Page 125 - Instant notes

Phase diagrams of mixtures 111

composition, x A, given by b would be present. If it were to lie closer to d, then more of
the phase of composition, x A, given by d would be present. The ratio of the number of
moles of the liquid of composition given by b, n b, to the liquid of composition given by d,
n d, for a mixture of overall composition given by e is therefore given by the Lever rule:

where¯denotes the length of a line, so that and are the lengths of the lines between
b and e and between e and d, respectively.
These rules are general, and can be applied to all two-phase regions in any two
component mixture. In all cases, the composition of the two phases can be found by
drawing a horizontal line (a line at constant T) through the state point corresponding to
the temperature and overall composition of the mixture. The compositions of the two
phases will then be determined by the mole fractions of the points where this line
intersects the boundaries of the two-phase region. The Lever rule can then be used to
determine the number of moles of each phase present.
On increasing the temperature, T, the miscibility (solubility) of A and B in each other
often increases. This is the case in Fig. 1a, demonstrated by the fact that the two lines
become closer on increasing T, indicating that more A is required to produce the
saturated solution of A in B, and more B is required to produce the saturated solution of
B in A. Eventually the two curves meet, at the temperature and mole fraction denoted by
T uc and x uc. These are known as the upper consolute temperature and the upper
consolute composition respectively. T uc is the temperature above which only one phase
is present, as liquids A and B are completely miscible at all compositions.
In contrast, if the solubility of A and B in each other increase as T decreases, then the
phase diagram takes the form shown in Fig. 1b. Now the lines get closer together as T
decreases, and at T lc and x lc, the curves meet at the lower consolute temperature and
lower consolute composition (a lower consolute point). The lower consolute
temperature is the temperature below which the liquids are completely miscible and there
is only one liquid phase at all compositions.
In one special case, experimentally found for mixtures of nicotine and water, there is a
range of temperature over which A and B are partially miscible. This results in both an
upper consolute temperature and composition, T uc and x uc, and a lower consolute
temperature and composition, T lc and x lc, as can be seen in Fig. 1c.

Solid-liquid phase diagrams

If two completely miscible liquids, A and B, are cooled sufficiently, then solid will start
to form. When the amount of solute in the solution is small (A in B or B in A), the
freezing temperature, the temperature at which this occurs, is decreased by the addition of
solute (see Topic D3). This produces two lines corresponding to the change in the
freezing points of A and B with x A which can be plotted on the solid-liquid phase diagram
(Fig. 2a).