Standard state based on mole fractions

Thus for a gas-phase species, or a species with a standard state based on mole fraction or molality, [9lna, (g)/9E]p . is zero and Eq. 12.1.5 becomes... 

This section considers the solubility of a solid nonelectrolyte. For the solution process B(s) B(sln), the general expression for the thermodynamic equilibrium constant is K = aB(sln)/aB(s). The activity of the pure solid is a (s) = 7b(s). Let us use a solute standard state based on mole fraction then the solute activity is flB(sln) = Fx,b yx,B JCb- From these relations, the solubility expressed as a mole fraction is... 

The equilibrium when two liquid phases are present is B(P) B(a), and the expression for the thermodynamic equilibrium constant, with the solute standard state based on mole fraction, is... 

Consider a two-component system of two equilibrated liquid phases, a and p. If we add a small quantity of a third component, C, it will distribute itself between the two phases. It is appropriate to treat C as a solute in both phases. The thermodynamic equilibrium constant for the equilibrium C(P) C(a), with solute standard states based on mole fraction, is... 

For a solution process in which species B is transferred from a gas phase to a liquid solution, find the relation between AsoiG° (solute standard state based on mole fraction) and the Henry s law constant A h,b. 

Relationships analogous to those given above may be derived in an exactb similar manner for the activities referred to mole fractions or molarities. As seen in 37c, the activities for the various standard states, based on the ideal dilute solution, can be related to one another by equation (37.7). The result is, however, applicable to a single molecular species the corresponding relationships between the mean ionic activity coefficients of a strong electrolyte, assumed to be completely ionized, are found to be... 

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. 

The sum on the right side includes only solute species whose standard states are based on concentration. The expression is simpler if all solute standard states are based on mole fraction or molality ... 

The LFER that results when correlating partitioning in the octanol-water system and the humic substances-water system Implies that the thermodynamics of these two systems are related. Hence, much can be learned about humic substances-water partitioning by first considering partitioning In the simpler octanol-water system. The thermodynamic derivation that follows is based largely on the approach developed by Chlou and coworkers (18-20), Miller et al. (21), and of Karickhoff (J, 22). In the subsequent discussion, we will adopt the pure liquid as the standard state and, therefore, use the Lewls-Randall convention for activity coefficients, l.e., y = 1 if the mole fraction x 1. 

AG°m (pure hquid is the standard state for each substance) is 1194 J tnol at 0°C. In a solution containing only the two isomers, equilibrium is attained when the mole fraction of the 3-methylhexane is 0.372. Is the equihbrium solution ideal Show the computations on which your answer is based. 

G. Scatchard, S. S. Prentice, and P. T. Jones, J. Am. Chem. Soc. 54, 2690 (1932). The activity coefficient of the solute is based on the unit molarity ratio standard state, whereas the activity coefficient of the solvent is based on the unit mole fraction standard state. 

The reference state of each component in a system may be defined in many other ways. As an example, we may choose the reference state of each component to be that at some composition with the condition that the composition of the reference state is the same at all temperatures and pressures of interest. For convenience and simplicity, we may choose a single solution of fixed composition to be the reference state for all components, and designate xf to be the mole fraction of the /cth component in this solution. If (Afikx) represents the values of the excess chemical potential based on this reference state, then (A/if x ) [T, P, x ] is zero at all temperatures and pressures at the composition of the reference state. That this definition determines the standard state is seen from Equation (8.71), for then... 

In the above relationship, the left-hand side term, being ratio of two activity coefficient terms is independent of the standard state chosen. The activity coefficient (generally termed /) based on 1 weight percent solution as the standard state, can very well be used. On the right hand side, N or the mole- (or atom-) fraction, would be replaced by weight per cent of solute and the interaction coefficient would have to absorb within it the corresponding conversion factor for composition. In this form, the interaction coefficient is generally represented by the symbol e . 

Although the standard state could be based on any reference behavior, for simplicity the choices are conventionally limited to one of two main types (Figure 2-1). One is the limiting behavior of a substance as it approaches zero mole fraction (condensed phase) or zero partial pressure (gas phase) this is called henryan reference behavior. The other is the limiting behavior of a substance as it approaches unit... 

