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3.012 PS 8 thermo solutions
Issued: 11.18.04
Due: 11.24.04
3.012
Fall 2004
THERMODYNAMICS
1. Analyzing binary phase diagrams. Shown on the following page is a hypothetical phase
diagram (Phase diagram 1) for a closed binary system (call the two components generically
A and B). The system behaves as a regular solution in the solid state, and an ideal
solution in the liquid state. Note that the qualitative shape of the molar free energy of the
RS
regular solution G curve vs. composition is similar to the shape of the molar free energy
change on mixing in the regular solution vs. composition ΔG mix,RS = G RS − G heter . Answer
the following questions about the given phase diagram:
(
)
€
a. Using the diagram and the given information, construct qualitatively reasonable
curves for the molar free energies of the
€ solution vs. composition (XB) for each phase
at the temperatures denoted T1 and T2 –you can use the ‘frames’ provided below or
sketch it on another sheet of paper. Mark the identity of each free energy curve you
draw.
See diagrams below.
b. On the free energy diagrams you have sketched, draw in any common tangents that
are present (qualitatively). Along the XB axis, mark the compositions that bound the
ends of each common tangent with a number or letter designation. Finally, mark
these same letters/numbers (from both free energy graphs) along the XB axis of the
phase diagram, to show where those compositions lie on the phase diagram. Use
vertical dashed lines to mark where these compositions intersect with features on the
phase diagram.
See diagrams below.
c. Write an expression for the phase fraction of each phase present at composition X’ at
temperature T1, in terms of the composition points you marked on the phase diagram
in part (b).
The phase fractions for the point (X’, T1) are obtained by applying the lever
rule to the tie line running from X1 to X2 at temperature T1:
f α1 =
X ' X 2 ( X 2 − X')
=
X1 X 2 ( X 2 − X1 )
fL=
X1 X ' ( X'−X1 )
=
X1 X 2 ( X 2 − X1 )
(where the X’s with bars are meant to represent line segments).
€
3.012 PS 8
1 of 4
11/24/04
d. Mark on your free energy diagram for temperature T1 the free energy change that
occurs if liquid with composition X’ transforms to the stable state.
See diagram below.
e. Now, let’s look at phase diagram 2 on page 4. This system also has regular solution
behavior in the solid state and ideal solution behavior in the liquid state. Use your
analyses of phase diagram 1 and your understanding of how the regular solution
mixing free energy varies with temperature to help you predict what the free energy
curves at temperatures T1, T2, and T3 must look like (qualitatively) to obtain this phase
diagram.
The diagrams are shown below. We see that this completely different phase diagram
arises from a rather simple conceptual change in the free energy behavior of the
material: The liquid free energy curve ‘passes through’ the regular solution free
energy curve as temperature drops before the regular solution phase separates at the
critical temperature (i.e., the liquid solidifies to solid α at a temperature above the
critical temperature of the solid α solution). In this situation, a eutectic diagram does
not form.
3.012 PS 8
2 of 4
11/24/04