Standard state for activity

Depending on the problem the following standard states for activity are used ... 

Standard states are either stated or implied in any quantitative discussion or tabulation of free energies, enthalpies, internal energies, or activities, but the following discussion will be based on the use of standard states for activities because of the much wider range of possibilities encountered. Standard states for tabulated free energies, etc. generally do not get any more complicated than the cases already dealt with in Chapter 7. 

The derivation of p a values from pH measurements is too familiar an exercise to be repeated here. Since pH values refer to a standard state for activities in methanol, the acidity constants will likewise have this reference state, i.e. they will be Ka values. When measurements are made in very dilute solutions (ca. 10" AO, not only are corrections to zero ionic strength small but ion-pairing of buffer salts becomes negligible. Some results of Juillard s measurements, based on the de Ligny standard buffers, are given in Appendix 3.5.5. 

We have now introduced three kinds of standard states for activity coefficients one based on pure-components ( 10.2.1), a second based on the solute-free Henry s law ( 10.2.3), and a third based on the reference-solvent Henry s law ( 10.2.4). The prin-... 

In any equilibrium state, both fi, and /t are absolute, finite quantities with a fixed difference between them. If the same standard state is chosen for each of these equations, then fi, — ji° is the same in each equation, and the activity would be the same in all phases at equilibrium. This would be nice, but it would mean using a vapor pressure as the standard state for activity in solids, or an ideal one molal solution standard state for activities in a gas, or perhaps an ideal gas at one bar for an aqueous solute. This would be not only inconvenient, but impossible in many cases. So we accept the small inconvenience of having different activities for the same species in different phases. 

In order to compare the thermodynamic parameters of different reactions, it is convenient to define a standard state. For solutes in a solution, the standard state is normally unit activity (often simplified to 1 M concentration). Enthalpy, internal energy, and other thermodynamic quantities are often given or determined for standard-state conditions and are then denoted by a superscript degree sign ( ° ), as in API", AE°, and so on. 

A chemical reaction can occur only if — AG > 0, i.e. if — AG is positive in addition the value a = is by definition the maximum activity for a condensed component where the pure phase is taken as standard state, thus A/i is always negative. This discussion will be restricted to gases where p 1 taking p = 1 atm (101 325 kN/m ) as the standard state for the gas, X, it is evident that A/ix is always a negative quantity or zero. 

Activity is a dimensionless quantity, and / must be expressed in kPa with this choice of standard state. It is inconvenient to carry f° = 100 kPa through calculations involving activity of gases. Choosing the standard state for a gas as we have described above creates a situation where SI units are not convenient. Instead of expressing the standard state as /° = 100 kPa, we often express the pressure and fugacity in bars, since 1 bar = 100 kPa. In this case, /0 — 1 bar, and equation (6.92) becomes4... 

E6.6 The partial pressure of Bri above a (. 1CCI4 +. v Bn) solution is 1.369 kPa. The composition of the solution is a = 0.0250. The vapor pressure of pure bromine at the same temperature is 28.4 kPa. Assume a Raoult s law standard state for bromine and calculate the activity coefficient of Br2 in the solution. 

We now have the foundation for applying thermodynamics to chemical processes. We have defined the potential that moves mass in a chemical process and have developed the criteria for spontaneity and for equilibrium in terms of this chemical potential. We have defined fugacity and activity in terms of the chemical potential and have derived the equations for determining the effect of pressure and temperature on the fugacity and activity. Finally, we have introduced the concept of a standard state, have described the usual choices of standard states for pure substances (solids, liquids, or gases) and for components in solution, and have seen how these choices of standard states reduce the activity to pressure in gaseous systems in the limits of low pressure, to concentration (mole fraction or molality) in solutions in the limit of low concentration of solute, and to a value near unity for pure solids or pure liquids at pressures near ambient. 

Note that the first term in the rate law could be written ArisflH2o but if water is the solvent, as we shall assume, its activity is unity. Were one to write the term as ki,[H20], this would be tantamount to adopting a nonconventional standard state for water, which is usually not advisable. With [OH- ] [(CH3)2CHBr], the reaction follows first-order kinetics with... 

