Materials 3C04 Assignment 5 Solutions 1. Gaskell 10.1 CaF2 and MgF2 are mutually insoluble in the solid state and form a simple binary eutectic system. Calculate the composition and temperature of the eutectic melt assuming that the liquid solutions are Raoultian. The actual eutectic occurs at X CaF2 = 0.45 and T = 1243 K.
Solution Eutectic melt – liquid that has the e utectic composition Liquid and solid solutions behave ideally. In an A-B binary system which contains a e utectic equilibrium and in which the extent of solid solution is negligibly small, the liquidus lines are determined by the conditions (in the liquidus-A melt) (in the liquidus-B melt) where the subscript m denotes melting. If the liquid solutions are ideal, a A = XA and aB = XB, that that is, is, aCaF2 = XCaF2 and aMgF2 = XMgF2, so (in the liquidus-A melt) (in the liquidus-A melt) The equations for the liquidus lines are
At the eutectic point, the A liquidus, B liquidus, and solidus lines intersect. Therefore XCaF2 = 1-XMgF2, and
Since
then
Solving the above equation for T gives T = 1329 1329 K. Also
2. Gaskell 10.2 Gold and silicon are mutually insoluble in the solid state and form a eutectic system with a eutectic temperature of 636 K and a eutectic composition of XSi = 0.186. Calculate the Gibbs free energy of the eutectic melt relative to (a) unmixed liquid Au and liquid Si, and (b) unmixed solid Au and solid Si. At 636 K
Solution The Gibbs free energy of the melt relative to the unmixed state is equal to the Gibbs free energy of mixing. (a)
(b)
3. Gaskell 10.3 Al2O3, which melts melts at 2324 K, and and Cr2O3, which melts at 2538 K, form complete complete ranges of solid and liquid liquid solutions. Assuming that
, and that the solid and liquid solutions in the system
Al2O3-Cr2O3 behave ideally, calculate a) The temperature at which equilibrium melting begins when an alloy of XAl2O3 = 0.5 is heated b) The composition of the melt that first forms c)
The temperature at which equilibrium melting is complete
d) The composition of the last-formed solid
Solution In the binary system Al 2O3-Cr2O3, which forms ideal liquid and solid solutions, solutions, the solidus line is given by
and the liquidus line is given by
a) Equilibrium melting begins when the solidus temperature is reached. The solidus temperature at XAl2O3 = 0.5 is found by solving the following equation for T :
b) The composition of the melt which first forms is given by the liquidus line at the temperature at which melting begins. The composition is found by solving the following equation for X Al2O3(l), where T = 241 2418 8 K:
c)
Equilibrium melting is complete when the liquidus temperature is reached. The liquidus temperature at XAl2O3 = 0.5 is found by solving the following equation for T :
d) The composition of the last-formed solid is given by the solidus line at the temperature at which melting is complete. The composition is found by solving the following equation for X Al2O3(s), where T = 244 2444 4 K:
4. Gaskell 10.5 SiO2, which melts melts at 1723°C, 1723°C, and TiO2, which melts at 1842°C, are immiscible immiscible in the solid state, state, and the SiO2-TiO2 binary system contains a monotectic equilibrium at 1794°C, at which temperature virtually pure TiO2 is in equilibrium with two liquids containing mole fractions of SiO 2 of 0.04 and 0.76. If, for the purpose of simple calculation, it is assumed that the com positions of the two liquids are XSiO2 = 0.24 and XSiO2 = 0.76 and that the liquid solutions are regular in their behaviour, what is the value of Ωl and at what temperature does the liquid immiscibility gap disappear? disappear?
Solution
M
At the composition of the double tangent to the ∆G curve,
Therefore at 2067 K,
Then
