Equilibrium/equilibria combining expressions

The equilibrium position for any reaction is defined by a fixed equilibrium constant, not by a fixed combination of concentrations for the reactants and products. This is easily appreciated by examining the equilibrium constant expression for the dissociation of acetic acid. 

You should be able to describe a system at equilibrium both qualitatively and quantitatively. Rigorous solutions to equilibrium problems can be developed by combining equilibrium constant expressions with appropriate mass balance and charge balance equations. Using this systematic approach, you can solve some quite complicated equilibrium problems. When a less rigorous an-... 

For gas-phase reactions, when combining Eqns. (5.4-8) and (5.4-9), the equilibrium constant is expressed as ... 

The description of the chain dynamics in terms of the Rouse model is not only limited by local stiffness effects but also by local dissipative relaxation processes like jumps over the barrier in the rotational potential. Thus, in order to extend the range of description, a combination of the modified Rouse model with a simple description of the rotational jump processes is asked for. Allegra et al. [213,214] introduced an internal viscosity as a force which arises due to a transient departure from configurational equilibrium, that relaxes by reorientational jumps. Thereby, the rotational relaxation processes are described by one single relaxation rate Tj. From an expression for the difference in free energy due to small excursions from equilibrium an explicit expression for the internal viscosity force in terms of a memory function is derived. The internal viscosity force acting on the k-th backbone atom becomes ... 

If the saturated solution is prepared by a method other than dissolution of CaF2 in pure water, there are no separate restrictions on [Ca2+] and [F—] the only restriction on the ion concentrations is that the equilibrium constant expression [Ca2+][F-]2 must equal the Ksp. That condition is satisfied by an infinite number of combinations of [Ca2+] and [F-], and therefore we can prepare many different solutions that are saturated with respect to CaF2. For example, if [F-] is 1.0 X 10 2 M, then [Ca2+] must be 1.5 x 10 6 M ... 

Notice the equilibrium constant for these combined reactions equals the product of the individual equilibrium constant expressions, and the [HA] terms... 

Law of definite proportion a given compound always contains exactly the same proportion of elements by mass. (2.2) Law of mass action a general description of the equilibrium condition it defines the equilibrium constant expression. (6.2) Law of multiple proportions when two elements form a series of compounds, the ratios of the masses of the second element that combine with one gram of the first element can always be reduced to small whole numbers. (2.2)... 

The solution of the equilibrium-constant expression and the pH are sometimes combined into one operation. The combined expression is termed the Henderson-Hasselbalch equation. 

Before we discuss redox titration curves based on reduction-oxidation potentials, we need to learn how to calculate equilibrium constants for redox reactions from the half-reaction potentials. The reaction equilibrium constant is used in calculating equilibrium concentrations at the equivalence point, in order to calculate the equivalence point potential. Recall from Chapter 12 that since a cell voltage is zero at reaction equilibrium, the difference between the two half-reaction potentials is zero (or the two potentials are equal), and the Nemst equations for the halfreactions can be equated. When the equations are combined, the log term is that of the equilibrium constant expression for the reaction (see Equation 12.20), and a numerical value can be calculated for the equilibrium constant. This is a consequence of the relationship between the free energy and the equilibrium constant of a reaction. Recall from Equation 6.10 that AG° = —RT In K. Since AG° = —nFE° for the reaction, then... 

Nickel that is more than 99.9% pure can be produced by the carbonyl process Impure nickel combines with CO at 50 C to produce Ni(CO)4( ). The Ni(CO)4 is then heated to 200°C, causing it to decompose back into Ni(s) and CO g). (a) Write the equilibrium-constant expression for the formation of Ni(CO)4 (b) Given the temperatures used for the steps in the carbonyl process, do you think this reaction is endothermic or exothermic (c) In the early days of automobiles, nickel-plated exhaust pipes were used. Even though the equilibrium constant for the formation of Ni(CO)4 is very small at the temperature of automotive exhaust gases, the exhaust pipes quickly corroded. Explain why this occurred. 

Plan We first must determine the number of moles of weak acid and strong base that have been combined. This will teU us how much of the weak acid s conjugate base has been produced, and we can solve for pH using the equilibrium-constant expression. 

