Ideal Gaseous Solutions

The important models with which we are concerned are the perfect gas, the perfect gas mixture and the ideal solution (gaseous, liquid or solid). These may be defined in either of two ways which are entirely equivalent (1) in terms of limiting experimental laws such as the gas equation and Raoult s law (2) in terms of expressions for the chemical potentials of the various components. These expressions are as follows ... 

Equations (11) and (12) are based on ideal solutions (gaseous and liquid, respectively) and are, therefore, only approximately accurate for real liquid or gaseous solutions. The inadequaey of the ideal solution model to predict real solution behavior is testimony to the need for the activity coefficient. 

All the above deals with gases and gas phase processes. We now turn to non-gaseous components of the system. There are many ways of expressing this. Probably the simplest is to consider an ideal solution of a solute in a solvent. If the solution is ideal, the vapour pressure of the solute is proportional to its concentration, and we may write p = kc, where c is the concentration and k is the proportionality constant. Similarly, = Arc , which expresses the fact that the standard pressure is related to a standard concentration. Thus we may write from equation 20.198 for a particular component... 

The summation is taken over all species (including inerts) present in the system. For gaseous mixtures that follow ideal solution behavior the partial molal quantities may be replaced by the pure component values. 

Consider a dilute ideal solution of the solute B (which could be gaseous, liquid, or solid at the temperature in question) in the solvent A. Suppose that more concentrated solutions do not behave ideally and, in particular, the state of pure liquid B cannot be attained by going to more and more concentrated solutions (e.g., by removing A by volatilization). It is possible to define a standard chemical potential pertaining to a hypothetical standard state of the ideal infinitely dilute solution as the limit ... 

The physical state of each substance is indicated in the column headed State as crystalline solid (c), liquid (liq), gaseous (g), or amorphous (amorp). Solutions in water are listed as aqueous (aq). Solutions in water are designated as aqueous, and the concentration of the solution is expressed in terms of the number of moles of solvent associated with 1 mol of the solute. If no concentration is indicated, the solution is assumed to be dilute. The standard state for a solute in aqueous solution is taken as the hypothetical ideal solution of unit molality (indicated as std state, m = 1). In this state the partial molal enthalpy and the heat capacity of the solute are the same as in the infinitely dilute real solution (aq. m). 

Amagat s law of additive volumes holds for all pressures, which means that the ideal-solution law holds for the gaseous mixture, but not necessarily for the pure gases per se, and therefore the fugacity (/) is given by... 

Before we begin to study reactions in detail we discuss the concept of mixing without a reaction taking place. We now give an example of the use of chemical potentials to study the thermodynamics of mixing for two ideal solutions designated A and B which can be either solid, liquid or gaseous ... 

Phase equilibria of vaporization, sublimation, melting, extraction, adsorption, etc. can also be represented by the methods of this section within the accuracy of the expressions for the chemical potentials. One simply treats the phase transition as if it were an equilibrium reaction step and enlarges the list of species so that each member has a designated phase. Thus, if Ai and A2 denote liquid and gaseous species i, respectively, the vaporization of Ai can be represented stoichiometrically as —Aj + A2 = 0 then Eq. (2.3-17) provides a vapor pressure equation for species i. The same can be done for fusion and sublimation equilibria and for solubilities in ideal solutions. 

Substituting the appropriate ideal expression for the activity of gaseous or dissolved species from Equation 14.8a or 14.8b leads to the forms of the mass action law and the equilibrium constant K already derived earlier in Section 14.3 for reactions in ideal gases or in ideal solutions. We write the mass action law for reactions involving pure solids and liquids and multiple phases by substituting unity for the activity of pure liquids or solids and the appropriate ideal expression for the activity of each gaseous or dissolved species into Equation 14.9. Once a proper reference state and concentration units have been identified for each reactant and product, we use tabulated free energies based on these reference states to calculate the equilibrium constant. 

Relative to the levels of the species we have been considering, water vapor is at a high concentration in the atmosphere. Liquid water, in the form of clouds and fog, is frequently present. Small water droplets can themselves be viewed as microscopic chemical reactors where gaseous species are absorbed, reactions take place, and species evaporate back to the gas phase. Droplets themselves do not always leave the atmosphere as precipitation more often than not, in fact, cloud droplets evaporate before coalescing to a point where precipitation can occur. In terms of atmospheric chemistry, droplets can both alter the course of gas-phase chemistry through the uptake of vapor species and act as a medium for production of species that otherwise would not be produced in the gas phase or would be produced by different paths at a lower rate in the gas phase (Fig. 10). Concentrations of dissolved species in cloud, fog, and rain droplets are in the micromolar range, and therefore one usually assumes that the atmospheric aqueous phase behaves as an ideal solution. 

