Real solutions activities

No actual solutions are ideal, and many solutions deviate from ideal-dilute behavior as soon as the concentration of solute rises above a small value. In thermodynamics we try to preserve the form of equations developed for ideal systems so that it becomes easy to step between the two types of system. This is the thought behind the introduction of the activity, flj, of a substance, which is a kind of effective concentration. The activity is defined so that the expression 

For ideal solutions, Uj = Xj, and the activity of each component is equal to its mole fraction. For ideal-dilute solutions using the definition in eqn 3.17, ag = [B]/c, and the activity of the solute is equal to the numerical value of its molar concentration. For o -ideal solutions we write 

An added advantage is that there are fewer equations to remember  

Because the solvent behaves more in accord with Raoult s law as it becomes 

Because the solute behaves more in accord with Henry s law as the solution 

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Chemical Potentials of Real Solutions. Activity, a and Activity Coefficients, f... 

In an ideal solution activities are equal to concentrations. In real solutions, activity coefficients are introduced to correct for the nonideal effects of the different solutes. 

In these equations addends J rdnic, and i T-lny. characterize deviation of the solutions from ideal and the work, which is necessary to expend in order to squeeze 1 mole of component i of the ideal solution into real solution. Activity coefficients can be greater or smaller than 1. When pressure of a gas solution or concentration of dissolved substances tends to 0, fugacity coefficients or activities coefficients approach 1. Even in diluted real solutions charged ions and dipole molecules experience electrostatic interaction, which shows up in a decrease of activities coefficient. Only in very diluted solutions this interaction becomes minuscule, and fugacity and activities values tend to values of partial pressure and concentration, respectively. Table 1.3 summarizes calculation formulae for activities values of groxmd water components under ideal and real conditions. 

For ideal solutions, the activity coefficient will be unity, but for real solutions, 7r i will differ from unity, and, in fact, can be used as a measure of the nonideality of the solution. But we have seen earlier that real solutions approach ideal solution behavior in dilute solution. That is, the behavior of the solvent in a solution approaches Raoult s law as. vi — 1, and we can write for the solvent... 

An ideal solution is an exception rather than the rule. Real solutions are, in general, nonideal. Any solution in which the activity of a component is not equal to its mole fraction is called non-ideal. The extent of the nonideality of a solution, i.e., the extent of its deviation from... 

Activity ax is termed the rational activity and coefficient yx is the rational activity coefficient This activity is not directly given by the ratio of the fugacities, as it is for gases, but appears nonetheless to be the best means from a thermodynamic point of view for description of the behaviour of real solutions. The rational activity corresponds to the mole fraction for ideal solutions (hence the subscript x). Both ax and yx are dimensionless numbers. 

It is generally observed that as the temperature increases, real solutions tend to become more ideal and r can be interpreted as the temperature at which a regular solution becomes ideal. To give a physically meaningful representation of a system r should be a positive quantity and larger than the temperature of investigation. The activity coefficient of component A for various values of Q AB is shown as a function of temperature for t = 3000 K and xA = xB = 0.5 in Figure 9.3. The model approaches the ideal model as T - t. 

For C crit t 161"6 ls a double root to the maximization equation, and there is an inflection point in the AG function (curve D on Figure 1). Since there is no activation barrier to opening up the etch pit, any pit nucleated at a dislocation should open up into a macroscopic etch pit. Similarly, for C < Ccr t, there are no real solutions and no maxima and minima in the A G function, and nucleated pits open up into etch pits. At 300°C, the calculated Ccr t for quartz equals 0.6CQ. 

The case of a real solution, shown by equation 2.62, is handled by introducing a dimensionless correction, called the activity coefficient (y ). In the high dilution limit (mi -> 0), Xi = 1, and equation 2.62 reduces to 2.61. The activity of species i is simply a, = ypnjma. 

