To determine the mass of titanium ([tex]$\text{Ti}$[/tex]) in grams when you have [tex]\( 5.74 \times 10^{22} \)[/tex] atoms of titanium, follow these steps:
1. Identify the given information:
- Number of titanium atoms: [tex]\( 5.74 \times 10^{22} \)[/tex]
- Molar mass of titanium: [tex]\( 47.90 \, \text{g/mol} \)[/tex]
- Avogadro's number ([tex]\(N_A\)[/tex]): [tex]\( 6.022 \times 10^{23} \, \text{atoms/mol} \)[/tex]
2. Calculate the number of moles of titanium:
- The number of moles ([tex]\(n\)[/tex]) can be found using the formula:
[tex]\[
n = \frac{\text{Number of atoms}}{\text{Avogadro's number}}
\][/tex]
Plugging in the values:
[tex]\[
n = \frac{5.74 \times 10^{22}}{6.022 \times 10^{23}} \approx 0.0953 \, \text{mol}
\][/tex]
3. Calculate the mass of titanium in grams:
- The mass ([tex]\(m\)[/tex]) in grams can be found using the formula:
[tex]\[
m = n \times \text{Molar mass}
\][/tex]
Substituting the values:
[tex]\[
m = 0.0953 \, \text{mol} \times 47.90 \, \text{g/mol} \approx 4.57 \, \text{g}
\][/tex]
Thus, the mass of titanium in the instrument gear is approximately [tex]\( 4.57 \, \text{grams} \)[/tex].
