The Ideal One Molal Standard State

Changing ideal to mean Raoultian ideality, the ideal one molal standard state would give 

Using a standard state of pure crystalline B we would get 

Obviously there are many possible choices if one knows the fugacities involved, but of course if we did know the fugacities, we would not use activities and their bothersome standard states at all, we would use the fugacities themselves. 

The real situation is of course that in this type of solution no fugacity data are available for the solute, but activity coefficients indicating the deviation from Henry s Law can be either calculated or measured, and they allow us to calculate exactly the same activities and hence ( — ) values using the ideal one molal standard state. For example at 0.5 and 1.0 molal the activity coefficient (7h) is 1.33 and 1.5 respectively, so the activities of B using the ideal one molal standard state (ub = are 0.67 

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Figure 12.4 corresponds to reality. The lengthy introduction by way of the fictitious Figure 12.3 is simply to emphasize that activities using the ideal one molal standard state are really no different from any other activities. They can be thought of as fugacity ratios, and they are simply another of the wide range of choices available for standard states. 

Suppose you have the activity of a constituent with respect to a particular standard state, but you need its activity using some other standard state. For example, you might know fcoi in a fluid, which is equivalent to knowing its aco2 using an ideal gaseous CO2 at T, 1 bar standard state, but you want to do speciation calculations so you need acoi using the ideal one molal standard state. 

Let s suppose that a measurement of quartz solubility has been used to obtain the free energy of formation (standard or apparent) of H4Si04 in the ideal one molal standard state. This number can then be used (with A/G° terms for the minerals) to calculate the equilibrium constant of the albite-nepheline reaction (equation (13.11)), giving the equilibrium silica concentration in a solution that may never have been experimentally determined, or perhaps never existed, and in which quartz is not stable. Thus knowing the solubility of quartz, one could in a similar way calculate the silica concentration in fluids in contact with a variety of mineral assemblages. 

As usual, it is best to see the truth of a relationship by nnderstanding it rather than by seeing no fault with its derivation. In this case this can be accomplished by realizing that in the ideal one molal standard state to which ArG° refers, the solute component HCl consists in solution entirely as H and Cl, therefore the G of component HCl has no choice but to be identical to Gh+ + Gci, from which it follows that A G° = 0. 

The activity thus allows calculation of the difference between the jXi in a solution and in the ideal one molal standard state at the same T and P as the solution. This sounds like a fairly esoteric thing to do, but because standard Gibbs energies of formation are determined for this ideal standard state (albeit at 25 °C, 1 bar), it is immensely useful, as we will see. 

Figure 8.7 Henryan activity of sucrose, Raoultian activity of water, and osmotic coefficients in sucrose solutions at 25°C. The inset is an enlargement of the region up to 1 molal, and the circle shows the ideal one molal standard state.

In summary, then, for dissolved substances we use the ideal one molal standard state for Gibbs energy, and the infinite dilution standard state for enthalpy, volume and heat capacity. What about entropy ... 

In summary, a value of 1.0 for 7 allows us to obtain values for the properties of the standard state, which we mentioned earlier is the essential factor in the choice of standard state. As for rn°, any value could be used, but none has any advantage over 1.0. Therefore the hypothetical ideal one molal standard state is in universal use for dilute solution (molality-based) activities. 

Although theoretically we could choose any value for m°, any choice except m° = 1 would introduce complications, and of course we want 7 = 1 so that the standard state lies on the tangent and refers to properties at infinite dilution. This leads to the adoption of the hypothetical ideal one molal standard state for aqueous solutes. If in Figure 8.4 we change the concentration scale to molality, and focus on the lower left corner of the diagram, we have Figure 8.5. We have assumed that B is a nonelectrolyte with v = l such as sucrose or oxygen, and the conversion is... 

The ideal one-molal (1 M) standard state is used for the ion For solutions that are very dilute in the ions (i.e., water without an added electrolyte), we can neglect the solution nonidealities and replace the activities with concentrations in terms of molalities. 

