Equilibrium constant, multiple reactions

Perform equilibrium measurements to establish the overall free energy of the reaction in solution, and, if possible, the internal equilibrium constants for reactions occurring at the active site. In addition, it is often possible to measure the equilibrium dissociation constants for the binding of any substrates that, by themselves, will bind and not react, such as the first substrate to bind in a multiple-substrate mechanism. 

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. 

If you want to prove to yourself that free energies sum, you can write the equilibrium expressions for the first and second reactions and multiply them together, and you ll get the equilibrium expression for the hydrolysis of ATP. Multiplication is equivalent to the addition of logarithms, so that when you multiply equilibrium constants, you re actually adding free energies (or vice versa). 

For multiple equilibria dissociation constants (such as polyprotic acids), K for the overall reaction is the product of the equilibrium constants for the individual reactions. Therefore,... 

Equations containing a number of solvent parameters in linear or multiple linear regression and expressing the effect of the solvent on the rate of the reaction or the thermodynamic equilibrium constant. See Ej Values Kamlet-Taft Solvent Parameter Koppel-Palm Solvent Parameter Z Value... 

The simplest one-constant limitation concept cannot be applied to all systems. There is another very simple case based on exclusion of "fast equilibria" A Ay. In this limit, the ratio of reaction constants Kij — kij/kji is bounded, 0equilibrium constant", even if there is no relevant thermodynamics.) Ray (1983) discussed that case systematically for some real examples. Of course, it is possible to create the theory for that case very similarly to the theory presented above. This should be done, but it is worth to mention now that the limitation concept can be applied to any modular structure of reaction network. Let for the reaction network if the set of elementary reactions is partitioned on some modules — U j. We can consider the related multiscale ensemble of reaction constants let the ratio of any two-rate constants inside each module be bounded (and separated from zero, of course), but the ratios between modules form a well-separated ensemble. This can be formalized by multiplication of rate constants of each module on a timescale coefficient fc,. If we assume that In fc, are uniformly and independently distributed on a real line (or fc, are independently and log-uniformly distributed on a sufficiently large interval) then we come to the problem of modular limitation. The problem is quite general describe the typical behavior of multiscale ensembles for systems with given modular structure each module has its own timescale and these time scales are well separated. 

You can calculate the equilibrium constant for a reaction, from the concentrations of reactants and products at equilibrium. In the following reaction, for example, A and B are reactants, C and D are products, and a, b, c, and d are stoichiometric coefficients (numbers showing mole multiples in a balanced equation) ... 

This calculation illustrates an important point about equilibrium constants although the AG ° values for two reactions that sum to a third are additive, the iTeq for a reaction that is the sum of two reactions is the product of their individual K, .q values. Equilibrium constants are multiplicative. By coupling ATP hydrolysis to glu-... 

STRATEGY The general procedure is like that set out in Toolbox 9.1. Write the chemical equation and the expression for its equilibrium constant, then set up an equilibrium table, using the concentrations immediately after addition of the reagent (before any reaction has taken place) as the initial concentrations. The initial concentration of the substance added is the sum of the initial and added amounts divided by the volume. Use a multiple of x to denote the change in molar concentration of each substance. Because a, product has been added, we expect that reaction will occur in the direction... 

After de Forcrand s Clapeyron, and Handa s methods, a third method for the determination of hydrate number, proposed by Miller and Strong (1946), was determined to be applicable when simple hydrates were formed from a solution with an inhibitor, such as a salt. They proposed that a thermodynamic equilibrium constant K be written for the physical reaction of Equation 4.14 to produce 1 mol of guest M, and n mol of water from 1 mol of hydrate. Writing the equilibrium constant K as multiple of the activity of each product over the activity of the reactant, each raised to its stoichiometric coefficient, one obtains ... 

The mass action law assumes that the reaction medium is homogeneous. In heterogeneous reactions (involving different substances in multiple phases), the densities and effective concentrations of pure condensed phases (liquids or solids) are constant. The concentrations of such species are set to unity in the equilibrium constant expression for such reactions. For example, given the following decomposition,... 

This chapter, after introducing the equilibrium constant, discusses briefly the rate of entropy production in chemical reactions and coupling aspects of multiple reactions. Enzyme kinetics is also summarized. 

Substituting the appropriate ideal expression for the activity of gaseous or dissolved species from Equation 14.8a or 14.8b leads to the forms of the mass action law and the equilibrium constant K already derived earlier in Section 14.3 for reactions in ideal gases or in ideal solutions. We write the mass action law for reactions involving pure solids and liquids and multiple phases by substituting unity for the activity of pure liquids or solids and the appropriate ideal expression for the activity of each gaseous or dissolved species into Equation 14.9. Once a proper reference state and concentration units have been identified for each reactant and product, we use tabulated free energies based on these reference states to calculate the equilibrium constant. 

If a second equation is subtracted from the first, the resulting equilibrium constant is that of the first divided by that of the second (subtracting a reaction is the same as adding the reverse reaction). The operations of addition and subtraction applied to chemical equations transform into multiplication and division of the equilibrium expressions and equilibrium constants. 

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