TYPES OF IDEAL SOLUTIONS 10.2.1. Ideal Gaseous Solutions... 

Despite the dissimilarities in these hypothetical models of ideal gaseous, liquid, and solid solutions, we see that they share a number of important properties. All ideal solutions, no matter what the phase, have no heat of mixing when prepared from their components, and total volumes must simply be the sum of the individual volumes of the components before mixing. 

Next we will derive equation (10.9) one final time, generalizing to all ideal solutions, whether gaseous, liquid, or solid. Recall that... 

The form of Eq. (11.14) suggests a generalization. Suppose we define an ideal mixture, or ideal solution, in any state of aggregation (solid, liquid, or gaseous) as one in which... 

Figure 5.5 Excess properties for gaseous mixtures of methane and sulfur hexafluoride at 60°C and 20 bar computed from the virial equation (5.3.3) using (5.3.9)-(5.3.11). Excess properties relative to Lewis-Randall ideal solution (5.1.6).

In 4.5.5 we computed residual properties for gaseous mixtures of methane and sulfur hexafluoride mixtures at 60°C and 20 bar. In 5.3.1 and 5.3.2 we computed excess properties for this same mixture. We can also compute residual properties for the ideal solution (Lewis-Randall standard state). Comparisons of these three kinds of difference measures are shown in Table 5.1 for equimolar mixtures. We see that the equimolar mixture of methane and sulfur hexafluoride exhibits positive deviations... 

Understanding the choice of standard states in a problem is critical to proper treatment. Sometimes the standard state is one which does not exist at all, but can be readily pictured, hypothetically. For example, most gas mixtiu es do not behave in an ideal fashion. The molecules occupy space (they are not point molecules) they will interact to some extent unless they are infinitely far apart Hence, the commonly used standard state for gaseous substances is defined as hypothetical partial pressure of one atmosphere. Hypothetical, that is, because at one atmosphere, real gases will require some correction in their free energy value to compensate for their volumes and interactions. Analogously, the standard state for solutes commonly used is hypothetical one molal cMwcentration i.e., the concentration of an ideal solute in an ideal solution that would result in the value of the standard free energy. In real solutions, a correction would have to be... 

This result may be compared with equation (3 72) which refers to an ideal gaseous solution and where // is the fugacity of the pure vapour component at the temperature and tote pressure under discussion. If it occurs that the liquid and vapour phases in equilibrium are both ideal solutions, and remain ideal up to we obtain... 

This difference of chemical potentials is equal to the difference in the free energy of formation of ammonia in a hypothetical ideal solution of unit total molality and the free energy of formation of gaseous ammonia at unit pressure. The latter quantity is known from measurements on gas equilibria to be —3976 cal mol at 25 C. Therefore the former quantity has the value —3976—2384=—6360calmol . Thus... 

A solution is a homogeneous mixture. For example, air is a gaseous solution of several gases, seawater is a liquid solution of sodium chloride and other materials. The component of a solution that is present in the greatest amount, and therefore determines the state of matter (solid, liquid or gas) of the solution, is called the solvent the other components are called solutes. A solution in which water is the solvent (e.g., seawater) is called an aqueous solution. An ideal solution is one for which both solvent and solutes obey RaoulLs law (see Section 4.4) at all concentrations. 

This general treatment is applicable to any ideal solutions whether these are gaseous, liquid, or solid, and it applies also to defect chemistry as we have examplified initially. 

For gaseous species satisfying Henry s law for the system under consideration, /g = Hi. Further, if the liquid phase behaves tts an ideal solution and the gas phase behaves as an ideal gets. 

A solution is a homogeneous mixture of two or more components (substances whose amounts can be independently varied). We ordinarily apply the name only to solid and liquid mixtures, although a gaseous mixture is also homogeneous. We begin with ideal solutions, which are defined to be solutions in which the chemical potential of each component is given for all compositions by the formula... 

The composition of the vapor phase at equilibrium with a liquid solution is not the same as the composition of the liquid solution. If an ideal gas mixture is at equilibrium with a two-component ideal solution, the mole fraction of component 1 in the gaseous phase is given by Dalton s law of partial pressures ... 

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