To determine the percentage of hydroxyapatite in tooth enamel, we can follow these detailed steps:
1. Calculate the moles of [tex]\( \text{CaCl}_2 \)[/tex] produced:
We start by using the given mass of [tex]\( \text{CaCl}_2 \)[/tex]: 0.975 grams. The molar mass of [tex]\( \text{CaCl}_2 \)[/tex] is approximately 110.98 g/mol.
[tex]\[
\text{Moles of } \text{CaCl}_2 = \frac{0.975 \text{ g}}{110.98 \text{ g/mol}} \approx 0.00879 \text{ mol}
\][/tex]
2. Relate the moles of [tex]\( \text{CaCl}_2 \)[/tex] to the moles of hydroxyapatite ( [tex]\( \text{Ca}_{10}( \text{PO}_4 )_6( \text{OH} )_2 \)[/tex] ):
From the balanced equation:
[tex]\[
\text{Ca}_{10}( \text{PO}_4 )_6( \text{OH} )_2 \rightarrow 10 \text{CaCl}_2
\][/tex]
1 mole of hydroxyapatite produces 10 moles of [tex]\( \text{CaCl}_2 \)[/tex].
Therefore:
[tex]\[
\text{Moles of hydroxyapatite} = \frac{\text{Moles of } \text{CaCl}_2}{10} = \frac{0.00879 \text{ mol}}{10} \approx 0.00088 \text{ mol}
\][/tex]
3. Calculate the mass of hydroxyapatite that reacted:
The molar mass of hydroxyapatite [tex]\( \text{Ca}_{10}( \text{PO}_4 )_6( \text{OH} )_2 \)[/tex] is approximately 1004.64 g/mol.
[tex]\[
\text{Mass of hydroxyapatite} = \text{Moles of hydroxyapatite} \times \text{Molar mass of hydroxyapatite}
\][/tex]
[tex]\[
\text{Mass of hydroxyapatite} = 0.00088 \text{ mol} \times 1004.64 \text{ g/mol} \approx 0.883 \text{ g}
\][/tex]
4. Calculate the percentage of hydroxyapatite in tooth enamel:
Given that the initial mass of the tooth enamel sample is 1.125 g, the percentage of hydroxyapatite is calculated as follows:
[tex]\[
\text{Percentage of hydroxyapatite} = \left( \frac{\text{Mass of hydroxyapatite}}{\text{Mass of tooth enamel}} \right) \times 100
\][/tex]
[tex]\[
\text{Percentage of hydroxyapatite} = \left( \frac{0.883 \text{ g}}{1.125 \text{ g}} \right) \times 100 \approx 78.45\%
\][/tex]
Therefore, the percentage of hydroxyapatite in the tooth enamel is approximately [tex]\( 78.45\% \)[/tex].
