phase diagram

FREEZING POINT DEPRESSION
Represents one line in phase diagram of a condensed system, the line corresponding to pure A. At the temperature Tm, the two phases, solid and liquid, are in equilibrium. Because we will be interested in the temperature region below Tm, let us take pure solid A as the standar state. For the reaction (or phase transformation) from solid A to liquid A, we can write the gibbs free energy change as follows :
At the melting temperature Tm, the two phases are in equilibrium : hence the value for the gibbs free energy change in the reaction is zero. The activity of the liquid is therefore l, the same as the solid. At temperatures lower than Tm, the value of gibbs free energy change for the melting of pure A can be written as
The term L (laten heat of fusion) is introduced for the enthalpy of melting to avoid confusion with the notation for mixing. For simplicity, assume that there is no difference in heat capacity between liquid and solid . in this case, the enthalpy change and entropy change of melting are each independent of temperature. Noting that at the melting temperature,
Based on this equation, it is apparent that the activity of pure liquid A is greater than one at temperatures below Tm, with the solid being considered the standard state. Note that the standard state is defined for each temperature.
Now let us deal with the addition of material B to A. at some temperature T (below Tm). The activity of A in an ideal A-B solution as a function of composition in shown in figure 9.2. consider the case in which A and B are immiscible in the solid state, but form ideal solutions in the liquid state. Liquid of composition is in equilibrium with pure solid A at temperature T. consider now the dissolving of pure, liquid A in the liquid solution.
The dissolution of pure, solid A in the liquid solution is the sum of the two processes above : the melting of pure A, and the dissolution of pure liquid A in the liquid solution.
The gibbs free energy change is the sum . furthermore. If the liquid solution is in equilibrium with the pure solid, then the G = 0 .
In the region of small Xb, therefore, the relationship between xb, the composition of the liquid, and the melting point depression, is

Where T = Tm – T. the melting point depression.

This expression can be plotted on a phase diagram in which temperature is the ordinate and composition is the abscissa. A region of such a diagram is shown as figure 9.3. in the portion of the diagram labeled liquid, the equilibrium phase is a liquid A-B solution. In the two phase region labeled “L + S” (liquid plus solid). Pure solid A is in equilibrium with a liquid solution of composition Xb.
As an example, let us calculate the lowering of the melting point of silver caused by the addition of one mole of lead. The conditions assumed in the derivation of eq.9.4 are followed by the silver-lead system, although there is small solubility of lead in solid silver.
For silver : Tm = 1234 K and L = 11.300
Actually the measured melting point depression is about 10K for an addition of one mole percent lead . considering the slight solubility of lead in silver, the agreement between the calculated and measured values is not bad.
Considering the phase rule in the liquid + solid region (condensed phases)
Because there are two components, A and B, and two phases, liquid and solid. There is only one degree of freedom. Once the temperature has been specified, the composition of the phases at equilibrium is specified. It is important to note that the relative amounts of the phases present (liquid and solid) are not determined by the phase rule. Only the composition of the phases is determined. We show next that if the overall composition of the A-B combination is given, the quantities of the various phases can be calculated using the lever rule.
THE LEVEL RULE
In the two phase region illustrated by figure 9.4. xb represents the overall composition of a system. At temperature T, the phase diagram tell us that the equilibrium liquid composition is xb. Given the overall composition and the composition of the two phases, we can calculate the relative quantities of fractions of liquid and solid using a mass balance.
SIMPLE EUTECTIC DIAGRAM
Consider a system in which materials A and B are immiscible in the solid state, but completely miscible in the liquid state. As shown in section 9.1. the addition of B to A lowers the melting point of A. the reverse is also true. The addition of A to B lowers the melting point of B. this relationship Is illustrated in figure 9.5. although the linear relationship between composition and temperature derived as eq.9.4. may no longer exist, the melting point depressions will continue. When the melting point depression lines intersect, the material will solidify totally into solid A and solid B (figure 9.5). the temperature at which the two curves intersect, called the eutectic temperature, is the lowest temperature at which a liquid solution of A and B may exist at equilibrium with solid A and solid B. the composition at which they intersect is the eutectic composition. At the eutectic point, the phase change may be represented by :
Liquid = solid A + solid B
According to the phase rule, there are two degrees of freedom in the liquid region ; that is, temperature and composition may be arbitrarily fixed. In the region labeled A + liquid, there is only one degree of freedom. Once the temperature has been fixed, the composition of each phase is fixed. The same is true of the region B + liquid.
At the eutectic temperature there are no degrees of freedom because there are two components (A and B) and three phases (solid A, solid B, liquid). Thus F = 0. if all three phases (liquid, solid A, and solid B) are present, one must, at equilibrium, be at the eutectic temperature and the liquid will have the eutectic composition.
COOLING CURVES
If a pure material-pure A. for example-is cooled from a temperature Tm (its melting temperature) to below Tm by removing thermal energy at a constant rate, the temperature of the material as a function of time follows a pattern illustrated in figure 9.6. assuming that equilibrium is maintained at all times. When material A is above T, in the liquid state, the removal of thermal energy lowers its temperature. When the melting point is reached, the removal of thermal energy result in solidification. During solidification, liquid A and solid A are in equilibrium and the temperature of the system does not change. This condition is called a thermal arrest in the cooling curve. Once all of material A has solidified, the temperature decrease resumes.
The same type of cooling curve, with a thermal arrest at the melting temperature, is observed if one cools a liquid of eutectic composition. As temperature drops, liquid will exist until the eutectic temperature is reached. At that temperature, all of the liquid solidifies into solid A and B. when all is solid, the cooling resumes as energy is removed from the system.
At compositions other than the eutectic composition, such as composition Xb in figure 9.7. the cooling rate of the material changes when temperature T is reached. At temperatures below T some pure, solid A is formed upon cooling, and the rate of temperature change is diminished, because to solidify A. energy must be removed from the system. After the material has reached the eutectic temperature, all the remaining liquid solidifies at a constant temperature, causing a thermal arrest at T (figure 9.8).