A bag contains a mixture of copper and lead BBs. The average density of the BBs is 9.60g/cm3.Assuming that the copper and lead are pure, determine the relative amounts of each kind of BB.
Express your answers numerically separated by a comma.
The density of Cu is 8.96 g/cm^3. The density of Pb is 11.4 g/cm^3. Let the mass fraction of copper, Cu = x, Then the mass fraction of Pb is (1-x). The algebraic expression for the "weighted" average density is:
8.96x + 11.4(1-x) = 9.60
Solve for x to get the mass fraction for copper. How would you get the mass fraction of Pb?
To determine the relative amounts of each kind of BB in the bag, we need to use the concept of density.
The density formula is given by:
Density = Mass / Volume
Let's assume the mass of copper BBs is "x" grams and the mass of lead BBs is "y" grams.
The volume of the copper BBs can be calculated using the density of copper, which is 8.96 g/cm^3:
Volume of copper BBs = Mass of copper BBs / Density of copper
Similarly, the volume of the lead BBs can be calculated using the density of lead, which is 11.34 g/cm^3:
Volume of lead BBs = Mass of lead BBs / Density of lead
Since the BBs are mixed together, the total volume of the BBs can be calculated by summing the volumes of copper and lead BBs:
Total Volume = Volume of copper BBs + Volume of lead BBs
Given that the average density of the BBs is 9.60 g/cm^3, we can express the total volume in terms of the average density and the total mass:
Total Volume = (Mass of copper BBs + Mass of lead BBs) / Average Density
Now we can set up two equations using the above information:
Equation 1: Volume of copper BBs + Volume of lead BBs = Total Volume
Equation 2: Mass of copper BBs + Mass of lead BBs = Total Mass
Substituting the volume equations into Equation 1, we get:
(Mass of copper BBs / Density of copper) + (Mass of lead BBs / Density of lead) = (Mass of copper BBs + Mass of lead BBs) / Average Density
To solve for the relative amounts of copper and lead BBs, we can rearrange Equation 2 to express the mass of copper BBs in terms of the mass of lead BBs:
Mass of copper BBs = Total Mass - Mass of lead BBs
Substituting this expression into Equation 1 and simplifying, we obtain:
(Mass of lead BBs / Density of lead) = Total Mass / Average Density - (Total Mass - Mass of lead BBs) / Density of copper
Now we can solve Equation 2 for the mass of lead BBs:
Mass of lead BBs = Total Mass * (Density of lead - Average Density) / (Density of lead - Density of copper)
Finally, we can substitute the values of the densities and the average density to calculate the relative amounts of copper and lead BBs.