Question $1($Precipitation from Supersaturated Solutions$)$

Consider the following equilibrium diagram $[$from Gaskell $2003]$:

The chemical reaction representing the dissolution of $A{l}_2{O}_3$ in alkaline solutions can be written:

$A{l}_2{O}_3+H_2$Oharr$2Al{O}_2^{-}+2H^{+}$

for which $G^0=166\mathrm{.}5k\frac{J}{m}ol$ at $298K.$

Figure $15\mathrm{.}17$ The solubility of $A{l}_2{O}_3$ in aqueous solution.

Show that the equation for the solubility curve in the alkaline region of the equilibrium diagram is given by: $lo{g}_{10}[Al{O}_2^{-}]=pH-14\mathrm{.}59.$

$($b$)$ Suppose that an experimental bauxite processing plant seeks to implement a modified Bayer process that precipitates alumina hydrate at $298K$ by reducing $pH$ through the addition of water to a saturated alkaline solution prior to seeding. Create a plot of the driving force for precipitation $G$ as a function of $pH$ upon rapid dilution of a saturated solution from an initial $pH=14$ down to $pH=7.$

Hint: Start by defining the $Al{O}_2^{-}$supersaturation as a function of $pH.$

$($c$)$ If the specific surface energy and molar volume of alumina hydrate $($gibbsite$)$ are given by $r=5{10}^{-6}\frac{J}{c}{m}^2$ and $V_m=32\mathrm{.}87c\frac{m^3}{m}ol,$ respectively, then plot the critical nucleus radius as a function of $pH$ for the scenario described in part $($b$).$

Question $2($Liquid phase sintering$)$

extra extra

$(3)$ Consider the dansification data shown in the labie at tigmt obtaned from observing the llquid phase sintering of aumins with MgO$-$Al$2$O$,-$SiO, $($MAS$)$ ternary outectic an $1600C.$ Doas the data better suppon diffusion or interface contral as the rabe Iming mechanim for densification? Justify your answer.

Explain the differences in the driving farce betwoen the two primary mechariams by which grain shape accommedation occurs $($T$.$e$.$ contact flastaning and Ostwakd ripening$)$ during solution$-$precipitation stage of Iquid phase sinaring$.$

Q$.$ Suppose the MAS vernary eutactic siquid $(148=72m\frac{N}{m}$ contacts $A{l}_{2}O,($Yer $=188m\frac{N}{m})$ with a wetting angla of $18.$ Calculate the critical value of no that would allow the iquid to spresd over the ceramic with a weting angle of $0".$

$\backslash$table$[[\backslash$table$[[$Sintering$],[$timp$($mini$)]],\backslash$table$[[$Rolative$],[$dansity$]]],[0\mathrm{.}0,0\mathrm{.}58],[0\mathrm{.}2,0\mathrm{.}65],[0\mathrm{.}5,0\mathrm{.}68],[1\mathrm{.}0,0\mathrm{.}70],[1\mathrm{.}5,0\mathrm{.}72],[2\mathrm{.}5,0\mathrm{.}75],[3\mathrm{.}5,0\mathrm{.}78],[4\mathrm{.}5,0\mathrm{.}80],[5\mathrm{.}6,0\mathrm{.}82],[70,0\mathrm{.}84],[9\mathrm{.}5,0\mathrm{.}88],[15\mathrm{.}0,0\mathrm{.}94],[20\mathrm{.}0,0\mathrm{.}99],[,]]$

Question $3($Particle packing$)$

$($a$)$ Estimate the maximum pocking fraction that would result from dense random packing of a mixture of $1$ pm and $10$ um spheres, as well as the appfopriate weight ratio of each type of sphere in the mixture.