Answer :
Sure! Let's break down the problem and go through each part step-by-step:
1. Weighing the Quarter:
- The weight of the quarter is given as 5.6 grams.
2. Mass of Copper and Nickel in the Quarter:
- The quarter is composed of 91.67% copper and 8.33% nickel by mass.
- To find the mass of copper:
[tex]\[
\text{Mass of Copper} = 0.9167 \times 5.6 = 5.13352 \, \text{grams}
\][/tex]
- To find the mass of nickel:
[tex]\[
\text{Mass of Nickel} = 0.0833 \times 5.6 = 0.46648 \, \text{grams}
\][/tex]
3. Moles of Copper and Nickel:
- Molar mass of copper = 63.55 g/mol.
- Molar mass of nickel = 58.69 g/mol.
- To calculate moles of copper:
[tex]\[
\text{Moles of Copper} = \frac{5.13352}{63.55} = 0.08078 \, \text{moles}
\][/tex]
- To calculate moles of nickel:
[tex]\[
\text{Moles of Nickel} = \frac{0.46648}{58.69} = 0.00795 \, \text{moles}
\][/tex]
4. Atoms of Copper and Nickel:
- Using Avogadro's number [tex]\(6.022 \times 10^{23}\)[/tex] atoms/mol.
- Number of atoms of copper:
[tex]\[
\text{Atoms of Copper} = 0.08078 \times 6.022 \times 10^{23} = 4.8645 \times 10^{22} \, \text{atoms}
\][/tex]
- Number of atoms of nickel:
[tex]\[
\text{Atoms of Nickel} = 0.00795 \times 6.022 \times 10^{23} = 4.7864 \times 10^{21} \, \text{atoms}
\][/tex]
5. Value of Metals in the Quarter:
- Conversion factor: 1 ounce = 28.3495 grams.
- Current prices: Copper = [tex]$0.29 per ounce, Nickel = $[/tex]0.44 per ounce.
- Value of copper in the quarter:
[tex]\[
\text{Value of Copper} = \left(\frac{5.13352}{28.3495}\right) \times 0.29 \approx 0.05257 \, \text{dollars}
\][/tex]
- Value of nickel in the quarter:
[tex]\[
\text{Value of Nickel} = \left(\frac{0.46648}{28.3495}\right) \times 0.44 \approx 0.00718 \, \text{dollars}
\][/tex]
- Total metal value:
[tex]\[
\text{Total Metal Value} = 0.05257 + 0.00718 = 0.05975 \, \text{dollars}
\][/tex]
6. Comparison with Face Value:
- The face value of a quarter is [tex]$0.25.
- Metal value ($[/tex]0.05975) is less than the face value ($0.25).
- Thus, the quarter is worth more in face value than in metal content. Selling it for its metal would be less than its worth as currency.
This breakdown provides the mass of each metal, the number of moles, the number of atoms, and the monetary comparison of metal value versus face value.
1. Weighing the Quarter:
- The weight of the quarter is given as 5.6 grams.
2. Mass of Copper and Nickel in the Quarter:
- The quarter is composed of 91.67% copper and 8.33% nickel by mass.
- To find the mass of copper:
[tex]\[
\text{Mass of Copper} = 0.9167 \times 5.6 = 5.13352 \, \text{grams}
\][/tex]
- To find the mass of nickel:
[tex]\[
\text{Mass of Nickel} = 0.0833 \times 5.6 = 0.46648 \, \text{grams}
\][/tex]
3. Moles of Copper and Nickel:
- Molar mass of copper = 63.55 g/mol.
- Molar mass of nickel = 58.69 g/mol.
- To calculate moles of copper:
[tex]\[
\text{Moles of Copper} = \frac{5.13352}{63.55} = 0.08078 \, \text{moles}
\][/tex]
- To calculate moles of nickel:
[tex]\[
\text{Moles of Nickel} = \frac{0.46648}{58.69} = 0.00795 \, \text{moles}
\][/tex]
4. Atoms of Copper and Nickel:
- Using Avogadro's number [tex]\(6.022 \times 10^{23}\)[/tex] atoms/mol.
- Number of atoms of copper:
[tex]\[
\text{Atoms of Copper} = 0.08078 \times 6.022 \times 10^{23} = 4.8645 \times 10^{22} \, \text{atoms}
\][/tex]
- Number of atoms of nickel:
[tex]\[
\text{Atoms of Nickel} = 0.00795 \times 6.022 \times 10^{23} = 4.7864 \times 10^{21} \, \text{atoms}
\][/tex]
5. Value of Metals in the Quarter:
- Conversion factor: 1 ounce = 28.3495 grams.
- Current prices: Copper = [tex]$0.29 per ounce, Nickel = $[/tex]0.44 per ounce.
- Value of copper in the quarter:
[tex]\[
\text{Value of Copper} = \left(\frac{5.13352}{28.3495}\right) \times 0.29 \approx 0.05257 \, \text{dollars}
\][/tex]
- Value of nickel in the quarter:
[tex]\[
\text{Value of Nickel} = \left(\frac{0.46648}{28.3495}\right) \times 0.44 \approx 0.00718 \, \text{dollars}
\][/tex]
- Total metal value:
[tex]\[
\text{Total Metal Value} = 0.05257 + 0.00718 = 0.05975 \, \text{dollars}
\][/tex]
6. Comparison with Face Value:
- The face value of a quarter is [tex]$0.25.
- Metal value ($[/tex]0.05975) is less than the face value ($0.25).
- Thus, the quarter is worth more in face value than in metal content. Selling it for its metal would be less than its worth as currency.
This breakdown provides the mass of each metal, the number of moles, the number of atoms, and the monetary comparison of metal value versus face value.
