Standard states fraction

For such components, as the composition of the solution approaches that of the pure liquid, the fugacity becomes equal to the mole fraction multiplied by the standard-state fugacity. In this case,the standard-state fugacity for component i is the fugacity of pure liquid i at system temperature T. In many cases all the components in a liquid mixture are condensable and Equation (13) is therefore used for all components in this case, since all components are treated alike, the normalization of activity coefficients is said to follow the symmetric convention. ... 

Activity coefficients are equal to 1.0 for an ideal solution when the mole fraction is equal to the activity. The activity (a) of a component, i, at a specific temperature, pressure and composition is defined as the ratio of the fugacity of i at these conditions to the fugacity of i at the standard state [54]. 

But that is not all. For dilute solutions, the solvent concentration is high (55 mol kg ) for pure water, and does not vary significantly unless the solute is fairly concentrated. It is therefore common practice and fully justified to use unit mole fraction as the standard state for the solvent. The standard state of a close up pure solid in an electrochemical reaction is similarly treated as unit mole fraction (sometimes referred to as the pure component) this includes metals, solid oxides etc. 

Figure 6.12 shows a graph of u and a2 as a function of mole fraction for mixtures of. yi H O +. Y2CCI4J12 at T = 308.15 K.. A Raoult s law standard state has been chosen for both components. The system shows negative deviation from Raoult s law over the entire range of composition, with it less than. Y and a2 less than. V2. so that all -r.i and 7R.2 are less than 1. 

Equations (7.93) and (7.94) are usually applied to mixtures of nonelectrolytes where Raoult s law standard states are chosen for both components. For these mixtures, Hi is often expressed as a function of mole fraction by the Redlich-Kister equation given by equation (5.40). That is... 

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. 

Standard state unit mole fraction in solution and unit coverage in monolayer. r=-15 °C. 

We connected our earlier definition of activity to a standard state of 1.0 bar or 1.0 M or a mole fraction of unity. None of these make much sense for electrons, but we may define electron... 

Equation 29 implies that is the chemical potential of a hypothetical solution in which XA = 1, but the vapor pressure over the solution still obeys Henry s law as extrapolated from infinite dilution. Thus the standard state is a hypothetical Henry s law solution of unit mole fraction. 

To evaluate the logarithm, we must measure the vapor pressure Pa of A in equilibrium with a solution where its mole fraction is XA in the limit where the solution becomes infinitely dilute. That is, in the limit of infinite dilution where y is 1, the free energy of solvation can be obtained from measurement of the solute vapor pressure (in the appropriate standard state units) over a solution of known concentration. 

The mole fraction X in the previous equation is replaced with a new unitless variable at, the species activity. The standard potentials pt° are defined at a new standard state a hypothetical one-molal solution of the species in which activity and molality are equal, and in which the species properties have been extrapolated to infinite dilution. 

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. 

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. 

Debye-Huckel effects are significant in the dilute range and are not considered, and (2) the usual composition scale for the solute standard state is molality rather than mole fraction. Both of these problems have been overcome, and the more complex relationships are being presented elsewhere (17). However, for most purposes, the virial coefficient equations for electrolytes are more convenient and have been widely used. Hence our primary presentation will be in those terms. 

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. 

Equation A1.3 shows that isotope effects calculated from standard state free energy differences, and this includes theoretical calculations of isotope effects from the partition functions, are not directly proportional to the measured (or predicted) isotope effects on the logarithm of the isotopic pressure ratios. Rather they must be corrected by the isotopic ratio of activity coefficients. At elevated pressures the correction term can be significant, and in the critical region it may even predominate. Similar considerations apply in the condensed phase except the fugacity ratios which define Kf are replaced by activity ratios, a = Y X and a = y C , for the mole fraction or molar concentration scales respectively. In either case corrections for nonideality, II (Yi)Vi, arising from isotope effects on the activity coefficients can be considerable. Further details are found in standard thermodynamic texts and in Chapter 5. 

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... 

The chemical potential of the polymer is affected by "impurities" such as solvents or copolymerized units. For an equilibrium condition in the presence of water as the diluent, the melting temperature of starch (Tm) would be lower because p in the presence of diluent is less than pi). For the starch-water system at equilibrium, the difference between the chemical potentials of the crystalline phase and the phase in the standard state (pure polymer at the same temperature and pressure) must be equal to the decrease in chemical potential of the polymer unit in solution relative to the same standard state (Flory, 1953). By considering the free energy of fusion per repeating unit and volume fraction of water (diluent), the... 

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. 

Solute in Solution. When the mole fraction scale is used, it is convenient to choose a standard state such that the activity would approach the mole fraction in... 

When data are available for the solute over the entire concentration range, from mole fraction 0 to 1, the choice of standard state, either the hypothetical unit mole fraction (Henry s law) or the actual unit mole fraction (Raoult s law), is arbitrary, but it is frequently easier to demonstrate Raoult s law as a limiting law than Heiuy s law. Figure 16.2 shows the relationships for activity and activity coefficient when Heiuy s law is used to define the standard state, and Figure 16.3 shows the same relationships when pure solute is chosen as the standard state. 

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. 

For solvents, 1, is equal to V because the standard state is the pure solvent, if we neglect the small effect of the difference between the vapor pressure of pure solvent and 1 bar. As the standard state for the solute is the hypothetical unit mole fraction state (Fig. 16.2) or the hypothetical 1-molal solution (Fig. 16.4), the chemical potential of the solute that follows Henry s law is given either by Equation (15.5) or Equation (15.11). In either case, because mole fraction and molality are not pressure dependent. 

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