First equation

The first equation (1) is the equation of state and the second equation (2) is derived from the measurement process. Finally, G5 (r,r ) is a row-vector that takes the three components of the anomalous ciurent density vector Je (r) = normal component of the induced magnetic field. This system is non hnear (bilinear) because the product of the two unknowns /(r) and E(r) is present. 

We have seen that equilibrium in an isolated system (dt/= 0, dF= 0) requires that the entropy Sbe a maximum, i.e. tliat dS di )jjy = 0. Examination of the first equation above shows that this can only be true if. p. vanishes. Exactly the same conclusion applies for equilibrium under the other constraints. Thus, for constant teinperamre and pressure, minimization of the Gibbs free energy requires that dGId Qj, =. p. =... 

Step 6 - the remaining unknowns are found by back substitution using the following formula from the ( - l)st to the first equation in turn... 

Solving several equations by the method of Gaussian elimination, one might divide the first equation by [ j. obtaining 1 in the [ j position. Multiplying ayi into the first equation makes an ay]. Now subtracting the first equation from the second, a zero is produced in the a2i position. The same thing can be done to produce a zero in the a. ] position and so on, until the first column of the eoefflcient matrix is filled with zeros except for the Un position. 

If you try to solve these n equations for all of the elements of the v veetor (vi...Vn), you ean eliminate one variable using one equation, a seeond variable using a seeond equation, ete., beeause the equations are linear. For example you eould solve for vi using the first equation and then substitute for vi in the seeond equation as you solve for V2, ete. Then when you eome to the nth equation, you would have n-1 of the variables expressed in terms of the one remaining variable, Vn. 

Multiplying the first equation by 0.11096 and subtracting the second equation gives... 

Solving the first equation for k, substituting into the second equation, and simplifying gives... 

Subtracting the second equation from the first equation and solving for [A]o gives... 

The first equation is an example of hydrolysis and is commonly referred to as chemical precipitation. The separation is effective because of the differences in solubiUty products of the copper(II) and iron(III) hydroxides. The second equation is known as reductive precipitation and is an example of an electrochemical reaction. The use of more electropositive metals to effect reductive precipitation is known as cementation. Precipitation is used to separate impurities from a metal in solution such as iron from copper (eq. 1), or it can be used to remove the primary metal, copper, from solution (eq. 2). Precipitation is commonly practiced for the separation of small quantities of metals from large volumes of water, such as from industrial waste processes. 

If the first equation is multiplied by aoo and the second by — 7i9 and the results added, we obtain... 

The latter two equations require that F is a function only of a., and therefore 9F/9a. = dP/dx. Inspection of the first equation shows one term which is a function only of A. and one which is only a function of y. This requires that both terms are constant. The pressure gradient —dP/dx is constant. The A.-component equation becomes... 

The first equation ignores the existence of the intermediate titanium oxides, which is reasonable for this analysis of die oxidation mechanism.)... 

There are two contradictory requirements here. The first is to keep the difference between Ci and C as small as possible so that it can be neglected. The second is to analyze these two only very slightly different concentrations with such precision that the difference will be significantly greater than the measurement error. This second need is for calculation of the rate of reaction, as shown in the first equation of this section. 

As has been described earlier, the stoichiometric reactions should be manipuitUed algebraically to retain the transferable species (H2S) only in the first equation. Therefore, HjS can be eliminated from Eq. (8.13) by subtracting (8.12) from (8.13) to get... 

The first equation may be applied to a control volume CV surrounding a gas turbine power plant, receiving reactants at state Rg = Ro and discharging products at state Py = P4. As for the combustion process, we may subtract the steady flow availability function for the equilibrium product state (Gpo) from each side of Eq. (2.47) to give... 

Again, a closure is needed. Even with a closure, the system of equations is not complete. A relation between the singlet function p(r) and the pair functions is needed. For this purpose the first equation of the BGY hierarchy may be used. Alternatively, one can apply the Lovett-Mou-Buff-Wertheim equation [100,101]... 

The first equation expresses a scaling idea as in Eq. (18), introducing the arbitrary factor 2/3 instead of unity for numerical convenience—the problem of statistical accuracy becomes easier, and numerical prefactors should not matter in statistical considerations. 

The total energy of a vessel s contents is a measure of the strength of the explosion following rupture. For both the statistical and the theoretical models, a value for this energy must be calculated. The first equation for a vessel filled with an ideal gas was derived by Brode (1959) ... 

Several points are worth noting about these formulae. Firstly, the concentrations follow an Arrhenius law except for the constitutional def t, however in no case is the activation energy a single point defect formation energy. Secondly, in a quantitative calculation the activation energy should include a temperature dependence of the formation energies and their formation entropies. The latter will appear as a preexponential factor, for example, the first equation becomes... 

The first equation states that cytotoxic cells grow only if helper cells, macrophages and the virus are all present. The second equation implies that, when the virus is not present, helper cells grow if macrophages and/or helper cells are present. The third equation implies that macrophages grow both when the virus is present and there is already a concentration of macrophages. The last equation describes the... 

Next, we substitute these dimensionless variables into the incompressible Navier-Stokes equations (equation 9.16). In Cartesian coordinates, the T component of the first equation reads... 

The first equation gives the diserete version of Newton s equation the second equation gives energy c onservation. We make two comments (1) Notice that while energy eouseivation is a natural consequence of Newton s equation in continuum mechanics, it becomes an independent property of the system in Lee s discrete mechanics (2) If time is treated as a conventional parameter and not as a dynamical variable, the discretized system is not tiine-translationally invariant and energy is not conserved. Making both and t , dynamical variables is therefore one way to sidestep this problem. 

The first equation gives an i value closest to 1.2. Actually about 20% of the oxalic add is ionized via the second equation. 

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