Soda and lime are added to a glass batch in the form of soda ash \(\left(\mathrm{Na}_{2} \mathrm{CO}_{3}\right)\) and limestone \(\left(\mathrm{CaCO}_{3}\right)\). During heating, these two ingredients decompose to give off carbon dioxide \(\left(\mathrm{CO}_{2}\right)\), the resulting products being soda and lime. Compute the weight of soda ash and limestone that must be added to \(125 \mathrm{lb}_{\mathrm{m}}\) of quartz \(\left(\mathrm{SiO}_{2}\right)\) to yield a glass of composition 78 wt\(\%\) \(\mathrm{SiO}_{2}, 17\) wt \(\% \mathrm{Na}_{2} \mathrm{O}\) and \(5 \mathrm{wt} \% \mathrm{CaO}\).

From the given percentages, we compute the weight of each component in the glass: - SiO2: 78 wt% of the glass - Na2O: 17 wt% of the glass - CaO: 5 wt% of the glass Let the total glass weight be \(G\) lb; then the weights of each component can be expressed in terms of \(G\). We are given that \(125\) lb of quartz is used as starting material which will provide SiO2. Hence, - SiO2: \(0.78G = 125\) - Na2O: \(0.17G\) - CaO: \(0.05G\) Solving for \(G\), the total glass weight, \(G = \frac{125}{0.78} \approx 160.26 \, \text{lb}\) Now we can find the weights of Na2O and CaO needed in the glass composition: - Na2O: \(160.26(0.17) \approx 27.24 \, \text{lb}\) - CaO: \(160.26(0.05) \approx 8.01 \, \text{lb}\)

The decomposition reactions for soda ash and limestone are given by: - Na2CO3 -> Na2O + CO2 - CaCO3 -> CaO + CO2 The molar masses of the compounds are (to one decimal place): - Na2O: \(62.0 \, \frac{\text g}{\text{mol}}\) - CaO: \(56.1 \, \frac{\text g}{\text{mol}}\) - Na2CO3: \(105.9 \, \frac{\text g}{\text{mol}}\) - CaCO3: \(100.1 \, \frac{\text g}{\text{mol}}\) Using stoichiometric ratios, we can calculate the needed amount of soda ash and limestone for the amounts of Na2O and CaO found previously. To obtain Na2O from Na2CO3: \(\frac{1 \, \text{mol } \text{Na}_{2}\text{O}}{62.0 \, \text{g } \text{Na}_{2}\text{O}} = \frac{1 \, \text{mol } \text{Na}_{2}\text{CO}_{3}}{105.9 \, \text{g } \text{Na}_{2}\text{CO}_{3}}\) The weight of soda ash needed to produce \(27.24\) lb of Na2O is: \(Weight_{Na_{2}CO_{3}} = \frac{105.9}{62.0} \cdot 27.24 \, \text{lb} \approx 46.01 \, \text{lb} \) To obtain CaO from CaCO3: \(\frac{1 \, \text{mol } \text{CaO}}{56.1 \, \text{g } \text{CaO}} = \frac{1 \, \text{mol } \mathrm{CaCO}_{3}}{100.1 \, \text{g } \mathrm{CaCO}_{3}}\) The weight of limestone needed to produce \(8.01\) lb of CaO is: \(Weight_{CaCO_{3}} = \frac{100.1}{56.1} \cdot 8.01 \, \text{lb} \approx 14.29 \, \text{lb} \)

Now we can calculate the total amount of soda ash and limestone that must be added to the quartz to yield the desired glass composition. Total weight needed of soda ash and limestone: \(Weight_{total} = Weight_{Na_{2}CO_{3}} + Weight_{CaCO_{3}} = 46.01 \, \text{lb} + 14.29 \, \text{lb} \approx 60.3 \, \text{lb}\) So to yield a glass of composition \(78 \% \mathrm{SiO}_{2}, 17 \% \mathrm{Na}_{2} \mathrm{O}\) and \(5 \% \mathrm{CaO}\), \(60.3\) lb of a mixture comprising soda ash and limestone must be added to \(125\) lb of quartz.