Here, the a s refer to the activities in the chosen arbitrary state. The concept of activity is presented separately in a later section. For the present, the activity of a species in a system may just be considered to be a function of its concentration in the system, and when the species is in a pure form (or in its standard state), its activity is taken to be unity. The activities ac, aD, aA, aB given above correspond to the actual conditions of the reaction, and these may or may not correspond to the state of equilibrium. Two special situations can be considered. In the first, the arbitrary states are taken to correspond to those for the system at equilibrium. Q would then become identical to the equilibrium constant K and, according to the Van t Hoff isotherm, AG would then be zero. In the second situation, all the reactants and the products are considered to be present as pure species or in their standard states, and aA, aB, ac, and aD are all equal to 1. Then (7=1 and the free energy change is given by... 

When the standard states for the solid and liquid species correspond to the pure species at 1 atm pressure or at a low equilibrium vapor pressure of the condensed phase, the activities of the pure species at equilibrium are taken as unity at all moderate pressures. Consequently, the gas phase composition at equilibrium will not be... 

If k is expressed in liters per mole per second, the standard state for the free energy and entropy of activation is 1 mole/liter. If the units of k are cubic centimeters per molecule per second, the corresponding standard state concentration is 1 molecule/cm3. The magnitudes of AG and AS reflect changes in the standard state, so it is not useful to say that a particular reaction is characterized by specific numerical values of these parameters unless the standard states associated with them are clearly identified. These standard states are automatically determined by the units chosen to describe the reactant concentrations in the phenomenological rate expressions. 

For protonation-dehydration processes, such as trityl cation formation from triphenylcarbinols, equation (24), the water activity has to be included if the formulation of the activity coefficient ratio term is to be the same as that in equation (7), which it should be if linearity in X is to be expected see equation (25). The excess acidity expression in this case becomes equation (26) this is a slightly different formulation from that used previously for these processes,36 the one given here being more rigorous. Molarity-based water activities must be used, or else the standard states for all the species in equation (26) will not be the same, see above. For consistency this means that all values of p/fR listed in the literature will have to have 1.743 added to them, since at present the custom... 

At equilibrium, all components of a mixture have the same molar free energy, i.e., the same chemical potential, in any phase in which they are present, and they have the same chemical potential as all other components. However it is not always convenient to use the same standard state for all components or even for the same component in all phases. Just as Equation 6 defines fugacity, Equation 7 or 8 defines activity. Furthermore, Equations 6-8 define / and a for all substances, not just gases. However we should keep in mind that we do not use the same standard state for a substance in all the phases, mixtures, or pure states in which it may occur or for all components of a mixture. 

The numerical value of the activity clearly depends upon the standard state, and one often encounters other choices for the standard state for solutes. For example, just as we obtained Equations 29 and 30 from Equation 22, we could have obtained similar looking equations from Equations 23 or Equation 24. However, the derivation requires a mention of... 

The case of liquid solutions is more complicated because the conventions vary. These are always stated in introductory chapters of the thermochemical databases and deserve a careful reading. In most tables and in the present book, it is agreed that the standard state for the solvent is the pure solvent under the pressure of 1 bar (which corresponds to unit activity). For the solute, the standard state may refer to the substance in a hypothetical ideal solution at unit molality (the amount of substance of solute per kilogram of solvent) or at mole fraction x = 1. 

The standard state for the mean ionic activity coefficient is Henry s constant H., f is the standard-state fugacity for the activity coefficient f- and x. is the mole fraction of electrolyte i calculated as though thi electrolytes did not dissociate in solution. The activity coefficient f is normalized such that it becomes unity at some mole fraction xt. For NaCl, xi is conveniently taken as the saturation point. Thus r is unity at 25°C for the saturation molality of 6.05. The activity coefficient of HC1 is normalized to be unity at an HC1 molality of 10.0 for all temperatures. These standard states have been chosen to be close to conditions of interest in phase equilibria. 

AV is then the excess molar volume of products over that of reactants, in their standard states. For dilute solutions, where activity corrections may be neglected, and where Kx is expressed in mole fraction units... 

ACTIVITIES, EXCESS GIBBS FUNCTIONS, AND STANDARD STATES FOR NON ELECTROLYTES... 

It can be seen from Figures 16.2 and 16.3 that the numerical values of the activity and activity coefficient of the solute are different for the two choices of standard state. The scale of activities, for example, is necessarily different. The activity coefficient at mole fraction X2(i) is given by the ratio N/M in both figures. Thus, when the standard state is chosen on the basis of Henry s law, the activity coefficients are less than 1, whereas when the pure solute is chosen as standard state, the activity coefficients all are greater than 1. 

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