Equilibria equations

A reaction s equilibrium equation is given directly from the form of the reaction and the value of the equilibrium constant. Hence, it is an easy matter to extend a reaction balancing program to report equilibrium lines. For example, the reaction, 

To use RXN to calculate the equilibrium lines above in general and specific forms, we type 

The long command tells the program to show the reaction s equilibrium constant versus temperature and calculate its equilibrium equation pH = causes the program to render the equation in terms of pH instead of log aH+. To find the equilibrium lines written in terms of pe and Eh, we type 

Many mineralogic reactions involve exchange of cations or anions. Hence, geochemists commonly need to determine equilibrium lines in terms of activity ratios. Consider, for example, the reaction at 25 °C between the clay kaolinite [Al2Si205(0H)4] and the mica muscovite. The RXN commands 

In our second example, we calculate the same ratio for the reaction between muscovite and potassium feldspar (KAlSiaOs maximum microcline in the database) in the presence of quartz  

Assuming linear equilibrium sorption conditions and if the decay constants for the i-th species are identical in the aqueous and sorbed phases (2. , = 2., = 2.,), the above equations reduce to that equations given by van Genuchten (1985) as follows  

The last divergence term appearing in (2.64) is frequently omitted when considering strong anchoring, for the reasons indicated earlier in this Section, since it can be transformed to a surface term via the divergence theorem (cf. de Gennes and Frost [110, p.l04]). The identity 

This allows the one-constant approximation in equation (2.64) to be expressed in an equivalent Cartesian components form as 

This means that, in Cartesian coordinates, the energy is calculated simply by summing the squares of the components riij. 

It is known from the previous Section that k = ku = 0 for both nematics and cholesterics, and so, by (2.37) i, Sq = 0 also. The invariance condition (2.6) must hold for cholesterics also, showing that the constants ki must again be of the form given in equation (2.22) but these values for ki must necessarily be inequivalent to those given in (2.47) because of the presence of enantiomorphy in cholesterics. Noting that kz must be zero, we are therefore forced to conclude that k2 0 for cholesterics. Hence, by (2.37)2, to 0 and, employing the notation of the previous Section, the energy for cholesterics based upon (2.44) and (2.48) can now be written as 

It is instructive at this point to insert the natural cholesteric state ric from equation (2.69) into Wchoi to find that 

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The most reliable estimates of the parameters are obtained from multiple measurements, usually a series of vapor-liquid equilibrium data (T, P, x and y). Because the number of data points exceeds the number of parameters to be estimated, the equilibrium equations are not exactly satisfied for all experimental measurements. Exact agreement between the model and experiment is not achieved due to random and systematic errors in the data and due to inadequacies of the model. The optimum parameters should, therefore, be found by satisfaction of some selected statistical criterion, as discussed in Chapter 6. However, regardless of statistical sophistication, there is no substitute for reliable experimental data. 

Using the equilibrium equations of the elasticity theory enables one to determine the stress tensor component (Tjj normal to the plane of translumination. The other stress components can be determined using additional measurements or additional information. We assume that there exists a temperature field T, the so-called fictitious temperature, which causes a stress field, equal to the residual stress pattern. In this paper we formulate the boundary-value problem for determining all components of the residual stresses from the results of the translumination of the specimen in a system of parallel planes. Theory of the fictitious temperature has been successfully used in the case of plane strain [2]. The aim of this paper is to show how this method can be applied in the general case. 

Using the equilibrium equations of the elasticity theory enables one to reduce these integrals to the ordinary Radon transform [1]. 

By substituting relations (26) into equations (24), (25) we obtain the general solution of the equilibrium equations... 

In integrated photoelasticity it is impossible to achieve a complete reconstruction of stresses in samples by only illuminating a system of parallel planes and using equilibrium equations of the elasticity theory. Theory of the fictitious temperature field allows one to formulate a boundary-value problem which permits to determine all components of the stress tensor field in some cases. If the stress gradient in the axial direction is smooth enough, then perturbation method can be used for the solution of the inverse problem. As an example, distribution of stresses in a bow tie type fiber preforms is shown in Fig. 2 [2]. 

Gordus, A. A. Ghemical Equilibrum 11. Deriving an Exact Equilibrium Equation, /. Chem. Educ. 1991, 68, 215-217. 

The simpler model can be derived to describe a shallow shell which is characterized by the closeness of the mid-surface to the plane. In other words, it is assumed that a = b = 1 and the coordinate system a, (5) coincides with the Descartes system X, X2- Then differentiating the fourth and the fifth equilibrium equations with respect to Xi and X2, respectively, and combining with the third equilibrium equation give... 

