Rule of multiple equilibria

The product expression, at the right, is clearly the equilibrium constant for the overall reaction, as it should be according to the rule of multiple equilibria. 

Because Equation (1) + Equation (2) = Equation (3), we have, according to the rule of multiple equilibria (Chapter 12),... 

Strategy Apply the rule of multiple equilibria to find K. Then work with the expression for K to find the molar solubility of AgCL... 

Words that can be used as topics in essays 5% rale buffer common ion effect equilibrium expression equivalence point Henderson-Hasselbalch equation heterogeneous equilibria homogeneous equilibria indicator ion product, P Ka Kb Kc Keq KP Ksp Kw law of mass action Le Chatelier s principle limiting reactant method of successive approximation net ionic equation percent dissociation pH P Ka P Kb pOH reaction quotient, Q reciprocal rule rule of multiple equilibria solubility spectator ions strong acid strong base van t Hoff equation weak acid weak base... 

A property of K that you will find very useful in this and succeeding chapters is expressed by the rule of multiple equilibria, which states... 

O Determine K for this system, applying the rule of multiple equilibria to the two-step process referred to above. (5 Using K, calculate the molar solubility, s, of Zn(OH)2 in acid at pH 5.0. 

Multiple equilibria, rule of A rule stating that if Equation 1 + Equation 2 = Equation 3, then K3 = K1X K2,370,439 Multiple proportions, 27... 

These equations do not imply any progression in the polymerization process and reduce to identities in the Fincham-Richardson formalism (cf table 6.3). Nevertheless, the experimental evidence of Mysen et al. (1980) and Virgo et al. (1980) may be explained by the progressive polymerization steps listed in table 6.3. These equilibria are consistent with the Fincham-Richardson formalism (of which they constitute simple multiples) and obey the proportionality rules of figure 6.4. 

Multiple equilibria pose a formidable problem for the rational-expectations argument. A convention equilibrium like rules of the road cannot emerge by rational expectations, if the situation offers no clue to what others will do. (I shall have more to say about clues later.) Multiple equilibria with different winners and losers, as in the second version of the erosion story, are even less hospitable to rational expectations. In this circumstance many things might happen. The situation might remain indefinitely out of equilibrium. The realization of one equilibrium rather than another could happen by accident. One set of individuals might be sufficiently powerful to impose the equilibrium that favored them over other people. What can be ruled out is the realization of one equilibrium by tacit coordination and rational anticipation. 

Nevertheless, the phase rule is extremely useful for yielding a physical understanding of polymorphic systems and for providing a physical interpretation of phase transformation phenomena. Its greatest power is in its ability to rule out the existence of simultaneous multiple equilibria that violate its fundamental equation, permitting more quantitative investigations to focus on the possible aspects of such systems. 

To predict binding equilibria, the first step is to devise an appropriate model and compute the binding polynomial. In this section we show a short-cut for writing the binding polynomial directly from the model, skipping the step of writing out the equilibria. We illustrate it first on simple cases, but the main power of the method is for describing more complex cases. The key to this approach is to use the addition and multiplication rules of probabihty (see Chapter 1). 

Preparation of virtual screening databases starts with standardization of the input SMILES. This procedure was originally developed to deal with databases from commercial suppliers. Preferred tautomeric forms are generated in this step and ionized species are neutralized. Ionization states are set in the second step for biased equilibria and multiple forms are enumerated in a third step to represent balanced equilibria. The model treats an equilibrium as balanced if the equilibrium constant associated with its defining rule is likely to be less than about 1.5 log units. 

A straightforward, but tedious, route to obtain information of vapor-liquid and liquid-liquid coexistence lines for polymeric fluids is to perform multiple simulations in either the canonical or the isobaric-isothermal ensemble and to measure the chemical potential of all species. The simulation volumes or external pressures (and for multicomponent systems also the compositions) are then systematically changed to find the conditions that satisfy Gibbs phase coexistence rule. Since calculations of the chemical potentials are required, these techniques are often referred to as NVT- or NPT- methods. For the special case of polymeric fluids, these methods can be used very advantageously in combination with the incremental potential algorithm. Thus, phase equilibria can be obtained under conditions and for chain lengths where chemical potentials cannot be reliably obtained with unbiased or biased insertion methods, but can still be estimated using the incremental chemical potential ansatz [47-50]. 

Gibbs phase rule (Equation 1) for the system (polymolecular P)-fLMWL requires some modification. First, phase equilibria are considered under constant (atmospheric) pressure, i.e. pressure is fixed and no longer a variable. Second, the degrees of polymerization Pi with i = 1,2,..., s — 1 and concentrations of components become variables. The number of degrees of freedom for the m-multiple critical points is cancelled by (m - -1) conditions of its existence (see Equation 46, including n = — 1), so there remains... 

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