Certainly! Let's solve this problem step-by-step.
### Step 1: Understand the problem
We need to find the mass of copper that has the same volume as 100.0 grams of gold. The given densities are:
- Density of gold ([tex]\(\rho_{\text{gold}}\)[/tex]) = 19.31 g/cm³
- Density of copper ([tex]\(\rho_{\text{copper}}\)[/tex]) = 8.94 g/cm³
- Mass of gold ([tex]\(m_{\text{gold}}\)[/tex]) = 100.0 grams
### Step 2: Calculate the volume of gold
First, we need to find the volume occupied by 100.0 grams of gold. The formula for volume using mass and density is:
[tex]\[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \][/tex]
For gold:
[tex]\[ V_{\text{gold}} = \frac{m_{\text{gold}}}{\rho_{\text{gold}}} \][/tex]
Plugging in the given values:
[tex]\[ V_{\text{gold}} = \frac{100.0 \, \text{g}}{19.31 \, \text{g/cm}^3} \][/tex]
This simplifies to:
[tex]\[ V_{\text{gold}} \approx 5.1787 \, \text{cm}^3 \][/tex]
### Step 3: Calculate the mass of copper
Now, we need to find the mass of copper that has the same volume. Using the same volume from gold and the density of copper, we use the formula:
[tex]\[ \text{Mass} = \text{Density} \times \text{Volume} \][/tex]
For copper:
[tex]\[ m_{\text{copper}} = \rho_{\text{copper}} \times V_{\text{gold}} \][/tex]
Substituting the values:
[tex]\[ m_{\text{copper}} = 8.94 \, \text{g/cm}^3 \times 5.1787 \, \text{cm}^3 \][/tex]
This gives us:
[tex]\[ m_{\text{copper}} \approx 46.2973 \, \text{g} \][/tex]
### Conclusion
Therefore, the mass of copper that has the same volume as 100.0 grams of gold is approximately 46.2973 grams.
