MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Thermodynamics of Materials

3.00 Fall 2000

W. Craig Carter

Department of Materials Science and Engineering

Massachusetts Institute of Technology

77 Massachusetts Ave.

Cambridge, MA 02139

Problem Set 9: Due Fri. Dec. 8, Before 5PM in 13-5049

Exercise 9.1

Consider a binary alloy comprised of two components $A$ and $B$. Both pure $A$ and pure $B$ solidify into an FCC structure that will be designated as the $\gamma$-phase.

The molar Gibbs Free Energies¹of the liquid and $\gamma$ solutions are given as:

$$\ensuremath{\overline{G^\gamma}}= (X_B - \frac{1}{4})^2$$ (1)

and

$$\ensuremath{\overline{G^{liq}}}= (X_B - \frac{3}{4})^2 + \frac{T - T_A}{T_B - T_A} - \frac{1}{2}$$ (2)

where $T$ is the temperature, $X_B$ is the composition, and $T_A$ and $T_B$ are constants.

1. Calculate the lowest and highest temperatures, $T_L$ and $T_H$, that an equilibrium liquid can co-exist with an equilibrium solid phase and calculate the corresponding compositions of the liquid and solid phase at $T_H$ and $T_L$.

2. For temperatures $T_1 = T_L + \frac{1}{3}(T_H - T_L)$ and $T_2 = T_L + \frac{2}{3}(T_H - T_L)$, calculate and plot $\mu_A$ as a function of $X_B$.

3. For temperatures $T_1 = T_L + \frac{1}{3}(T_H - T_L)$ and $T_2 = T_L + \frac{2}{3}(T_H - T_L)$, calculate the equilibrium composition of the liquid and the $\gamma$ phase.

4. For the composition $X_B = \frac{1}{2}$, plot $\mu_A$ as a function of $T$, $T_L < T < T_H$.

5. Sketch the phase diagram of this alloy.