Unit of the equilibrium constants

Irrespective of the reaction under study and the expression of the law of mass action used, the corresponding equilibrium constant has no unit, because it is always defined by an exponential. This means, in particular, that the law of mass action expressed in partial pressures must contain only the pressure ratios. Similarly, in terms of concentrations, as we have seen, they play a part in the ratios of concentrations. It is true that the denominator of those fractions is often 1 (normal pressure of 1 bar and reference concentration 1 mole/1). 

However, it must be remembered that the equilibrium constant depends on the convention chosen to define the solution. 

By applying relation [A 1.22], by deriving relation [3.2] in relation to temperature  

Similarly, by deriving relation [3.14], we obtain the following, in the case of gases  

For perfect gases, the enthalpy is independent of the pressure and the internal energy does not depend on the volume  

---

We have seen that the (apparent) units of the equilibrium constant depend on the specific reaction. For example, for the reaction... 

If one takes into account the concentration units of the equilibrium constant, the temperature dependence of the rate constant is related to the enthalpy of activation by... 

Solubility equilibria are described quantitatively by the equilibrium constant for solid dissolution, Ksp (the solubility product). Formally, this equilibrium constant should be written as the activity of the products divided by that of the reactants, including the solid. However, since the activity of any pure solid is defined as 1.0, the solid is commonly left out of the equilibrium constant expression. The activity of the solid is important in natural systems where the solids are frequently not pure, but are mixtures. In such a case, the activity of a solid component that forms part of an "ideal" solid solution is defined as its mole fraction in the solid phase. Empirically, it appears that most solid solutions are far from ideal, with the dilute component having an activity considerably greater than its mole fraction. Nevertheless, the point remains that not all solid components found in an aquatic system have unit activity, and thus their solubility will be less than that defined by the solubility constant in its conventional form. 

Equilibrium constants are dimensionless numbers, yet the concentrations used in an equilibrium constant expression have units. To understand this, we need to explore the reaction quotient Q, introduced in Chapter 14. In Section 16-1 we explore in detail the link between Q and Keq. Here we use Q to address the issue of concentration units and the equilibrium constant. 

Each differential equation contains a flow term identified by Q/V (flow rate/reactor volume) and also a reaction term which can be identified by a rate of reaction or equilibrium constant (k, K, k ). These reaction and equilibrium constants are functions of temperature which, in this study, was fixed. The viscosity dependence of the equilibrium constant (relating reactive species to total polymer) shown in Equations 6 and 7 was observed experimentally and is known as the Trommsdorf effect (6). Table I lists values and units of all parameters in Equations 1-7. 

Equilibrium constants do not have units because in the strict thermodynamic definition of the equilibrium constant, the activity of a component is used, not its concentration. The activity of a species in an ideal mixture is the ratio of its concentration or partial pressure to a standard concentration (1 M) or pressure (1 atm). Because activity is a ratio, it is unitless and the equilibrium constant involving activities is also unitless. 

In the case of a bidentate ligand the ratio KMLJKWL)i has units of concentration in mol 1 1. If one assumes that the equilibrium constants for formation of the monoco-ordinated complexes ML and ML L are equal,2 combination of the equilibrium constants for (26) and (27) shows that the ratio... 

The Gibbs energy changes of the reaction are given at the temperatures in the table. The units are cal/gmol dimethyl ether. Confirm the tabulated values of the equilibrium constant, K, and of the fractional conversion, x, of the methanol. 

Finally, it is not appropriate to derive thermodynamic properties of solid solutions from experimental distribution coefficients unless it can be shown independently that equilibrium has been established. One possible exception applies to trace substitution where the assumptions of stoichiometric saturation and unit activity for the predominant component allow close approximation of equilibrium behavior for the trace components (9). The method of Thorstenson and Plummer (10) based on the compositional dependence of the equilibrium constant, as used in this study, is well suited to testing equilibrium for all solid solution compositions. However, because equilibrium has not been found, the thermodynamic properties of the KCl-KBr solid solutions remain provisional until the observed compositional dependence of the equilibrium constant can be verified. One means of verification is the demonstration that recrystallization in the KCl-KBr-H20 system occurs at stoichiometric saturation. 

