For ideal solutions

The entropy of a solution is itself a composite quantity comprising (i) a part depending only on tire amount of solvent and solute species, and independent from what tliey are, and (ii) a part characteristic of tire actual species (A, B,. ..) involved (equal to zero for ideal solutions). These two parts have been denoted respectively cratic and unitary by Gurney [55]. At extreme dilution, (ii) becomes more or less negligible, and only tire cratic tenn remains, whose contribution to tire free energy of mixing is... 

A solution which obeys Raoult s law over the full range of compositions is called an ideal solution (see Example 7.1). Equation (8.22) describes the relationship between activity and mole fraction for ideal solutions. In the case of nonideal solutions, the nonideality may be taken into account by introducing an activity coefficient as a factor of proportionality into Eq. (8.22). 

Since the 0 s are fractions, the logarithms in Eq. (8.38) are less than unity and AGj is negative for all concentrations. In the case of athermal mixtures entropy considerations alone are sufficient to account for polymer-solvent miscibility at all concentrations. Exactly the same is true for ideal solutions. As a matter of fact, it is possible to regard the expressions for AS and AGj for ideal solutions as special cases of Eqs. (8.37) and (8.38) for the situation where n happens to equal unity. The following example compares values for ASj for ideal and Flory-Huggins solutions to examine quantitatively the effect of variations in n on the entropy of mixing. 

Evaluate ASj for ideal solutions and for athermal solutions of polymers having n values of 50, 100, and 500 by solving Eqs. (8.28) and (8.38) at regular intervals of mole fraction. Compare these calculated quantities by preparing a suitable plot of the results. 

We express the calculated entropies of mixing in units of R. For ideal solutions the values of are evaluated directly from Eq. (8.28) ... 

At equilibrium, a component of a gas in contact with a liquid has identical fugacities in both the gas and liquid phase. For ideal solutions Raoult s law applies ... 

For ideal solutions (7 = 1) of a binary mixture, the equation simplifies to the following, which appHes whether the separation is by distillation or by any other technique. 

When Eq. (4-282) is applied to XT E for which the vapor phase is an ideal gas and the liquid phase is an ideal solution, it reduces to a veiy simple expression. For ideal gases, fugacity coefficients and are unity, and the right-hand side of Eq. (4-283) reduces to the Poynting factor. For the systems of interest here this factor is always veiy close to unity, and for practical purposes <1 = 1. For ideal solutions, the activity coefficients are also unity. Equation (4-282) therefore reduces to... 

Equation (13-50) is used to calculate, from the previous stage, the (f/d) ratio on each stage in the rectifying section. The assumed temperature and phase-rate-profile assumptions conveniently fix all the A values for ideal solutions. The calculations are started by writing the equation for stage N ... 

Deviations in which the observed vapor pressure are smaller than predicted for ideal solution behavior are also observed. Figure 6.8 gives the vapor pressure of. (CHjCF XiN +. viCHCfi at T — 283.15 K, an example of such behavior,10 This system is said to exhibit negative deviations from Raoult s law. 

Therefore, it is a sufficient condition for ideal solution behavior in a binary mixture that one component obeys Raoult s law over the entire composition range, since the other component must do the same. 

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... 

The activity is a measure of the tendency of a substance to react relative to its reacting tendency in the standard state. Here we relate activity to c/c for ideal solutions. For ideal gases and ideal solvents, the activity approaches P/P and X, respectively. Although c is taken to be 1.0 M, Equation (8) works best when c is much less than 1.0 M. 

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. 

For ideal solutions, the activity coefficients are unity, giving ... 

Vapour pressure osmometry is the second experimental technique based on colligative properties with importance for molar mass determination. The vapour pressure of the solvent above a (polymer) solution is determined by the requirement that the chemical potential of the solvent in the vapour and in the liquid phase must be identical. For ideal solutions the change of the vapour pressure p of the solvent due to the presence of the solute with molar volume V/1 is given by... 

Before closing this section we note that even in nonideal solutions we can use the standard state of Equation 16 for the solute. Since Equation 16 only holds for ideal solutions, one generalizes to obtain48... 

For ideal solutions Q. is zero and there are no extra interactions between the species that constitute the solution. In terms of nearest neighbour interactions only, the energy of an A-B interaction, mab, equals the average of the A-A, uAA, and B-B, wBB, interactions or... 

Background If a nonvolatile solid is dissolved in a liquid, the vapor pressure of the liquid solvent is lowered and can be determined through the use of Raoult s Law, Pi = X 0. Raoult s Law is valid for ideal solutions wherein AH = 0 and in which there is no chemical interaction among the components of the dilute solution (see Figure 1). 

The constant (Kp) is also known as the distribution coefficient or the partition coefficient. Interestingly, this particular relation [Eq. (a)) was originally derived for ideal solutions only, but it caters for a fairly good description of the behavioural pattern of a number of real-extraction-systems encountered in the analysis of pharmaceutical substances. However, the Partition Law offers the following two limitations, namely ... 

For solutions obeying Henry s law, as for ideal solutions, and for solutions of ideal gases, the chemical potential is a linear function of the logarithm of the composition variable, and the standard chemical potential depends on the choice of composition variable. The chemical potential is, of course, independent of our choice of standard state and composition measure. 

If the activity coefficients are known (unity for ideal solution behavior), this coupled set of first-order differential equations can be solved numerically to obtain the radius and composition as functions of time. 

For ideal solutions (x = 0). the melting temperature of the crystallizable polymer from solutions follows as... 

K being a constant which is usually determined experimentally during cell calibration. Lj is the heat of evaporation of the solvent, the density of the solution, and c the polymer concentration. Finally, because the given deviation is valid only for ideal solutions but only real solutions can be studied in practice, the above equation is developed in a power law series with respect to c ... 

For ideal solutions the relationship between the chemical potential and the concentration is given by the expression... 

Figure 3-36 The dependence of interdiffusivity on composition for two models (Equations 3-137c versus 3-138b) for ideal solutions and concentration-independent T>a and X>b- The solid curve is for interdiffusion of two ions of identical charge. The dashed curve is for interdiffusion of neutral atomic species such as in an alloy.

For ideal solutions the osmotic pressure is simply given by CkT. Hence, the osmotic pressure difference between the mid-plane region and the... 

In general, if rarh = 1 and if the chain termination constants for oxidations of A alone and B alone are the same ( ideal reactivities), then the total rates of oxidation of mixtures at constant rate of initiation are a linear function (for ideal solutions) of the volume % of B in the A—B feed. Russell (30,31, 32) and Alagy and co-workers (1, 2,3,4, 8, 33) have shown for several systems that when B has a higher termination constant than A and when B is sufficiently reactive, B can reduce (sometimes fourfold) the rate of oxidation to less than the ideal rate. A secondary objective was to see if there were any other important abnormalities (particularly rates much greater than ideal) in co-oxidations of hydrocarbons. 

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