Binary ideal solutions

Given below is a typical form of phase diagram for an ideal binary solution ... 

Show that the mole fraction of B in the vapor phase above an ideal binary solution consisting of components A and B with vapor pressures PJ and Pjf is yB = PB(P — P )/P(Pb P%)> when the solution vapor pressure is P. 

The vapour pressure of an ideal binary solution of two components A and B is shown in Fig. 6. It is clear from the graph that the curve of the partial pressure of each component against its mole fraction in the solution is a straight line and the total vapour pressure of the solution for a given concen-... 

In contrast to a perfect solution, a solution is called an ideal solution, if Eq. 8.1 is valid for solute substances in the range of dilute concentrations only. Moreover, the unitary chemical potential p2(T,p) of solute substance 2 is not the same as the chemical potential p2( T,p) of solute 2 in the pure substance p2(T,p) p2(T,p) Henry s law. For the main constituent solvent, on the other hand, the unitary chemical potential p[( T,p) is normally set to be equal to f l p) in the ideal dilute solution p"(T,p) = p°(l p). The free enthalpy per mole of an ideal binary solution of solvent 1 and solute 2 is thus given by Eq. 8.10 ... 

For a perfect binary solution the free enthalpy (Gibbs energy) of mixing per mole has been given in Eq. 8.7. We extend this equation 8.7 to a non-ideal binary solution by using the activity coefficients and y2 as shown in Eq. 8.17 ... 

In this section we shall always define the activity coefficients with respect to the symmetrical reference system. Comparing Eq. 8.7 and Eq. 8.17, we define the excess free enthalpy (excess Gibbs energy) gE per mole of a non-ideal binary solution as Eq. 8.18 ... 

The difference in thermodynamic functions between a non-ideal solution and a comparative perfect solution is called in general the thermodynamic excess function. In addition to the excess free enthalpy gE, other excess functions may also be defined such as excess entropy sE, excess enthalpy hE, excess volume vE, and excess free energy fE per mole of a non-ideal binary solution. These excess functions can be derived as partial derivatives of the excess free enthalpy gE in the following. 

This excess enthalpy hE corresponds to the heat of mixing of the non-ideal binary solution at constant pressure. Namely, hE = xf + x2h with - ht -h - -RT2(dlny JdT), where hf is the partial molar heat of mixing of substance i, ht is the partial molar enthalpy of i in the non-ideal binary solution, and h° is the molar enthalpy of pure substance i. Remind ourselves that the reference system for the activity coefficients is symmetrical. 

This excess volume vE is the difference between the mean molar volume of the non-ideal binary solution, v"° dcal = V+ n2), and the mean molar volume of the perfect binary solution vper/ = XjV + x2v2 (the sum of the volume of the two pure substances before mixing... 

Type I isotherms result where there is essentially no affinity for the surface by the two molecules. Depending upon the distribution coefficient, the adsorption isotherms for ideal binary solutions can be shown in Figure 9.12. Here the distribution coefficient, K, as defined in equation 9.42 determines the shape of these isotherms. 

Here T, is the excess number of moles of compound i at the smface and /u., is the chemical potential of the ith component. For an ideal binary solution at constant temperatme, the smface composition of the solution can be estimated from the smface tensions of the two components by the equation... 

It is of interest to note that the bubble-point line for an ideal binary solution is a linear function of composition. Hiis follows from Raoult s Law since in this case the bubble-point pressure is given by... 

These equations may be used to compute the liquid and vapor compositions of non-ideal binary solutions. 

Now, we calculate the surface tension of an ideal binary solution from Eq. (6.30). Since the system is in equilibrium, it holds... 

One mole of component A and two moles of component B are mixed at 27°C to form an ideal binary solution. Calculation ofDVmix, DHmix DGmix andT>Smix. (R = 8.314 JK-1 mol-1)... 

Fig. 9.6. (a) Variation of the molar free energy (G) of an ideal binary solution A-B. Tangent... 

Fig. 15.2. Gibbs free energy of dissolution for ideal binary solutions from 0 to 2000 K. Gideai diaaoVn and Gidaal soi n decreasB markedly at higher T, making solutions more stable.

Symmetric, siightty non-ideal binary solutions ( Regular )... 

In an ideal binary solution, defined by (4.437), we obtain the Svedbeig formula for measuring of molar mass M (usually of macromolecular substance) in centrifuge with i above (see (4.292)-(4.294))... 

Example 1.6 Surface Tension of Ideal Binary Solutions... 

As an example we discuss the surface composition of an ideal binary solution. For such a solution at a constant temperature the Gibbs equation can be expressed as follows ... 

For an ideal binary solution, the total vapor pressure (Pj) can be determined by combining Equations 9.7 and Raoult s Law (Equation 9.8) ... 

On the molecular level, the interactions between A and B molecules in an ideal binary solution are identical to the interactions between A molecules and the interactions between B molecules. Although no solution is strictly ideal, many come close. For example, benzene (CsHg) and toluene (CyHg) have very similar structures, and therefore very similar intermolecular interactions, and are known to form a very nearly ideal solution when mixed ... 

The plots of the total vapor pressure as functions of the mole fraction of A in both the liquid and vapor phases are shown in Figure 9.12(a) and (b), respectively. The combined plot shown in Eigure 9.12(c) is a liquid-vapor phase diagram for an ideal binary solution at a fixed temperature T—often called a pressure-composition diagram. At any pressure and composition above the upper curve (the liquid line) the mixtnre is a liquid. Below the lower curve (the vapor line), the mixture is entirely vapor. The region between the two curves is a region of phase coexistence, that is, both liquid and vapor phases are present in the system. 

It is worthwhile to consider these equations for the following cases (a) a pure phase, (b) an ideal binary solution, and (c) a nonideal binary solution. 

At constant temperature, p° and pi are constant hence, for an ideal binary solution at constant temperature, the partial pressures and the total pressure are linear functions of the molar composition (Xi or Xj). Example 1.7 in Section 1.3.4 illustrates Raoult s law behavior and deviations from it. 

---