The obtained model (1.15)-(1.17) of a shallow shell can be simplified once more. The values of kiQi, kiQ2 are small enough very often, and they can be omitted. By doing so, we obtain the simplified model of a shallow shell consisting of the following equilibrium equations ... 

We call a plate the shallow shell when k =k2 = 0. This implies that the plate mid-surface coincides with the plane z = 0, and the plate is limited by the two parallel planes z = h, z = —h and a boundary contour. Let us redenote the horizontal and vertical displacements of the plate mid-surface by u = ui, u = U2, w. In this case, the plate horizontal and vertical displacements are not coupled. Indeed, it follows from (1.18), (1.19), that U = (ui,U2) is described by the following equilibrium equations ... 

To find the normal displacements w we should consider the equilibrium equation... 

Substituting the moments into the equilibrium equation, we obtain the equation for an isotropic viscoelastic plate,... 

In this case we cannot directly substitute Mij into the equilibrium equation as it was done for the previous elastic and inelastic models. So w, Mij cannot be found in consecutive order, in general. 

To derive the last relation, we take the equilibrium equations... 

This form of (2.224) is more convenient for further consideration and, moreover, (2.225), (2.226) imply the following equilibrium equations in in the sense of distributions ... 

By varying the test function W K, one can deduce that the variational inequality (2.265) is equivalent to the equilibrium equations in flc. 

We shall consider an equilibrium problem with a constitutive law corresponding to a creep, in particular, the strain and integrated stress tensor components (IT ), ay(lT ) will depend on = (lT, w ), where (lT, w ) are connected with (IT, w) by (3.1). In this case, the equilibrium equations will be nonlocal with respect to t. 

In accordance with (3.53) the functional II/(x) + Ilg( ) is coercive and weakly lower semicontinuous on the space H, consequently, the problem (3.48) (or the problem (3.54)) has a solution. The solution is unique. Note that the equilibrium equations... 

In this section we derive a nonpenetration condition between crack faces for inclined cracks in plates and discuss the equilibrium problem. As it turns out, the nonpenetration condition for inclined cracks is of nonlocal character. This means that by writing the condition at a fixed point we have to take into account the displacement values both at the point and at the other point chosen at the opposite crack face. As a corollary of this fact, the equilibrium equations hold only in a domain located outside the crack surface projection on the mid-surface of the plate. This section follows the papers (Khludnev, 1997b Kovtunenko et ah, 1998). 

Here i —> i is a continuous convex function describing the plastic yield condition. The equations (5.7) provide a decomposition of the strain tensor Sij u) into a sum of an elastic part aijuicru and a plastic part ij, and (5.6) are the equilibrium equations. 

In fact, by the second Korn inequality this scalar product induces a norm which is equivalent to the norm given by (5.3). Hence, because (/, p) = 0 for all p G R fl), the identity (5.29) actually holds for every u G Therefore, the equilibrium equations... 

The dependence of solutions to (5.79)-(5.82) on the parameters a, 5 is not indicated at this step in order to simplify the formulae. Note that boundary conditions (5.81) do not coincide with (5.71) the conditions (5.81) can be viewed as a regularization of (5.71) connected with the proposed regularization of the equilibrium equations (5.68). Also, the artificial initial condition for o is introduced. 

Here i —> i is the convex and continuous function describing a plasticity yield condition. The function w describes vertical displacements of the plate, rriij are bending moments, (5.139) is the equilibrium equation, and equations (5.140) give a decomposition of the curvatures —Wjj as a... 

Here i —> i is the convex and continuous function describing a plasticity yield condition, the dot denotes a derivative with respect to t, n = (ni,ri2) is the unit normal vector to the boundary F. The function v describes a vertical velocity of the plate, rriij are bending moments, (5.175) is the equilibrium equation, and equations (5.176) give a decomposition of the curvature velocities —Vij as a sum of elastic and plastic parts aijkiirikiy Vijy respectively. Let aijki x) = ajiki x) = akuj x), i,j,k,l = 1,2, and there exist two positive constants ci,C2 such that for all m = rriij ... 

Note that from (5.247) we obtain two equilibrium equations... 

Equation (4-272) for each species, giving (tt — )N phase-equilibrium equations. 

If species i is an element, AG/ is zero. There are N equilibrium equations (Eqs. [4-355]), one for each chemical species, and there are w material-balance equations (Eqs. [4-353]), one for each element—a total of N + to equations. The unknowns in these equations are the (note that y, = of which there are N, and the Xi, of which... 

The dynamic material-balance and phase equilibrium equations corresponding to this description are as follows ... 

Using the simple radial equilibrium equation, the eomputation of the axial veloeity distribution ean be ealeulated. The aeeuraey of the teehniques depends on how linear F /r is with the radius. 

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