Equilibrium constant of a reaction

Example 8.1 Equilibrium constant of a reaction Consider a homogeneous reaction between species B and P B = E An ideal solution of 1L containing 1 mol of B initially has the concentration 1M. Define the equilibrium constant K. The Gibbs energy of the reaction is 

Using the extent of reaction e, we find NB = 1 — e and NP = e. With these amounts, the relation above becomes 

At equilibrium (dG/de)P = 0, and the equilibrium extent of reaction is seq then we find 

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The equilibrium constant of a reaction is the product of compositions of reactants and products each raised to its stoichiometric coefficient. Hence, for the reaction described by Eq. (8.2) one may write... 

An enzyme—usually a large protein—is a substance that acts as a catalyst for a biological reaction. Like all catalysts, an enzyme doesn t affect the equilibrium constant of a reaction and can t bring about a chemical change that is otherwise unfavorable. An enzyme acts only to lower the activation energy for a reaction,... 

The equilibrium constant of a reaction can be related to the changes in Gibbs Free Energy (AG), enthalpy (AH) and entropy (AS) which occur during the reaction by the mathematical expressions ... 

We are free to choose either K or Kc to report the equilibrium constant of a reaction. However, it is important to remember that calculations of an equilibrium constant from thermodynamic tables of data (standard Gibbs free energies of formation, for instance) and Eq. 8 give K, not Kc. In some cases, we need to know Kc after we have calculated K from thermodynamic data, and so we need to be able to convert between these two constants. 

The equilibrium constant of a reaction contains information about the equilibrium composition at the given temperature. However, in many cases, we know only the initial composition of the reaction mixture and are given apparently incomplete information about the equilibrium composition. In fact, the missing information can usually be inferred by using the reaction stoichiometry. The easiest way to proceed is to draw up an equilibrium table, a table showing the initial composition, the changes needed to reach equilibrium in terms of some unknown quantity x, and the final equilibrium composition. The procedure is summarized in Toolbox 9.1 and illustrated in the examples that follow. 

The equilibrium constant of a reaction is an exponential function of the standard emf of the corresponding cell. We can expect a cell reaction with a large positive emf to have a strong tendency to take place, and therefore to produce a high proportion of products at equilibrium. Therefore, we expect K > 1 when ° > 0 (and often K 1). The opposite is true for a cell reaction with a negative standard emf. 

The equilibrium constant of a reaction can be calculated from standard potentials by combining the equations for the balf-reactions to give the cell reaction of interest and determining the standard potential of the corresponding cell. 

The Van t Hoff isotherm establishes the relationship between the standard free energy change and the equilibrium constant. It is of interest to know how the equilibrium constant of a reaction varies with temperature. The Varft Hoff isochore allows one to calculate the effect of temperature on the equilibrium constant. It can be readily obtained by combining the Gibbs-Helmholtz equation with the Varft Hoffisotherm. The relationship that is obtained is... 

A rate of reaction usually depends more strongly on temperature than on concentration. Thus, in a first-order (n = 1) reaction, the rate doubles if the concentration is doubled. However, a rate may double if the temperature is raised by only 10 K, in the range, say, from 290 to 300 K. This essentially exponential behavior is analogous to the temperature-dependence of the vapor pressure of a liquid, p, or the equilibrium constant of a reaction, K. In the former case, this is represented approximately by the Clausius-Clapeyron equation,... 

The equilibrium constant of a reaction is defined in terms of the standard chemical potentials of reactants and products and thus can be expressed in terms of activities ... 

Catalysts are very effective at stabilizing the transition state, and enzymatic reactions often proceed at rates that are 10 -10 times faster than their uncatalyzed counterparts. Because a catalyst returns to the same chemical form after each catalytic round, the catalyst cannot alter the equilibrium constant of a reaction. Thus, an enzyme must accelerate the S P interconversion by the equivalent factor representing the acceleration of P S. 

Isotopic labels (and especially enriched materials) have proven crucial in the investigation of the mechanisms of homogeneously catalyzed reactions [130]. Further, isotope effects on the rate or the equilibrium constant of a reaction can be diagnostic, and structural information can be provided by isotope-induced changes in the chemical shifts of neighbouring nuclei, and/or alterations in the coupling pattern of the detected spectra. The isotope- and position-specific information inherent to NMR techniques are ideally suited for the analysis of isotope effects in catalysis [131]. 

The acid dissociation constant is simply the equilibrium constant of a reaction in which an acid is mixed with water cmd from which the water concentration has been removed. The water concentration is removed because the concentration of water is a constant in dilute solutions, and a better indicator of acidity is the concentration of the dissociated products divided by the concentration of the acid reactant. The general form of the acid dissociation constant is therefore... 

The thermodynamic feasibility of a reaction with precipitating products can be assessed by comparing the mass action ratio to the equilibrium constant of a reaction. This requires estimation of solubilities, using melting points of reactants in combination with the reaction equilibrium constant [39]. 

The two thermodynamic equations that are most useful for simple kinetic and binding experiments are (1) the relationship between the Gibbs free energy change and the equilibrium constant of a reaction. 

As an alternative to laboratory solubility measurements, solubility product constants (KSp), which are derived from thermodynamic data, can be used to calculate the solubility of solids in water (Table 2.9). Each solubility product constant describes a disassociation of a solid in water and calculates the activities or concentrations of the dissolution products in the saturated solution. The solubility product constant or another equilibrium constant of a reaction may be derived from the Gibbs free energy of the reaction (AG"K) as shown in the following equation ... 

When a reaction is fast enough to be considered at chemical equilibrium (e.g. the shift reaction), the production rate can be computed through the equilibrium constant. Referring to expression (3.19), the equilibrium constant of a reaction k. can be expressed as ... 

According to the principles of statistical mechanics the equilibrium constant of a reaction... 

A chemical reaction proceeds in the direction of decreasing its affinity and reaches equilibrium at which the affinity vanishes. The equilibrium is thus the state at which the unitary affinity of the reaction equals minus the affinity of mixing of the reaction system. The equilibrium constant of a reaction is accordingly an exponential function of the unitary affinity of the reaction. This chapter discusses the role of the unitary affinity in reaction equilibrium... 

The relation between the standard free energy and the equilibrium constant of a reaction can be readily derived. This equation is important because it displays the energetic relation between products and reactants in terms of their concentrations. At equilibrium, A G = 0. Equation 1 then becomes... 

Powering pathways. Compare the stoichiometries of glycolysis and gluconeogenesis (Sections 16.1.8 and 16.3.6). Recall that the input of one ATP equivalent changes the equilibrium constant of a reaction by a factor of about 10 (Section 14.1.3). By what factor do the additional high phosphoryl-transfer potential compounds alter the equilibrium constant of gluconeogenesis ... 

Therefore, a is an electronic descriptor of the substituent estimated by measured rate or equilibrium constants of a reaction, under the control parameter q. 

The gas constant R appears in numerous expressions, including the ideal gas law—Eq. [1-29]—and the equation relating an equilibrium constant of a reaction to its free energy—Eq. [1-12]. Equivalent values of R are... 

A catalyst works by providing an alternative reaction mechanism that competes with the uncatalyzed mechanism. A catalyst cannot alter the equilibrium constant of a reaction, so it must increase the rate of both the forward and the reverse reaction. Although a catalyst cannot change the equilibrium constant, it can, in some cases, change tire composition of the mixture at equilibrium however, the effect is usually very small and should be ignored for the MCAT. For the MCAT, catalysts do not change the equilibrium composition. 

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