The solid curve in Figure 16.1 shows the activity of the solvent in a solution as a function of the mole fraction of solvent. If the solution were ideal, Equations (14.6) and (16.1) would both be applicable over the whole range of mole fractions. Then, fli =Xi, which is a relationship indicated by the broken line in Figure 16.1. Also, because Equation (16.1) approaches Equation (14.6) in the limit as Xj 1 for the real solution, the solid curve approaches the ideal line asymptotically as Xj 1. 

Thus, the ideal solution is a reference for the solvent in a real solution, and the activity coefficient of the solvent measures the deviation from ideality. 

We have pointed out that a concentration m2(o of the solute in the real solution may have an activity of 1, which is equal to the activity of the hypothetical 1-molal standard state. Also, Hm2, the partial molar enthalpy of the solute in the standard state, equals the partial molar enthalpy of the solute at infinite dilution. We might inquire whether the partial molar entropy of the solute in the standard state corresponds to the partial molar entropy in either of these two solutions. 

We can summarize our conclusions about the thermodynamic properties of the solute in the hypothetical 1-molal standard state as follows. Such a solute is characterized by values of the thermodynamic functions that are represented by p2. 77m2. and 5m2- Frequently a real solution at some molality m2(j) also exists (Fig. 16.4) for which p.2 = that is, for which the activity has a value of 1. The real solution for which // i2 is equal to H 2 is the one at infinite dilution. Furthermore, 5 n,2 has a value equal to 5 2 for some real solution only at a molahty m2(k) that is neither zero nor m2( j). Thus, three different real concentrations of the solute exist for which the thermodynamic qualities p,2, //mi. and S a respectively, have the same values as in the hypothetical standard state. 

In general, activity is merely an alternative way to express chemical potential. The general objective is to express fil in a form that emulates the ideal gas expression (6.55), but with the actual vapor pressure PL of component i (rather than that assumed from Dalton s law). The trick will be to choose a standard-state divisor in (6.55) that makes this expression valid for the components of a real solution. 

Equation 6-24 and the equations that follow from it apply to molal activities. However, the concentration can be substituted for activity in very dilute solution where the behavior of the dissolved molecules approximates that of the hypothetical ideal solution for which the standard state is defined. For any real solution, the activity can be expressed as the product of an activity coefficient and the concentration (Eq. 6-26). 

It is unlikely that any real solution could possess the stringent qualiLcations that deLne the ideal solution. Indeed, the ideal solution exists only when a solute is dissolved in itself as the liquid solvent. The development of a theory for nonideality amounts to quantitatively estimating an activity coefLcient for the solute in the nonideal solution. Irrespective of the nature of the nonideal solution... 

These interactions exist in real systems for which the activity has the physical meaning of the effective concentration. Thus, only for dilute real solutions does A = a. In mixtures, the activity coefficient is usually, but not always, less than one and is affected by all the species in the multicomponent mixture. 

As real solutions may depart from ideality in a number of different ways, the activity coefficients in the above equations may take various expressions. We here discuss one... 

The basis of the ideal solution model is that the thermodynamic activities of the components are the same as their mole fractions. Implicit in this assumption is the idea that the activity coefficients are equal to unity. This is at best an approximation and has been found to be invalid in most cases. Solutions in which activity coefficients are taken into account are referred to as "real" solutions and are described by equation 3.4. 

Thus activity, av is effective concentration at which an ideal solution has the same chemical 0 potential as the real solution. 

Figure 39.2 The graphs show plots of (a) /r(i)

The solutions are considered so dilute that the effect of ionic strength can be neglected (ideal solution). Mathematical expressions derived so far in this chapter use a molar concentration term. If the chemical activity deviates from ideal solution behavior, the ionization of a weak acid or weak base may be given in terms of activity rather than molar concentration to account for interactions in the real solution as follows ... 

For dissolved substances (nonelectrolytes), present in solutions in lesser quantities, the state of a hypothetical ideal solution with a molality m — 1 or a molarity c = 1 is usually chosen as the standard state. These standard states are derived by extrapolation from the reference state of an infinitely diluted real solution, in which the activity of the solute equals the molality or molarity, to the region of the ideal solution of unit concentration. 

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