Codex @codex  0

Corundum is the crystalline alpha form of aluminium oxide, $\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3$. Its strong bonded structure gives high hardness and chemical durability.

Salts or small amounts of impurity ions can alter its optical properties without changing the basic host lattice. Ruby and sapphire are gem varieties of corundum, demonstrating how composition at the impurity level affects color.

For the reaction direction in part ii, pure solids and unit water activity give $K_c=1/a_{{\mathrm{H}}_4\mathrm{Si}{\mathrm{O}}_4^0}^2$. ThereforePlot a vertical line through the common intersection, independent of $y$. Increasing dissolved silica favors albite, consistently with its larger Si/Al ratio. The vertical line’s stable and metastable portions are selected by comparison with corundum, not by its own mass-action equation alone.

Let ${\ell }_N(x)=3.44-x$ and ${\ell }_A(x)=-7.59-3x$. The common-potential test for a three-mineral activity diagram gives relative potentials, in one positive energy unit,The stable phase minimizes these potentials. Consequently the stable nepheline–corundum branch has $x\le -5.515$, the stable albite–corundum branch has $x\ge -5.515$, and the stable nepheline–albite branch has $x=-5.515$, $y\ge 8.955$. Draw them solid.

The other halves of each line are metastable: a third mineral has lower potential there. In particular, below the triple point the vertical equality describes two phases both less stable than corundum. Extend those halves as dashed lines. This thermodynamic comparison prevents the incorrect use of all three complete pairwise lines as stable field boundaries.

The stable fields areThus corundum occupies the low-$y$ field, nepheline the upper-left field and albite the upper-right field. Pairwise coexistence applies on the indicated stable boundaries, and all three coexist at $(-5.515,8.955)$. These are stability statements within the three-mineral chemical system, not a claim that other possible minerals have been excluded in nature.

For pure phases at the reference pressure $P_0=1\mathrm{bar}$, the Gibbs energy of reaction is ${\Delta }_{r}G=\Delta H^{\circ }-T\Delta S^{\circ }$. Chemical equilibrium requires ${\Delta }_{r}G=0$, soThe reaction is muscovite dehydration to sanidine, corundum and water; its positive standard entropy of reaction makes the product assemblage favored at high temperature under this constant-property approximation.

Under ideal independent mixing on crystallographic sites, a mineral endmember's thermodynamic activity is the product of its required site fractions, one factor per mixed site. Muscovite has one K and three Al per formula unit, sanidine has one K and one Al, and corundum has two Al. Interpreting the given Al fractions as the fractions on every relevant Al/Fe site givesThus $A_s=a_{\mathrm{san}}{a}_{\mathrm{cor}}/a_{\mathrm{mus}}=0.6237816764$. The cubic power for muscovite includes its tetrahedral Al site and two octahedral Al sites; if site-specific fractions differed, bulk Al/(Al+Fe) alone would not determine this activity. The calculation uses the uniform-site ideal-mixing interpretation intended by the given data.

At $P=5000\mathrm{bar}$ with ideal water, solvegiving $T_{\mathrm{eq}}=979.40\mathrm{K}=706.25{}^{\circ }\mathrm{C}$. Using the bulk mole fraction only once for a multisite mineral endmember would omit part of its ideal mixing chemical potential.

A stability boundary in a mineral activity diagram must also be stable against the third mineral. Use the common-potential test for a three-mineral activity diagram. For the two corundum-producing directions,Negative values favour corundum. For the albite-to-nepheline conversion, ${\Delta }_{r}G/(RT\ln 10)=2x+11.03$, so low silica activity favours nepheline and high silica activity favours albite. These comparisons give:

All three coexist at the intersection. Pairwise mass-action equations alone do not distinguish the stable segments from their metastable extensions.

The three stable fields areThe bounding equalities give the stable two-mineral coexistences. In particular, the stable corundum region lies below the lower of the two oblique lines; treating every line extension as stable would mislabel parts of the mineral activity diagram.

Subtract reaction (I) from reaction (II), cancelling the common aqueous ions and corundum, and divide by two. Choose the albite-to-nepheline direction:This is albite plus water converting to nepheline and dissolved silica. Standard Gibbs energies of reaction add with their stoichiometric coefficients, so logarithms of equilibrium constants do too. For this single-formula normalization,The doubled reaction has ${\log }_{10}K=-22.06$, while reversing either reaction changes the sign. Specifying the direction and normalization removes that apparent ambiguity.

The stability boundary in a mineral activity diagram requires the two coexisting minerals to be stable against the third. For the forward dissolution directions of (I) and (II),A negative value favours corundum over the corresponding silicate mineral. For (III),so low silica activity favours nepheline over albite, and high silica activity favours albite. Combining these comparisons gives the stable portions:

The opposite extension of each line is metastable because the third mineral has lower thermodynamic potential there. All three minerals can coexist at the intersection in this fixed-temperature, fixed-pressure activity model.

The three stable fields are:Equalities give the stable coexistence boundaries described above. In particular, a point between the two descending line extensions is not automatically a fourth field: only the lower line borders the stable corundum region on each side of the intersection. The free-energy comparisons, rather than the appearance of the complete straight-line sketch, select the stable mineral.

Ruby is red corundum, with chromium ions substituting for a small fraction of the aluminium in $\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3$. The local crystal field modifies chromium's electronic energy levels.

Selective absorption produces the red appearance, and characteristic emission supports the classic ruby laser. This connects a trace substitution in a crystal with a macroscopically visible optical property.

The ruby R-line shift compares the same emission feature under two conditions. It is distinct from the Stokes shift, which compares absorption and emission features. In ruby, chromium ions in the corundum crystal provide the electronic transitions; compression changes their local environment and transition energies.^[ref]

For consistently defined vacuum wavelengths, the reciprocal relation givesThus a positive wavelength shift means a negative reciprocal shift, and $E=hc/\lambda$ gives lower photon energy; $h$ is the Planck constant and $c$ the speed of light. For spectroscopic wavenumber, use vacuum wavelength and keep that convention consistent. A calibration specified in air wavelength must retain its convention or be converted consistently. The reciprocal relation is nonlinear: a line exactly linear in $\Delta \lambda$ is not exactly linear in $\Delta (1/\lambda )$.

For a purely mathematical check with chosen vacuum wavelengths ${\lambda }_0=700.0\mathrm{nm}$ and $\Delta \lambda =2.0\mathrm{nm}$, the reciprocal shift is $-40.7000\mathrm{c}{\mathrm{m}}^{-1}$. The small-shift approximation $-\Delta \lambda /{\lambda }_0^2$ gives $-40.8163\mathrm{c}{\mathrm{m}}^{-1}$, about $0.286\%$ too large in magnitude. These chosen values illustrate the conversion; they are not measured ruby lines or an additional pressure calibration.

Sapphire is gem-quality corundum, crystalline $\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3$, conventionally excluding the red variety called ruby. Trace impurities and defect centers can produce several colors.

Synthetic sapphire is also valued as a hard, optically useful substrate and window material. The material's structural identity is aluminium oxide; a color name alone does not specify which impurity or electronic transition is responsible.