How much copper wire can be made from copper ore?

Question

Copper can be drawn into thin wires. How many meters of 34-gauge wire (diameter = 6.304 x 10^-3 in) can be produced from copper in 5.01 lbs of covellite, an ore of copper that is 66% Cu by mass? Copper has a density of 8.95 g/mL.

Answer 1

Hi Angela.

So you are starting with an ore of copper and trying to figure out the length of pure copper wire you can make.

The strategy I will use is to convert the mass of the ore into its equivalent mass of pure copper. Then, using the density, I will figure out the volume of pure copper that we have. A wire is essentially a cylinder with a round cross section but usually with a much greater lenght. Using the formula for the volume of a cylinder and the diameter given, we will figure out how long the wire is.

5.01 lbs of ore. The since the density of the copper is given in g/ml, I will eventually have to change the mass units to be the same. I find it easiest to work in grams, rather than pounds, so lets convert into grams.

2.2 lbs = 1000 grams

This means that 1000 grams / 2.2 lbs = 1

I can always multiply an equation by 1 and not change it, though it will seem to change because the units change.

5.01 * (1000 g / 2.2 lb) = 2277.27 g

Since only 66% of this is copper, if we purify it by smelting, we would have:

2277 g * 0.66 = 1503 g of pure copper.

(Remember that when multiplying by a percentage, you move the decimal 2 places.)

Density is the mass divided by the volume.

D = m / V

Our mass, m, is 1503 g and the density given is 8.95 g / ml.

Rearranging our equation, we get V = m / Density

or

V = 1503 g / (8.95 g/ml) = 167.93 ml

The volume, V, of a cylinder is given by:

V = pi * r^2 * h, where r is radius and h is height, or in our case, h is the length of the wire. (pi is approximately 3.1415.)

We don't have the radius, but we do have the diameter. The radius is half the diameter

r = diameter/2 = 6.304 x 10^-3 in /2 = 3.152 x 10^-3 in

I can see we will have an answer in either inches or cm. When I see ml, I think cm^3. I prefer to work in cm, so I will convert that radius to cm.

2.54 cm = 1 in

3.152 x 10^-3 in * 2.54 cm/ 1 in = 8.006 x 10^-3 cm

We are almost there.

From our cylinder equation, solving for h, the length of the wire, we get

h = V / (pi* r^2)

h = (167.93 cm ^ 3) / (3.1415 * (8.006 x 10^-3 cm)^2)

(I replaced ml with cm^3.)

h = 834,000 cm or 8,340 m.

The smallest number of significant figures used was 2 (66%), so I will round my answer to 8,300 m.

Answer 2

How much copper wire can be made from copper ore?

Copper can be drawn into thin wires. How many meters of 34-gauge wire (diameter = 6.304 x 10^-3 in) can be produced from copper in 5.01 lbs of covellite, an ore of copper that is 66% Cu by mass? Copper has a density of 8.95 g/mL.

The simplest way to solve this problem is using dimensional analysis, as follows:

%Cu * Weight ore * grams /pound * density ore * metric volume * inch / cm conversion cubed

0.66 * 5.01 lb Cu * 454 g / 1 lb Cu * 1 ml / 8.95 g * 1 cm³ / 1 ml * (1 inch / 2.54 cm)³ = 10.24 in³

The volume of a wire extruded in a cylindrical shape is, π r² L , where L is the length of the copper wire,

and π r² is the area of the circular face. Since we are given diameter instead of radius, use π d² / 4

To finish, simply divide the volume of 10.24 in³ by the area π d² / 4

L = 10.24 in³ / π 0.006304² / 4

L = 10.24 in³ * 4 / π 0.006304²

L = 328078 in.