence of the surface tension of polar and nonpolar fluids [54]. Widom and Tavan have modeled the surface tension of liquid He near the X transition with a modified van der Waals theory [55]. 

The right-hand equalities in tliese two equations arise directly from equation (A2.1.33) and equation (A2.1.34). 

Combining tliis witli the Omstein-Zemike equation, we have two equations and tluee unknowns h(r),c(r) and B(r) for a given pair potential u r). The problem then is to calculate or approximate the bridge fiinctions for which there is no simple general relation, although some progress for particular classes of systems has been made recently. 

Thus, only two of the five quantities Itl lJ-l-Utl-UlMfi lare independent. We choose the number of down spins [i] and nearest-neighbour pairs of down spins [ii] as the independent variables. Adding and subtracting the above two equations. 

Here the thennal diflfiisivity Dj. = tc/fp Cp. These two equations couple only to the longitudinal part T = V-iJ of the fluid velocity. From equation (A3.3.17) it is easy to see diat T satisfies... 

Since the potential depends only upon the scalar r, this equation, in spherical coordinates, can be separated into two equations, one depending only on r and one depending on 9 and ( ). The wave equation for the r-dependent part of the solution, R(r), is... 

These two equations are a special case of the corresponding tliree-dimensional definition, connnon in XRD, with the surface nonnal n replacing the third lattice vector c. 

One writes equations which T and T" are expected to obey. For example, in the early development of these methods [80], the Scln-ddinger equation itself was assumed to be obeyed, so = E V and = E are the two equations (note that, in the IP and EA cases, the latter equation, and the associated Hamiltonian //, refer to one fewer and one more electrons than does the reference equation ft V = E V). 

The acceleration can then be recalculated at the new corrected positions to give new velocities. The method then iterates over the two Equations (7.28) and (7.29). Two or three passes are usually required to achieve consistency, with a force evaluation at each step. The computational demands of this scheme mean that it is now rarely used, though it does give accurate solutions of the equations of motion. 

Sinee F(t) is only a funetion of time t, and F(qj) is only a funetion of the spatial eoordinates qj, and beeause the left hand and right hand sides must be equal for all values of t and of (qj, both the left and right hand sides must equal a eonstant. If this eonstant is ealled E, the two equations that are embodied in this separated Sehrodinger equation read as follows ... 

C3i2 (Note There are now two equations with three unknowns.) Seeond, root two eigenveetor two ... 

The two equations couple because the alpha Fock matrix depends on both the alpha and the beta solutions, C and cP (and sim ilarly for the beta Fock matrix). The self-consistent dependence of the Fock matrix on molecular orbital coefficients is best represen ted, as before, via the den sity matrices an d pP, wh ich essen -tially state the probability of describing an electron of alpha spin, and the probability of finding one of beta spin ... 

Comparing the core-core repulsion of the above two equations with those in the MNDO method, it can be seen that the only difference is in the last term. The extra terms in the AMI core-core repulsion define spherical Gaussian functions — the a, b, and c are adjustable parameters. AMI has between two and four Gaussian functions per atom. 

ZINDO/S is different from ZINDO/I because they use different algorithms in computing the Coulomb integrals. Hence the two equations used in the mixed model in ZINDO/1 are also employed... 

Alternatively, equations 3.11 or 3.12 can be solved for the amounts of both the analyte and the interferent. To do so, however, we must obtain two independent values for Sjneas- Using a concentration method as an example, gives two equations... 

This is an example of a direct analysis in which the two analytes are determined without a prior separation. The weight of the original precipitate and the ignited precipitate are given by the following two equations... 

Substituting the equations for g MgO and g AI2O3 into the equation for the combined weights of MgO and AI2O3 leaves us with two equations and two unknowns. 

Before adding together these two equations, the second equation must be multiplied by 5 so that the log terms can be combined thus... 

These two equations can be combined by recognizing that a conservation of mass for iodine requires that... 

The shift in the voltammogram for a metal ion in the presence of a ligand may be used to determine both the metal-ligand complex s stoichiometry and its formation constant. To derive a relationship between the relevant variables we begin with two equations the Nernst equation for the reduction of O... 

For two successive measured mass-to-charge ratios m and m two equations can be written,... 

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