At this point we have shown how the HKF model develops expressions for the standard state parameters V° and C° and hence S°, H°, and G° at high temperatures and pressures. The standard state universally used is the ideal one molal solution, which means that these parameters refer to the properties of ions or electrolytes in infinitely dilute solutions. You might suppose that therefore they would not be of much... 

Because partial molar volume, enthalpy, and heat capacity are the same anywhere on the Henry s law tangent, including both the state of infinite dilution and the ideal one molal solution, either of these states can serve as the standard state for these properties. We have chosen to say that the infinitely dilute solution is the standard state, but many treatments prefer to say that the standard state for these properties, as well as for the Gibbs energy and entropy, is the ideal one molal solution. For some reason, these treatments (e.g., Klotz, 1964, p. 361) then define the reference state for enthalpy, volume and heat capacity... 

The standard state of H20 is the pure liquid at 25 °C. and 1 atm. The standard states of the other species are ideal one molal solution. 

At this point the expression is perfectly general, valid for any conditions, and K is calculable from equation (13.8) if we know the standard state free energies of the three constituents. If the system we are considering is simply quartz in water at T and P, and if we define our standard states to be pure quartz and pure water at T and P, and ideal one molal H4Si04 at T and P, then both quartz and water have activities of 1.0, and... 

For non-electrolytes, we saw that the next step was to define a standard state such that f° was the fugacity of the solute in an ideal one molal solution. Another way of saying this is that f° is the (Henry s Law) constant of proportionality for /ab cx toab... 

The expression RTlnfi gives the difference between at T, P, and of ideal gas i at T, 1 bar, just as RTln m y ) gives the difference between /x. at T, P, and /Uj in an ideal one molal solution at T, P. / is no more a relative value than is So fugacities do not have standard states any more than corrected concentrations have standard states. 

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. 

For the ideal gaseous standard state, is evidently the molar enthalpy of an ideal gas. For standard states based on Henry s law, where y 1 as X ot m 0,lTi is the partial molar enthalpy of the solute in the hypothetical pure substance having yg = 1 or the hypothetical ideal one molal solution respectively. Substances in these strange states have partial molar enthalpies (and volumes) equal to that at infinite dilution, hence providing a method of measurement. This can be seen by considering Equations (8.38) and (8.39), which show that 71° becomes equal to // when y is 1.0. Therefore for Henryan standard states where y, -> 1 as X or m 0, must be the partial molar enthalpy of i at infinite dilution, and for Raoultian standard states where y, 1 as Xj -> 1, //° must be the partial molar enthalpy (the molar enthalpy) of pure i (confirming what we stated by simple inspection, above). 

On the other hand the standard state of a dissolved substance, which behaves in the solvent as a nonelectrolyte, is defined as its state in a hypothetical ideal solution containing one mole of the substance in 1000 grams of solvent (i. e. with a concentration in terms of molality, to = 1). In this solution some properties of the dissolved substance are the same as in an infinitely diluted solution In other cases the hypothetical standard state of the dissolved substance is used in which 1 mole of the substance is contained in one liter of the ideal solution (i. o. the concentration is expressed in terms of molarity, c — 1). 

It is of advantage to choose as the standard state of the undissociated part of the electrolyte its hypothetical unionized state in an ideal solution with the molality m = 1 (or molarity c = 1), and to consider as the standard state of the dissociated part of the substance its hypothetical completely ionized state in an ideal solution with the ion molality m+ = 1 and m — 1. If the chemical potential j.°ab corresponds to the first mentioned standard state, and the potential iA + + Xb to the second one, the difference in the standard free energy A0° between both states is expressed by the equation ... 

