Question

A typical formulation for window glass is 75% SiO₂, 15% Na₂O, and 10.% CaO by mass. What masses of sand (SiO₂), sodium carbonate, and calcium carbonate must be combined to produce 1.00 kg of glass after carbon dioxide is driven off by thermal decomposition of the carbonates?

Short answer

The mass of sand, sodium carbonate and calcium carbonate is 0.75 kg, 0.26 kg, and 0.18 kg, respectively.

Step-by-step solution

Determination of mass of sand (SiO2)

The reaction for the decomposition of sodium carbonate is,

${\mathbf{Na}}_{\mathbf{2}}{\mathbf{CO}}_{\mathbf{3}}\left(s\right)\mathbf{\to }{\mathbf{Na}}_{\mathbf{2}}\mathbf{O}\left(s\right)\mathbf{+}{\mathbf{CO}}_{\mathbf{2}}\left(g\right)\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{1}\mathbf{\right)}$

The reaction for the decomposition of calcium carbonate is,

${\mathbf{CaCO}}_{\mathbf{3}}\left(s\right)\mathbf{\to }\mathbf{CaO}\left(s\right)\mathbf{+}{\mathbf{CO}}_{\mathbf{2}}\left(g\right)\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{2}\mathbf{\right)}$

The percentage of SiO₂ in glass is 75%.

The mass of SiO₂ in 1.00 kg of glass is,

$$\begin{gathered} \mathbf{=}\frac{\mathbf{75}}{\mathbf{100}}{\mathbf{SiO}}_{\mathbf{2}}\mathbf{\times }\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{kg}\mathbf{\times }\frac{\mathbf{1000}\mathbf{g}}{\mathbf{1}\mathbf{kg}} \\ \mathbf{=}\mathbf{750}\mathbf{g} \\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{75}\mathbf{kg}\mathbf{}{\mathbf{SiO}}_{\mathbf{2}} \end{gathered}$$

Hence, the mass of the sand is 0.75 kg.

Determination of mass of sodium carbonate

The percentage of Na₂O in glass is 15%.

The mass of Na₂O in 1.00 kg of glass is,

$$\begin{gathered} \mathbf{=}\frac{\mathbf{15}}{\mathbf{100}}{\mathbf{Na}}_{\mathbf{2}}\mathbf{O}\mathbf{\times }\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{kg}\mathbf{\times }\frac{\mathbf{1000}\mathbf{g}}{\mathbf{1}\mathbf{kg}} \\ \mathbf{=}\mathbf{150}\mathbf{g} \\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{15}{\mathbf{kgNa}}_{\mathbf{2}}\mathbf{O} \end{gathered}$$

From equation (1), it can be concluded that 1 mol of sodium carbonate produces 1 mol of Na₂O.

So, the mass of sodium carbonate is,

$$\begin{gathered} \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{15}\mathbf{kg}\mathbf{\times }\frac{\mathbf{1000}\mathbf{g}}{\mathbf{1}\mathbf{kg}}\mathbf{\times }\frac{\mathbf{1}{\mathbf{molNa}}_{\mathbf{2}}\mathbf{O}}{\mathbf{61}\mathbf{.}\mathbf{98}{\mathbf{gNa}}_{\mathbf{2}}\mathbf{O}}\mathbf{\times }\frac{\mathbf{1}{\mathbf{molNa}}_{\mathbf{2}}{\mathbf{CO}}_{\mathbf{3}}}{\mathbf{1}{\mathbf{molNa}}_{\mathbf{2}}\mathbf{O}}\mathbf{\times }\frac{\mathbf{105}\mathbf{.}\mathbf{99}{\mathbf{Na}}_{\mathbf{2}}{\mathbf{CO}}_{\mathbf{3}}}{\mathbf{1}{\mathbf{molNa}}_{\mathbf{2}}{\mathbf{CO}}_{\mathbf{3}}}\mathbf{\times }\frac{\mathbf{1}\mathbf{kg}}{\mathbf{1000}\mathbf{g}} \\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{26}{\mathbf{kgNa}}_{\mathbf{2}}{\mathbf{CO}}_{\mathbf{3}} \end{gathered}$$

Hence, the mass of sodium carbonate is 0.26 kg.

Determination of mass of calcium carbonate

The percentage of CaO in glass is 10%.

The mass of Na₂O in 1.00 kg of glass is,

$$\begin{gathered} \mathbf{=}\frac{\mathbf{10}}{\mathbf{100}}\mathit{C}\mathit{a}\mathit{O}\mathbf{\times }\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{kg}\mathbf{\times }\frac{\mathbf{1000}\mathbf{g}}{\mathbf{1}\mathbf{kg}} \\ \mathbf{=}\mathbf{100}\mathbf{g} \\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{10}\mathbf{kgCaO} \end{gathered}$$

From equation (1), it can be concluded that 1 mol of calcium carbonate produces 1 mol of CaO.

So, the mass of calcium carbonate is,

$$\begin{gathered} \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{10}\mathbf{kg}\mathbf{\times }\frac{\mathbf{1000}\mathbf{g}}{\mathbf{1}\mathbf{kg}}\mathbf{\times }\frac{\mathbf{1}{\mathbf{molNa}}_{\mathbf{2}}\mathbf{O}}{\mathbf{56}\mathbf{.}\mathbf{08}\mathbf{gCaO}}\mathbf{\times }\frac{\mathbf{1}{\mathbf{molCaCO}}_{\mathbf{3}}}{\mathbf{1}\mathbf{molCaO}}\mathbf{\times }\frac{\mathbf{100}\mathbf{.}\mathbf{09}\mathit{C}\mathit{a}{\mathbf{CO}}_{\mathbf{3}}}{\mathbf{1}\mathbf{mol}\mathit{C}\mathit{a}{\mathbf{CO}}_{\mathbf{3}}}\mathbf{\times }\frac{\mathbf{1}\mathbf{kg}}{\mathbf{1000}\mathbf{g}} \\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{18}{\mathbf{kgCaCO}}_{\mathbf{3}} \end{gathered}$$

Hence, the mass of calcium carbonate is 0.18 kg.