In order to verify the values obtained, we can also determine the values of the standard entropy when we know the equilibrium constant at the desired temperature and the standard enthalpy of reaction at the same temperature. Then by combining expressions [4.1] and [4.2], we obtain ... 

We have assessed the qualitative correctness of reaction (21.19). To calculate the quantitative extent of the dissolution of CaCOg(s) in rainwater is somewhat more difficult. The calculation centers on the combined equilibrium constant expression for reaction (21.19), X = 2.6 X 10 , and is affected by the partial pressure of atmospheric CO2 in equilibrium with rainwater. Typical data are given in Chapter 16, Practice Example A, page 782. 

Combining the virial theorem at equilibrium with the expression for the total electronic energy... 

Taking the natural logaritlnn of (A3.1.54), we see that In+ In has to be conserved for an equilibrium solution of the Boltzmaim equation. Therefore, Incan generally be expressed as a linear combination with constant coefficients... 

Equilibrium constants for complexation reactions involving solids are defined by combining appropriate Ksp and K expressions. Eor example, the solubility of AgCl increases in the presence of excess chloride as the result of the following complexation reaction... 

Combined Pore and Solid Diffusion In porous adsorbents and ion-exchange resins, intraparticle transport can occur with pore and solid diffusion in parallel. The dominant transport process is the faster one, and this depends on the relative diffusivities and concentrations in the pore fluid and in the adsorbed phase. Often, equilibrium between the pore fluid and the solid phase can be assumed to exist locally at each point within a particle. In this case, the mass-transfer flux is expressed by ... 

The phase rule is a mathematical expression that describes the behavior of chemical systems in equilibrium. A chemical system is any combination of chemical substances. The substances exist as gas, liquid, or solid phases. The phase rule applies only to systems, called heterogeneous systems, in which two or more distinct phases are in equilibrium. A system cannot contain more than one gas phase, but can contain any number of liquid and solid phases. An alloy of copper and nickel, for example, contains two solid phases. The rule makes possible the simple correlation of very large quantities of physical data and limited prediction of the behavior of chemical systems. It is used particularly in alloy preparation, in chemical engineering, and in geology. 

The ionic product varies with the temperature, but under ordinary experimental conditions (at about 25 °C) its value may be taken as 1 x 10 14 with concentrations expressed in molL-1. This is sensibly constant in dilute aqueous solutions. If the product of [H + ] and [OH-] in aqueous solution momentarily exceeds this value, the excess ions will immediately combine to form water. Similarly, if the product of the two ionic concentrations is momentarily less than 10-14, more water molecules will dissociate until the equilibrium value is attained. 

The equilibrium equation for the first step is shown in Scheme 5-2. Introducing the equilibrium constant Kw of water (Kw = [H+][0H ]/[H20] leads to the equation shown in Scheme 5-3. ATW can be combined with the constant K (defined by Kx = K[KW) to give the equation of Scheme 5-4. In the same way, the second step can be expressed as in Scheme 5-5. 

The main polymerization method is by hydrolytic polymerization or a combination of ring opening as in (3.11) and hydrolytic polymerization as in (3.12).5,7 9 11 28 The reaction of a carboxylic group with an amino group can be noncatalyzed and acid catalyzed. This is illustrated in the reaction scheme shown in Fig. 3.13. The kinetics of the hydrolytic polyamidation-type reaction has die form shown in (3.13). In aqueous solutions, die polycondensation can be described by second-order kinetics.29 Equation (3.13) can also be expressed as (3.14) in which B is die temperature-independent equilibrium constant and AHa the endialpy change of die reaction5 6 812 28 29 ... 

You should verify for yourself that the three expressions in the first line do combine to give the final expression.) Because step 2 is slow relative to the fast pre-equilibrium, we can make the approximation fc,[01[0 ] [02 [0, or equivalently by canceling the 0, 2fO J [Pg.673]

One may combine the appropriate expressions for G for equilibria involving reactants in different phases to obtain a general expression, which relates the equilibrium constant to the... 

The second law of thermodynamics states that the total entropy of a system must increase if a process is to occur spontaneously. Entropy is the extent of disorder or randomness of the system and becomes maximum as equilibrium is approached. Under conditions of constant temperature and pressure, the relationship between the free energy change (AG) of a reacting system and the change in entropy (AS) is expressed by the following equation, which combines the two laws of thermodynamics ... 

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