The standard state is here also a hypothetical one it is equivalent to a 1 molal solution in which the solute has some of the partial molar properties, e.g., heat content and heat capacity, of the infinitely dilute solution. It has been referred to as the hypothetical ideal 1 molal solution At high dilutions the molality of a solution is directly proportional to its mole fraction ( 32f), and hence dilute solutions in which the activity of the solute is equal to its molality also satisfy Henry s law. Under such conditions, the departure from unity of the activity coefiicient 7 , equal to a2/m, like that of 7n, is a measure of the deviation from Henry s law. 

The standard state of a substance is a reference state that allows us to obtain relative values of such thermodynamic quantities as free energy, activity, enthalpy, and entropy. All substances are assigned unit activity in their standard state. For gases, the standard state has the properties of an ideal gas, but at one atmosphere pressure. It is thus said to be a hypothetical state. For pure liquids and solvents, the standard states are real states and are the pure substances at a specified temperature and pressure. For solutes In dilute solution, the standard state is a hypothetical state that has the properties of an infinitely dilute solute, but at unit concentration (molarity, molality, or mole fraction). The standard state of a solid is a real state and is the pure solid in its most stable crystalline form. 

The standard state for a pure liquid or solid is taken to be the substance in that state of aggregation at a pressure of 1 bar. This same standard state is also used for liquid mixtures of those components that exist as a liquid at the conditions of the mixture. Such substances are sometimes referred to as liquids that may act as a solvent. For substances that exist only as a solid or a gas in the pure component state at the temperature of the mixture, sometimes referred to as substances that can act only as a solute, the situation is more complicated, and standard states based on Henry s law may be used. In this case the pressure is again fixed at 1 bar, and thermal properties such as the standard-state enthalpy and heat capacity are based on the properties of the substance in the solvent at infinite dilution, but the standard-state Gibbs energy and entropy are based on a hypothetical state.of unit concentration (either unit molality or unit mole fraction, depending on the form of Henry s law used), with the standard-state fugacity at these conditions extrapolated from infinite-dilution behavior in the solvent, as shown in Fig. 9.1-3a and b. Therefore just as for a gas where the ideal gas state at 1 bar is a hypothetical state, the standard state of a substance that can only behave as a solute is a hypothetical state. However, one important characteristic of the solute standard state is that the properties depend strongly upon the solvent. used. Therefore, the standard-state properties are a function of the temperature, the solute, and the solvent. This can lead to difficulties when a mixed solvent is used. 

These weird standard states have one very attractive feature, which is that because they all have the same value of the activity of A would always be the same in all three phases at equilibrium. The three standard states could also coexist at equilibrium, if they could exist at all. As mentioned earlier, there is no reason why other concentrations or pressures could not be chosen for the standard states, that is, other than one molal or one bar, as long as ideal behavior is still part of the definition. But these other concentrations or pressures would then appear in all activity calculations and all equilibrium constants, and we would have to give up the convenience of being able to think of gaseous activities as approximate or thermodynamic pressures, and of aqueous activities as approximate or thermodynamic concentrations. It seems generally more convenient to add a little diversity to standard states, and keep activity expressions simple, as is the present custom. 

This table contains selected values for the p/C, standard molar enthalpy of reaction AH°, and standard molar heat-capacity change A C° for the ionization reactions of 64 buffers many of which are relevant to biochemistry and to biologyd The values pertain to the temperature T = 298.15 K and the pressure p = 0.1 MPa. The standard state is the hypothetical ideal solution of unit molality. These data permit one to calculate values of the pK and of AH° at temperatures in the vicinity T (274 K to 350 K) of the reference temperature 0 = 298.15 K by using the following equations ... 

Exceptions are ions, which cannot exist outside of the water solution in pure form. That is why free enthalpy of their formation is compared under conditions of some standard solution. As such was selected a single-molal water solution of one ion with properties of ideal, i.e., infinitely diluted, under standard conditions. In other words, the standard state of dissolved ions is considered rmder conditions of their interaction only with solvent, which is considered a pure substance. 

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