Composition of a barium-borate glass-ceramic in weight percent: 22-Mec-B8 Engineering Materials, December 2016, Q2

Question 2: Composition of a barium-borate glass-ceramic in weight percent (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 of 8.5 mole per cent TiO₂ nucleating agent in 91.5 mole per cent of the barium-borate glass BaO·4B₂O₃, with atomic weights read from the periodic table.

Given data
Quantity	Value
Glass former	BaO·4B₂O₃
Nucleating agent	TiO₂, 8.5 mole%
Atomic weight of Ba	137.33
Atomic weight of B	10.81
Atomic weight of O	16.00
Atomic weight of Ti	47.87

Find. The composition of the resulting glass-ceramic in weight percent. The question asks for the atomic weights “of each component element,” so the answer is reported on the oxide basis (which is how a batch is actually weighed out) and again on the elemental basis (which is what those atomic weights directly give).

Approach. Take a convenient basis of 100 moles of batch, convert each constituent from moles to mass through its formula weight, and express each mass as a percentage of the total. The whole calculation is one mole-to-mass conversion carried out three ways.

Build the formula weights from the atomic weights. Adding the constituent atoms, $$M_{\text{BaO}} = 137.33 + 16.00 = 153.33$$ $$M_{\text{B}_{2}\text{O}_{3}} = 2(10.81) + 3(16.00) = 69.62$$ so the glass itself, one BaO with four B₂O₃, weighs $$M_{g} = 153.33 + 4(69.62) = \boxed{431.81\ \text{g/mol}}$$ and the nucleating agent weighs $M_{\text{TiO}_{2}} = 47.87 + 2(16.00) = 79.87$ g/mol. The glass unit is 5.4 times heavier than the titania unit, and that single ratio governs everything that follows.
Choose a basis and convert moles to mass. Taking 100 mol of batch, of which 8.5 mol is TiO₂ and the remaining 91.5 mol is glass: $$m_{g} = 91.5(431.81) = 39\,510.62\ \text{g}$$ $$m_{t} = 8.5(79.87) = 678.90\ \text{g}$$ and the batch total is $m_{tot} = 40\,189.51$ g. Every percentage below is a fraction of this one number.
Report the two batch constituents in weight percent. Dividing through, $$\text{wt\% glass} = \frac{39\,510.62}{40\,189.51}\times100, \; \text{wt\% TiO}_{2} = \frac{678.90}{40\,189.51}\times100$$ which gives $$\boxed{98.31\ \text{wt\% BaO}\cdot4\text{B}_{2}\text{O}_{3} \; \text{and} \; 1.69\ \text{wt\% TiO}_{2}}$$ The contrast with the 8.5 mole per cent charged is the point of the question: a noticeable molar addition is a very small weight addition, because the heavy component is the one present in the majority.
Break the glass into its own oxides. Each of the 91.5 mol of glass carries one mole of BaO and four of B₂O₃, so $$\text{wt\% BaO} = \frac{91.5(153.33)}{40\,189.51}\times100 = 34.91\%$$ $$\text{wt\% B}_{2}\text{O}_{3} = \frac{91.5\times4\times69.62}{40\,189.51}\times100 = 63.40\%$$ Together with the 1.69 per cent TiO₂ these sum to 100.00 per cent, which is the arithmetic check.

Composition of the glass-ceramic on an oxide basis. Boron oxide dominates the mass even though barium is by far the heaviest element present, because there are four B₂O₃ units for every BaO.

The oxide statement above is what a batch house would weigh out. The question, however, asks for the atomic weights of each component element, so the same 40 189.51 g is now split element by element.

Count the atoms of each element in 100 mol of batch. The glass contributes 91.5 mol Ba, $91.5\times4\times2 = 732$ mol B and $91.5(1 + 12) = 1189.5$ mol O; the nucleating agent contributes 8.5 mol Ti and 17 mol O, for 1206.5 mol O in all.
Convert to mass and divide. Multiplying each count by its atomic weight, $$m_{\text{Ba}} = 12\,565.70,\; m_{\text{B}} = 7\,912.92,\; m_{\text{Ti}} = 406.90,\; m_{\text{O}} = 19\,304.00\ \text{g}$$ These sum to 40 189.51 g, matching the batch total exactly, so no atom has been lost. Expressed as percentages, $$\boxed{31.27\%\ \text{Ba},\; 19.69\%\ \text{B},\; 1.01\%\ \text{Ti},\; 48.03\%\ \text{O}}$$

Oxygen is very nearly half the mass of the material, which is characteristic of every oxide glass and is worth carrying as a sanity check: an elemental analysis of a borate or silicate that does not return roughly 45 to 55 per cent oxygen has an arithmetic error in it. The titanium content, at barely one per cent by weight, is equally typical — a nucleating agent has only to seed crystallisation, not to build the network, so it is added in small amounts. On the reheat, the TiO₂ phase-separates into a dense population of nuclei on which the barium-borate crystals then grow, converting the parent glass into a fine-grained glass-ceramic with far better strength and thermal-shock resistance.

Composition of the glass-ceramic in weight percent
Basis	Constituent	Weight percent
Batch constituents	BaO·4B₂O₃ glass	98.31%
Batch constituents	TiO₂	1.69%
Oxides	BaO	34.91%
	B₂O₃	63.40%
	TiO₂	1.69%
Elements	Ba	31.27%
	B	19.69%
	Ti	1.01%
	O	48.03%

Topic. Glass-ceramics: batch composition arithmetic, and the role of a nucleating agent in controlled crystallisation.

Key relations. The formula weight of a compound oxide is the sum of its constituent atomic weights; on a basis of $N$ total moles the weight percent of constituent $i$ is $100\,n_{i}M_{i}/\sum_{j}n_{j}M_{j}$. Mole percent and weight percent coincide only when the formula weights are equal, which they almost never are.

Why this works. A batch is specified in moles because crystallisation chemistry counts formula units, but it is weighed out in kilograms, so every real glass calculation is a conversion between the two. Choosing 100 moles as the basis turns the stated mole percentages directly into mole quantities and reduces the whole problem to one division. The larger lesson is the size of the discrepancy: 8.5 mole per cent TiO₂ is only 1.69 weight per cent, because the borate glass unit is 431.81 g/mol against 79.87 for the titania. A glass-ceramic is then made by melting and forming the material as an ordinary glass and reheating it in two stages — first to a nucleation hold, where the TiO₂ separates out as a dense population of nuclei, then to a higher growth hold, where crystals grow on those nuclei until the body is 90 per cent or more crystalline with a grain size of a micrometre or less. The result keeps the pore-free formability of glass while gaining the strength, stiffness and thermal-shock resistance of a fine-grained ceramic.

Common pitfall. Treating the stated 8.5 mole per cent as 8.5 weight per cent, which overstates the titania content by a factor of five. The second is to forget the coefficient 4 in BaO·4B₂O₃ and use 222.95 g/mol for the glass. The third is to answer only on the oxide basis when the question explicitly asks for the atomic weights of each component element; giving both bases, and showing that they reconcile to the same total mass, costs three lines and removes all ambiguity.

Source. Shackelford, Introduction to Materials Science for Engineers, 8th ed., Ch. 12 (glasses and glass-ceramics, nucleation and controlled crystallisation); Askeland & Wright, Ch. 15 §15.5.

Question 2 of 8: Composition of a barium-borate glass-ceramic in weight percent

Question 2: Composition of a barium-borate glass-ceramic in weight percent (20 marks)