Composition of a barium-borate glass-ceramic: 22-Mec-B8 Engineering Materials, May 2013, Q5

Question 5: Composition of a barium-borate glass-ceramic (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A batch consisting of the parent glass BaO·4B₂O₃ with TiO₂ added as a nucleating agent at 12 mol %, so the batch is 88 mol % BaO·4B₂O₃ and 12 mol % TiO₂. Atomic weights from the periodic table:

Given data — atomic weights
Element	Ba	B	O	Ti
Atomic weight (g mol⁻¹)	137.33	10.81	16.00	47.87

Find. The composition of the resulting glass-ceramic expressed in weight percent.

Approach. Take a convenient basis of 100 mol of batch, convert each constituent from moles to mass through its formula weight, and divide by the total mass; the answer is reported on the oxide basis the batch is written in and then, since the question asks for the molecular weights of each component element, on an elemental basis as well.

Build the formula weights from the atomic weights. The parent glass is one BaO unit plus four B₂O₃ units:$$M_{\mathrm{BaO}} = 137.33 + 16.00 = 153.33\ \text{g}\,\text{mol}^{-1}$$$$M_{\mathrm{B_2O_3}} = 2(10.81) + 3(16.00) = 69.62\ \text{g}\,\text{mol}^{-1}$$so that the glass former as a whole weighs$$M_{\mathrm{BaO\cdot 4B_2O_3}} = 153.33 + 4(69.62) = 431.81\ \text{g}\,\text{mol}^{-1}$$and the nucleating agent weighs $M_{\mathrm{TiO_2}} = 47.87 + 2(16.00) = 79.87\ \text{g}\,\text{mol}^{-1}$.
Convert the mole basis to a mass basis. Working with 100 mol of batch, so 88 mol of glass and 12 mol of TiO₂:$$m_{\text{glass}} = 88 \times 431.81 = 37\,999\ \text{g}, \qquad m_{\mathrm{TiO_2}} = 12 \times 79.87 = 958\ \text{g}$$The batch therefore weighs$$m_{\text{total}} = 37\,999 + 958 = 38\,958\ \text{g}$$
Express the two batch constituents in weight percent. Dividing each mass by the total,$$\mathrm{wt\%_{BaO\cdot 4B_2O_3}} = \frac{37\,999}{38\,958}\times 100 = 97.54\ \%, \qquad \mathrm{wt\%_{TiO_2}} = \frac{958}{38\,958}\times 100 = 2.46\ \%$$The first point of the calculation is already visible here: 12 mol % of TiO₂ is only about 2.5 wt %, because the glass-former formula unit is more than five times heavier than a TiO₂ unit. Mole percent and weight percent are very different quantities whenever the constituents differ in molar mass.
Resolve the glass former into its two oxides. Each mole of the parent glass carries one mole of BaO and four of B₂O₃, so on the oxide basis$$\mathrm{wt\%_{BaO}} = \frac{88(153.33)}{38\,958}\times 100 = 34.64\ \%$$$$\mathrm{wt\%_{B_2O_3}} = \frac{88(4)(69.62)}{38\,958}\times 100 = 62.90\ \%$$together with the 2.46 % TiO₂ found above. These three sum to 100.00 %, which is the arithmetic check.
Express the composition on an elemental basis. The question directs the candidate to the atomic weights of the component elements, so the composition is also required element by element. Counting atoms in 100 mol of batch: 88 mol Ba, 88×8 = 704 mol B, 12 mol Ti, and for oxygen 88×(1 + 12) + 12×2 = 1144 + 24 = 1168 mol O. Multiplying by the atomic weights and dividing by the 38 958 g total gives$$\boxed{\ \mathrm{Ba}\ 31.02\ \%,\quad \mathrm{B}\ 19.53\ \%,\quad \mathrm{Ti}\ 1.47\ \%,\quad \mathrm{O}\ 47.97\ \%\ }$$and these also sum to 100.00 %.

The glass-ceramic is thus overwhelmingly a borate glass by weight, with barium contributing the single largest elemental share after oxygen, and titanium present at under 1.5 wt % — a reminder that a nucleating agent works catalytically, by providing sites, and need not be present in structurally significant amounts.

Composition of the glass-ceramic in weight percent
Basis	Weight percent
Batch constituents	BaO·4B₂O₃ 97.54 %	TiO₂ 2.46 %
Oxides	BaO 34.64 %	B₂O₃ 62.90 %	TiO₂ 2.46 %
Elements	Ba 31.02 %	B 19.53 %	Ti 1.47 %	O 47.97 %

Topic. Batch calculations for glasses and glass-ceramics: converting between mole percent and weight percent, and the role of a nucleating agent.

Key relations. For a batch of constituents i present at n_i moles with formula weights M_i, the weight percent of constituent i is 100 n_iM_i divided by the sum over all constituents of n_jM_j. The elemental weight percent follows from the same expression once each formula unit is decomposed into its atoms.

Why this works. A glass is a frozen liquid: it has no long-range order, so its composition is a bookkeeping statement about which atoms are present rather than about which crystalline phases exist. That is what makes a straight mole-to-mass conversion legitimate. Converting a glass to a glass-ceramic means deliberately crystallising it in a controlled way, and the difficulty is that homogeneous nucleation in a viscous melt is extremely slow. A nucleating agent such as TiO₂ (or ZrO₂, or P₂O₅) is added because it phase separates first and provides a very high density of heterogeneous nucleation sites, so that on the subsequent controlled heat treatment the glass crystallises to a uniform fine grain size throughout its volume rather than growing a few coarse crystals inward from the surface. The fine, pore-free polycrystalline result is what gives glass-ceramics their strength and thermal-shock resistance.

Common pitfall. Reporting the mole percentages as though they were weight percentages — here that would give TiO₂ as 12 % rather than 2.46 %, an error of nearly five times. The second pitfall is losing the factor of four on B₂O₃, or forgetting that each B₂O₃ unit carries three oxygens and two borons when moving to the elemental basis.

Source. Shackelford, Introduction to Materials Science for Engineers, Ch. 12 (ceramics and glasses: glass-ceramics); Askeland & Wright, Ch. 14; Callister & Rethwisch, Ch. 13.

Question 5 of 8: Composition of a barium-borate glass-ceramic

Question 5: Composition of a barium-borate glass-ceramic (20 marks)