Temperature dependence equilibrium constants

Various amines find application for pH control. The most commonly used are ammonia, morpholine, cyclohexylamine, and, more recently AMP (2-amino-2-methyl-l-propanol). The amount of each needed to produce a given pH depends upon the basicity constant, and values of this are given in Table 17.4. The volatility also influences their utility and their selection for any particular application. Like other substances, amines tend towards equilibrium concentrations in each phase of the steam/water mixture, the equilibrium being temperature dependent. Values of the distribution coefficient, Kp, are also given in Table 17.4. These factors need to be taken into account when estimating the pH attainable at any given point in a circuit so as to provide appropriate protection for each location. 

Like all equilibrium constants, the magnitude of the equilibrium constant Ks depends quite strongly on temperature, according to... 

The p/<, of a base is actually that of its conjugate acid. As the numeric value of the dissociation constant increases (i.e., pKa decreases), the acid strength increases. Conversely, as the acid dissociation constant of a base (that of its conjugate acid) increases, the strength of the base decreases. For a more accurate definition of dissociation constants, each concentration term must be replaced by thermodynamic activity. In dilute solutions, concentration of each species is taken to be equal to activity. Activity-based dissociation constants are true equilibrium constants and depend only on temperature. Dissociation constants measured by spectroscopy are concentration dissociation constants." Most piCa values in the pharmaceutical literature are measured by ignoring activity effects and therefore are actually concentration dissociation constants or apparent dissociation constants. It is customary to report dissociation constant values at 25°C. 

The equilibrium constant K depends on temperature and an approximate expression is /C = exp(1.876 3110/T), where Tis in K. Equations 3-78, 3-80a, and 3-80b can be solved simultaneous numerically (e.g., Zhang et al., 1991a,b) for three unknowns, CH20 CH20m 1-oh- If equilibrium is not reached, then... 

This equation shows that the equilibrium constant K depends only on temperature, because AG° is a function of temperature only. 

The rate constant at the temperature of maximum broadening was found to be virtually independent of the equilibrium constant but dependent on the chemical-shift difference between the exchanging sites according to the relationship... 

Whereas the equilibrium constant itself depends on the temperature only, the conversion at equilibrium depends on the composition of the original reaction mixture and, in general, on the pressure. If the equilibrium constant is very high, the reaction may be treated as being irreversible. If the equilibrium constant is low, however, it may be possible to obtain acceptable conversions only by using high or low pressures. Two important examples are the reactions ... 

We turn our attention in this chapter to systems in which chemical reactions occur. We are concerned not only with the equilibrium conditions for the reactions themselves, but also the effect of such reactions on phase equilibria and, conversely, the possible determination of chemical equilibria from known thermodynamic properties of solutions. Various expressions for the equilibrium constants are first developed from the basic condition of equilibrium. We then discuss successively the experimental determination of the values of the equilibrium constants, the dependence of the equilibrium constants on the temperature and on the pressure, and the standard changes of the Gibbs energy of formation. Equilibria involving the ionization of weak electrolytes and the determination of equilibrium constants for association and complex formation in solutions are also discussed. 

There is an equilibrium between the different sites that determines the observed distribution. For example, the ratio of the sites with the cation vacancy nearby to those with the vacancy distant will depend upon the concentration of vacancies by the law of mass action. One can write the conventional equilibrium relationships given by mass action considerations for equilibria between vacancies and probe ion sites with local and distant vacancy compensation. The equilibrium constant will depend on the temperature under which the equilibrium was established. Since all of the sites can be observed by site selective laser spectroscopy, one can measure the equilibrium distributions directly. We find that the sites and their distributions are described excellently by the mass action relationships of conventional equilibria. This work is described in more detail elsewhere (5,6). 

Equilibrium constants are dependent upon the temperature of the system. Formation of ammonia is exothermic—heat is released as the reaction occurs N2 + 3H2 <= 2NH3 + 93.7 kilojoules of energy. Therefore, cooling the reaction mixture favors the formation of even more ammonia. 

The logarithm of the equilibrium constant, nKp depends upon the difference of an entropy term, and an enthalpy term which becomes relatively less important as temperature, T is increased. 

Because NO formation involves an alteration of n, the equilibrium constant K2 depends not only on the temperature, but also on the total pressure p. 

Through this work Setser and Rabinovitch " were the first to articulate clearly the fundamental agenda for experimental studies to determine rate constants for various one-center and two-center epimerization events, and use the experimental facts to discriminate among alternative mechanistic models. But in practice, synthetic and analytical limitations restricted their study to the 1-methyl-2,3-d2-cyclopropane isomers so that only one rate constant could be measured, a rate constant for approach of all isomers to equilibrium. The temperature dependence of this rate constant, 2(ki + k2 + k 12 + 23) to the activation parameters log = 15.35, = 60.5 kcal moT Thus, both practical and conceptu-... 

The numerical value of kt in (2-3) depends on how activity is defined and on the units in which concentration is expressed (molarity, mole fraction, partial pressure). Measurement of the absolute activity, or chemical potential, of an Individual ion is one of the classical unsolved problems. Since we cannot measure absolute ion activity, we are then necessarily interested in the next best—comparative changes in activities with changing conditions. To obtain comparative values numerically, we measure activity with respect to an arbitrarily chosen standard state under a given set of conditions of temperature and pressure, where the substance is assigned unit activity. The value of ki in (2-3) thus depends on the arbitrary standard state chosen accordingly, the value of the equilibrium constant also depends on the choice of standard states. 

