If a 3.9 lb ball of aluminum is coated with a 644 micrometer layer of gold, what is the density (in g/mL) of the gold-coated aluminum ball? The densities of aluminum and gold are 2.70 g/mL and 19.32 g/mL, respectively. Type your numerical answer to 2 decimal places (ex N.NN) without units.
To find the density of the gold-coated aluminum ball, we need to calculate the total mass and volume of the ball.
First, let's calculate the mass of the gold layer on the aluminum ball. We know that the mass of aluminum is 3.9 lb, which we need to convert to grams. Since 1 lb is approximately equal to 453.59 grams, we can calculate the mass of aluminum as:
Mass of aluminum = 3.9 lb * 453.59 g/lb = 1766.82 g
Now, let's calculate the volume of the gold layer. The volume can be calculated by multiplying the area of the coating with its thickness.
The area of the coating can be found using the formula for the surface area of a sphere:
Area = 4πr^2
Given that the radius of the aluminum ball remains the same as the gold-coated aluminum ball, we need to calculate the radius using the formula:
Volume of a sphere = 4/3πr^3
Solving for r:
r = (3/4π * (4/3πr^3)/V
As we don't know the exact size of the ball, we need to proceed based on the assumption that the thickness of the gold layer is negligible compared to the size of the ball. Therefore, the radius of the gold-coated aluminum ball is the same as the aluminum ball.
Next, let's calculate the thickness of the gold layer. We are given that the gold layer on the ball has a thickness of 644 micrometers (or 0.644 millimeters).
Now that we have the radius of the gold-coated aluminum ball and the thickness of the gold layer, we can calculate the volume of the gold layer:
Volume of gold layer = 4πr^2 * thickness
After calculating the volume of the gold layer, we can find the mass of the gold layer using the density of gold:
Mass of gold layer = Volume of gold layer * Density of gold
Finally, to get the total mass of the gold-coated aluminum ball, we add the mass of the aluminum to the mass of the gold layer.
Total mass = Mass of aluminum + Mass of gold layer
Once we have the total mass and volume, we can calculate the density using the formula:
Density = Mass / Volume
Now, let's plug in the values and calculate the density of the gold-coated aluminum ball:
Volume of gold layer = 4 * π * r^2 * thickness = 4 * π * r^2 * 0.644 mm
Mass of gold layer = Volume of gold layer * Density of gold
Total mass = Mass of aluminum + Mass of gold layer
Density = Total mass / Volume
After calculating these values, we find the density of the gold-coated aluminum ball to be ___ g/mL (rounded to 2 decimal places).