Question

How do I determine the relative amounts of copper and lead BBs in a mixture given their average density and the densities of pure copper and lead?

Answer 1

To determine the relative amounts of copper and lead BBs in a mixture based on their average density, you can use the following approach. Let's denote:

- \( \rho_{\text{avg}} \): average density of the mixture (given)
- \( \rho_{\text{Cu}} \): density of pure copper
- \( \rho_{\text{Pb}} \): density of pure lead
- \( x \): the fraction (by mass) of copper in the mixture
- \( y \): the fraction (by mass) of lead in the mixture

Since the fractions of the components must add up to 1, we have:
\[
x + y = 1
\]

From the definition of average density in a mixture, we can express the average density in terms of the densities and the mass fractions:
\[
\rho_{\text{avg}} = x \cdot \rho_{\text{Cu}} + y \cdot \rho_{\text{Pb}}
\]

Now substituting \( y = 1 - x \) into the equation gives:
\[
\rho_{\text{avg}} = x \cdot \rho_{\text{Cu}} + (1 - x) \cdot \rho_{\text{Pb}}
\]

Expanding this yields:
\[
\rho_{\text{avg}} = x \cdot \rho_{\text{Cu}} + \rho_{\text{Pb}} - x \cdot \rho_{\text{Pb}}
\]
\[
\rho_{\text{avg}} = \rho_{\text{Pb}} + x \cdot (\rho_{\text{Cu}} - \rho_{\text{Pb}})
\]

Rearranging to solve for \( x \):
\[
\rho_{\text{avg}} - \rho_{\text{Pb}} = x \cdot (\rho_{\text{Cu}} - \rho_{\text{Pb}})
\]
\[
x = \frac{\rho_{\text{avg}} - \rho_{\text{Pb}}}{\rho_{\text{Cu}} - \rho_{\text{Pb}}}
\]

Given that \( y = 1 - x \), you can find \( y \) by substituting \( x \) back in:
\[
y = 1 - \frac{\rho_{\text{avg}} - \rho_{\text{Pb}}}{\rho_{\text{Cu}} - \rho_{\text{Pb}}}
\]

### Steps to follow:
1. Determine the density values:
- Obtain the average density of the mixture \( \rho_{\text{avg}} \).
- Know the densities of pure copper (\( \rho_{\text{Cu}} \)) and of pure lead (\( \rho_{\text{Pb}} \)).

2. Calculate the fraction \( x \) of copper in the mixture using the formula:
\[
x = \frac{\rho_{\text{avg}} - \rho_{\text{Pb}}}{\rho_{\text{Cu}} - \rho_{\text{Pb}}}
\]

3. Calculate the fraction \( y \) of lead in the mixture:
\[
y = 1 - x
\]

4. Optionally, you can express \( x \) and \( y \) as percentages if needed.

This will give you the relative amounts of copper and lead in the mixture!