Ideal-gas tables of thermodynamic properties derived from statistical mechanics are based on the thermodynamic temperatures (as well as on the values of the physical constants used) and are hence independent of any practical temperature scale. The enthalpy of formation, Gibbs energy of formation, and logarithm of the equilibrium constant might depend on temperature-adjusted data. 

The stability of the complex ML is characterized by the equilibrium constant for which various terms are employed stability constant, binding constant, association constant, affinity constant or /i) or dissociation constant (Kj, reciprocal of Kg). From the thermodynamic point of view, the true equilibrium constant (that depends only on temperature) must be written with activities. But we will consider only dilute solutions so that the activity coefficients can be approximated to 1 (reference state solute at infinite dilution). Activities can then be replaced by molar fractions (dimensionless quantities), but in solution they are generally replaced by molar concentrations ... 

Sieverts Law c = Ks4p, where c is the subsurface concentration (solubility) of the dissolved atom in the solid metal, P is the partial pressure of the diatomic gas (sometimes replaced by the fugacity, j), and Ks is the solubility constant (temperature dependent), which is the chemical equilibrium constant between the molecular species in the gas phase and the atomic species within the metal lattice. This empirical relation was first demonstrated by Sieverts in 1929 for the solubility of hydrogen in iron. Departures from this law occur at high gas pressures and/or high concentrations of dissolved atoms. 

Equilibrium constants usually depend on the temperature and the general point can be made that ... 

The equilibrium constant Keq depends on the nature of the reactants and products, the temperature, and the pressure (particularly In reactions Involving gases). Under standard physical conditions (25 "C and 1 atm pressure, for biological systems), the Keq Is always the same for a given reaction, whether or not a catalyst Is present. 

At low concentrations the equilibrium constant k depends on the temperature (and perhaps pH, ionic simegth, or pressure) but not on the concentrations of the other solutes. Then the solute velocity for linanr systems is... 

If an inert diluent is added to a reacting mixture, it may change the equilibrium state of the system, not as a result of a change in the-value of the equilibrium constant (which depends only on the standard states and temperature), but rather as a result of the change in the concentration, and hence the activity, of each reacting species. This effect is illustrated in the following example. 

Note the absence of the subscript a on in the equation above, indicating that this is not a true equilibrium constant. Instead, K a is referred to as an apparent equilibrium constant and is a measured quantity that depends on the solution conditions (ionic strength, pH, etc.), unlike the thermodynamic equilibrium constant, which depends only on the standafSli tate and temperature. This apparent equilibrium constant is one of the concentration chemical equilibrium ratios defined in Table 13.1-3. As... 

Although the special ratio of products to reactants defined by the equilibrium expression is constant for a given reaction system at a given temperature, the equilibrium concentrations will not always be the same. Table 13.1 gives three sets of data for the synthesis of ammonia, showing that even though the individual sets of equilibrium concentrations are quite different for the different situations, the equilibrium constant, which depends on the ratio of the concentrations, remains the same (within experimental error). Note that subscript zeros indicate initial concentrations. 

The equation can be normalized, because F(l) will refer to the sum of the partial pressures. The equilibrium constant is dependent on the temperature. 

An issue that is not adequately addressed by most models (EQ and NEQ) is that of vapor and liquid flow patterns on distillation trays or maldistribution in packed columns. Since reaction rates and chemical equilibrium constants are dependent on the local concentrations and temperature, they may vary along the flow path of liquid on a tray, or from side to side of a packed column. For such systems the residence time distribution could be very important, as well as a proper description of mass transfer. On distillation trays, vapor will rise more or less in plug flow through a layer of froth. The liquid will flow along the tray more or less in plug flow, with some axial dispersion caused by the vapor jets and bubbles. In packed sections, maldistribution of internal vapor and liquid flows over the cross-sectional area of the column can lead to loss of interfacial area. 

Before a reaction-equilibrium calculation can be performed, we must select an appropriate standard state for each species. Moreover, we must clearly distinguish quantities, such as fugacities and activities, that depend on the final equilibrium state (T, P, x ), from those quantities, such as equilibrium constants, that depend only on the equilibrium temperature T, the standard-state pressures P , and the phase. Typically, the standard-state pressure and phase are chosen according to whether the real substance is gas, liquid, or solid at the equilibrium conditions. Those three possibilities are discussed, in turn, here, and each discussion culminates with a particular expression for the activity. Those expressions can be used either in the stoichiometric development, via (10.3.14), or in the nonstoichiometric development, via (10.3.38). We emphasize that when we use the stoichiometric approach, the standard states used for the fugacities must be consistent with those associated with the equilibrium constant. 

The conversions are limited by the position of the chemical equilibrium. The temperature dependency of the equilibrium constant Ki for converting CO and H2 can be expressed by the following equation [3.11] ... 

Since the equilibrium constant K depends on the temperature, a way to change its value is to change T. In particular, K increases with T for endothermic reactions and decreases for exothermic reactions according the well-known Van t Hoff equation ... 

For reversible reactions, an increase or decrease in temperature tends to directly influence the equilibrium constant, but depends on whether the reaction is exothermic or endothermic. When the reaction is exothermic (AH < 0), an increase in temperature favours the reverse reaction, since fC < < 1. For endothermic reactions, the opposite is valid. 

But i/k depends only on temperature. The equilibrium constant also depends only on temperature ... 

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