Equilibrium constant in terms of activity

The formulation of transition state theory has been in terms of reactant and transition state concentrations let us now define an equilibrium constant in terms of activities. 

Strictly, we should formulate all equilibrium constants in terms of activities rather than concentrations, so Equation (7.32) describes Ksp for dissolving partially soluble AgCI in water ... 

GIBBS FUNCTION AND THE EQUILIBRIUM CONSTANT IN TERMS OF ACTIVITY... 

Baes and Mesmer [161] have suggested that McDowell and Johnston s data can be satisfactorily explained in terms of only the tetrahydroxy species if one allows for the probable effect of increasing ionic strength. However, this statement is presumably incorrect as McDowell and Johnston reported their equilibrium constant in terms of activities, and kinetic measurements on the complexation reactions of Cu(II) at pH > 13 indicate that two different hydroxy species exist [165]. 

Although it is proper to write equilibrium constants in terms of activities, the complexity of manipulating activity coefficients is a nuisance. Most of the time, we will omit activity coefficients unless there is a particular point to be made. Occasional problems will remind you how to use activities. 

The equilibrium constant given by Eq. (9) using a values obtained from Eq. (8) differs from K, the true equilibrium constant in terms of activities, owing to the omission of activity coefficients (y ) from the numerator of Eq. (9) and the approximations inherent in Eq. (8). At the very low ionic concentrations encountered in the dissociation of a weak electrolyte, a simple extrapolation procedure can be developed to obtain from the values of Since y is an excellent approximation, it follows that... 

The activities now have an extra significance, because taken together they define the equilibrium condition, and are fixed among themselves. If T is fixed, then so is AG°. This means that the activity term must also be constant it is now equal to the equilibrium constant in terms of activities, K. 

Equilibrium constant in terms of activities ionization (dissociation) constant of an acid. 

The expressions, Ky, Kp, Kc, all have the mathematical form of the equilibrium constant in eq. (14. 7). with activity replaced by y., Pi, or c. and are often called equilibrium constants. As we see here, they are indeed related to the equilibrium constant. However, the term equilibrium constant should be strictly reserved for the expression that gives the equilibrium constant in terms of activities. This constant is dimensionless and a function of temperature only. The derivative forms, Ky, Kp, may have units, and are functions of pressure, if v is not zero. These expressions maybe used in calculations but should not be confused with the thermodynamic quantity that we call equilibrium constant. 

Our goal is to write an expression for the rate constant k,. We begin by writing the thermodynamic equilibrium constant in terms of activities a and activity coefficients y ... 

Note that we have also specified these equilibrium constants in terms of the activity of the associated defects. We can also write thermodynamic equations for these defects ... 

These quantities are not in general dimensionless. One can define in an analogous way an equilibrium constant in terms of fugacity Kf, etc. At low pressures Kp is approximately related to K by the equation K Kp/(p )lv, and similarly in dilute solutions Kc is approximately related to K by K Kc/(c )lv however, the exact relations involve fugacity coefficients or activity coefficients [24]. 

The first equation expresses the equilibrium constant in terms of quantities that can be obtained from tabulated values (formation enthalpy and Gibbs free energy, heat capacities). The second equation expresses the equilibrium constant in terms of the activity of species, and ultimately, in terms of mol fractions. We must remember that activities and formation properties of each species must refer to the same standard state. 

The form taken by the equilibrium constant depends on the type of expression which is substituted in the above equation for the purpose of expressing the chemical potentials in terms of the composition this in its turn dex>ends on additional physical knowledge concerning whether or not the real system in question may be approximately represented by means of a model, such as the perfect gas or the ideal solution. If the system does not approximate to either of these models it is still possible, of course, to formulate an equilibrium constant in terms of fugacities or in terms of mole fractions and activity coefficients. However, this isapurely formal process the fugacities and activity coefficients are themselves defined in terms of the chemical potentials and therefore the knowledge contained in equation (10 1) is in no way increased, but is obtained in a more convenient form. 

The criteria for co-crystal thermodynamic stability were presented in Section 11.3. An important feature of co-crystals is that they coexist in equilibrium with solution. This occurs when their molar free energy or chemical potential is equal to the sum of the chemical potentials of each co-crystal component in solution. Thus, the individual component chemical potentials in a solution saturated with co-crystal can vary as long as their sum is constant. In terms of activities, it is the activity product that is constant. 

The availability of powerful computers and advanced computational methods to treat problems in chemistry opens the possibility for predicting rates of reactions. As explained earlier, equilibrium thermodynamics has provided a rigorous basis for the prediction of maximum conversion levels and the conditions under which they are achieved. The Arrhenius equation served as a tool for rationalizing rate constants in terms of activation energies and preexponentials. These parameters, however, could not be predicted on the basis of molecular properties of the reacting species until the concept of the transition state evolved, around 1935. Gas-phase kinetics in particular established a fundamental understanding of the Arrhenius parameters. We treat the transition-state theory in Chapter 4. 

Consider again Reaction (7.1), but let us now relax the assumption that the activation of A is rate determining. The approach to equilibrium for the fuU reaction is shown in Equation (7.5), and we can write the equilibrium constant in terms of equilibrium constants for the two elementary steps ... 

The equilibrium constant in Eq. 2 is defined in terms of activities, and the activities are interpreted in terms of the partial pressures or concentrations. Gases always appear in K as the numerical values of their partial pressures and solutes always appear as the numerical values of their molarities. Often, however, we want to discuss gas-phase equilibria in terms of molar concentrations (the amount of gas molecules in moles divided by the volume of the container, [I] = j/V), not partial pressures. To do so, we introduce the equilibrium constant Kt., which for reaction E is defined as... 

The feed stream consists of 60 mole percent hydrogen, 20% nitrogen, and 20% argon. Calculate the composition of the exit gases, assuming equilibrium is achieved in the reactor. Make sure that you take deviations from the ideal gas law into account. The equilibrium constant expressed in terms of activities relative to standard states at 1 atm may be assumed to be equal to 8.75 x 10 3. The fugacity of pure H2 at 450 °C and 101.3 MPa may be assumed to be equal to 136.8 MPa. 

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 ... 

Ang= number of moles of products less that of reactants. When the equilibrium constant is expressed in terms of activities, the constant Ka is obtained which is valid for all gases and solutions. 

The derivation of the law of mass action from the second law of thermodynamics defines equilibrium constants K° in terms of activities. For dilute solutions and low ionic strengths, the numerical values of the molar concentration quotients of the solutes, if necessary amended by activity coefficients, are acceptable approximations to K° [Equation (3)]. However, there exists no justification for using the numerical value of a solvent s molar concentration as an approximation for the pure solvent s activity, which is unity by definition.76,77... 

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