In many instances, especially when the solute has a limited solubility in the solvent, as is often the case for solid solutes, particularly electrolytes or where dilute solutions are under consideration, it is more convenient to choose an entirely different standard state. Three such states are used one is based on compositions expressed in mole fractions, but the others are more frequently employed, because the compositions of dilute and moderately dilute solutions are usually stated in terms of molality, i.e., moles per 1000 g. solvent, or in terms of molarity, i.e., moles per liter of solution. ... 

The activity coefficients given by the Debye-Hiickel treatment presumably represent deviations from the dilute solution behavior, i.e., from Henry s law, and are consequently based on the standard state which makes the activity of an ion equal to its mole fraction at infinite dilution ( 37b, III B). In the experimental determination of activity coefficients, however, it is almost invariably the practice to take the activity as equal to the molarity or the molality at infinite dilution. The requisite corrections can be made by means of equation (39.13), but this is unnecessary, for in solutions that are sufficiently dilute for the Debye-Hackel limiting law to be applicable, the difference between the various activity coefficients is negligible. The equations derived above may thus be regarded as being independent of the standard state chosen for the ions, provided only that the activity coefficients are defined as being unity at infinite dilution. 

For many solutions, it is not possible to vary the composition of the components over the whole range of mole fractions. This is obviously true of solutions made up of a solid and a liquid. For these systems it is better to choose a standard state which is based on the properties of a dilute solution. This leads to the definition of an ideally dilute solution. Such a system is easily defined on a molecular basis as one in which the solute molecule only comes in contact with solvent molecules, and never with another solute molecule. In the previous discussion of regular solutions it was concluded that, when the two components are of equal size, the coordination number for the other molecules around a central one is twelve. This suggests that an ideally dilute solution must have a solute mole fraction which is less than 1/13, that is, 0.08. 

Dalton s law is used to express partial pressures in terms of mole fractions and total pressure. Total pressure must be expressed in atmospheres if equilibrium, p IS Calculated via the dimensionless thermodynamic equation for / equilibrium, /, which is based on standard-state enthalpies and entropies of reaction at 298 K. Furthermore, the dimensions of / equilibrium, p are the same as those of / s°tondard state P (atm) for this problem]. The latter equilibrium constant, which is based on standard-state fugacities of pure components at 1 atm and 298 K, has a magnitude of 1 when total pressure in the kinetic rate law is expressed in atmospheres. Hence,... 

Figure 5.6 Schematic of the composition dependence of the fugacity /j and activity coefficient Yj in a binary mixture at fixed T and P. This activity coefficient is based on the Lewis-Randail standard state (5.4.11) and therefore satisfies the pure-fluid (5.4.12) and dilute-solution (5.4.13) limits. Note that the fugacity of the ideal-solution (broken line) is linear in the mole fraction and that, in the Lewis-Randail standard state, f = j.

When we choose a standard state, we are merely identifying a particular ideal solution on which to base an activity coefficient. The standard state may be real or hypothetical, so long as it is weU-defined and so long as a value for its fugacity can be obtained. Ultimately the choice of standard state is made for computational convenience normally this means either that reliable models for y, exist, or else that the value of Yi is close to unity over the states of interest. When neither of these conditions pertain, we should consider changing the standard state. In many situations the appropriate choice is one of the possibilities discussed in 10.2.1-10.2.3 however, when the mole fraction is not a convenient measure of composition, such as occurs for mixtures of electrolytes or of polymers, then other standard states may be preferred. 

We can use a Raoultian standard state (pure water) for the solvent, but its deviation from ideal behavior, whether based on a mole fraction or a molality scale, is often converted to the osmotic coefficient , which does not actually have a standard state. It is an absolute system property. 

For a given reaction at a given temperature, we can derive relations between values of K that are based on different solute standard states. In the limit of infinite dilution, each solute activity coefficient is unity, and at the standard pressure each pressure factor is unity. Under these conditions of infinite dilution and standard pressure, the activities of solute B on a mole fraction, concentration, and molality basis are therefore... 

The volume indications always refer to the standard state (1.013 bar, 273.15 K) parts by volume are based on ideal gas volumes (mole fractions). 

The three contributions of special significance here are those of Frank and Evans, Claussen and Polglase, and N6methy and Scherage. The calculation is based on the transfer of one mole of hydrocarbon from the pure liquid hydrocarbon to dilute solution in water. The standard states were taken as mole fraction unity (Nhc = 1) in the liquid and the solution, and the pressure 1 atm in the gaseous state ... 

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