The reader should refer to the original tables for the reference material on which the thermochemical data are based. The reference state used in Chapter 1 was chosen as 298 K consequently, the thermochemical values at this temperature are identified from this listing. The logarithm of the equilibrium constant is to the base 10. The unit notation (J/K/mol) is equivalent to (JK mol ). Supplemental thermochemical data for species included in the reaction listing of Appendix C, and not given in Table A2, are listed in Table A3. These data, in combination with those of Table A2, may be used to calculate heats of reaction and reverse reaction rate constants as described in Chapter 2. References for the thermochemical data cited in Table A3 may be found in the respective references for the chemical mechanisms of Appendix C. 

The form of the equilibrium constant in Equation (10.21) is different from that presented in introductory courses. It has the advantages that 1) it is explicit that Kp is a dimensionless quantity 2) it is explicit that the numerical value of Kp depends on the choice of standard state but not on the units used to describe the standard state pressure the equilibrium constant has the same value whether P° is expressed as 750.062 Torr, 0.98692 atm, 0.1 MPa, or 1 bar. 

Calculate the number of acetic acid molecules adsorbed per gram of charcoal (N) and the corresponding equilibrium acid concentration (C). Plot N against C and C/N against C. Determine the surface area per gram of charcoal assuming that one adsorbed acetic acid molecule occupies an area of 21A. Estimate the value of the equilibrium constant K with the correct units. 

The magnitude of the equilibrium constant, K, is a measure of the strength of interaction between the two molecules. The equilibrium constant has been determined for a variety of complexes (3). The rate of formation of this complex is usually much higher than the rate of polymerization. The polymerization proceeds by adding donor-acceptor units to the growing chain. 

The exact definition of the equilibrium constant given by IUPAC requires it to be defined in terms of fugacity coefficients or activity coefficients, in which case it carries no units. This convention is widely used in popular physical chemistry texts, but it is also common to find the equilibrium constant specified in terms of molar concentrations, pressure or molality, in which cases the equilibrium constant will carry appropriate units. 

However, measurements of pAR in this case lead to a lesser dependence of the equilibrium constant upon carbocation stability than pAR. Guthrie has calculated relative values of pAR and pAR and shown that an unfavorable geminal interaction between Cl and CF3 reduces the difference between ArCH2 and ArCHCF on the pAR scale by about 7 log units compared with pAR. This implies that replacing CH3 by CF3 in the p-methoxybenzyl cation decreases pAR by 14 units. Based on the value of pAR = -8.7 for the -methoxybenzyl cation, pAR for the a-CF3 cation should be close to -23.5. 

The pH scale is like a ruler marked from 0 to 14. This number span reflects the values of the equilibrium constant for the dissociation of water mentioned earlier in this unit. Recall the following equation. 

Since log K for this reaction is positive, ArG° is negative, and the products are more stable than the reactants in the Standard State. The situation in a state other than the Standard State may be evaluated, as usual, by a consideration of the equilibrium constant itself. In the present case (assuming unit water activity)... 

This procedure illustrates the arbitrariness in the specification of K. The numerical value of the equilibrium constant would have been different if some other sets of units had been employed. However, once the K value is determined for a particular physical situation, this quantity can be used to determine x under any other set of conditions, provided P is expressed in atmospheres and the same temperature prevails. It is this feature that renders the equilibrium constant such a useful quantity. 

Equation (3.10.5) or (3.10.7) is the fundamental relation of interest, the van t Hoff equation it is evident that from the enthalpy change accompanying a unit advancement of the reaction of interest under standard conditions, one can obtain the rate of change of the equilibrium constants Kq with alterations in temperature. 

This situation may be characterized by (among others) use of the equilibrium constant Km specified by Eq. (3.7.8b). It is conventional either to ignore the product term f7] [aJ"(T,P) l 1 as being equal to (at unit pressure) or close to unity, or to absorb this constant factor into the equilibrium constant as well. This then gives rise to the expression... 

From Eqs. (IV.3.4) and (IV.3.7) we see that at equilibrium the ratio of the rates of forward and reverse reactions is a function only of the equilibrium constant divided into the equilibrium-constant expression in concentration units ... 

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. 

We return to Eq. (3.7.10) to note the very fundamental interrelation between (i) the free energy change per unit advancement of the reaction, as specified by u/A/ = 0, when all participating species are isolated and maintained at standard conditions, and (ii) the natural logarithm of the equilibrium constant pertaining to the reaction in question. 

---