Earth's oceans have an average depth of 3800 m, a total area of 3.63 108 km2, and an average concentration of dissolved gold of 5.8 10-9 g/L.
(a) How many grams of gold are in the oceans?
(b) How many cubic meters of gold are in the oceans?
Assuming the price of gold is $392.00/troy oz, what is the value of gold in the oceans (1 troy oz = 31.1 g; d of gold = 19.3 g/cm3)?
AND
Copper can be drawn into thin wires. How many meters of 34-gauge wire (diameter = 6.304 10-3 in) can be produced from the copper in 5.09 lb of covellite, an ore of copper that is 66% copper by mass? (Hint: Treat the wire as a cylinder: V of cylinder = πr2h; d of copper = 8.95 g/cm3.)
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To solve the first question, we need to find the total amount of gold in the oceans. We can do this by multiplying the average concentration of dissolved gold by the total volume of the oceans.
(a) To find the grams of gold in the oceans:
- Convert the average depth of the oceans from meters to centimeters: 3800 m = 3800 * 100 = 380,000 cm.
- Calculate the total volume of the oceans using the formula: volume = area * depth. Convert the area from km^2 to cm^2: 3.63 * 10^8 km^2 = 3.63 * 10^8 * (10^5)^2 = 3.63 * 10^18 cm^2.
- Multiply the volume by the average concentration of dissolved gold: volume * concentration = 3.63 * 10^18 cm^2 * 380,000 cm * 5.8 * 10^-9 g/L = 7.72132 * 10^22 g.
Therefore, there are approximately 7.72132 * 10^22 grams of gold in the oceans.
(b) To find the cubic meters of gold in the oceans:
- Convert the average depth of the oceans from meters to cubic meters: 3800 m = 3800 m * (10^-3 m/cm) = 3.8 * 10^0 m.
- Calculate the total volume of the oceans using the formula: volume = area * depth. Convert the area from km^2 to m^2: 3.63 * 10^8 km^2 = 3.63 * 10^8 * (10^3)^2 = 3.63 * 10^14 m^2.
- Multiply the volume by the average concentration of dissolved gold: volume * concentration = 3.63 * 10^14 m^2 * 3.8 * 10^0 m * 5.8 * 10^-9 g/L = 7.09848 * 10^2 m^3.
Therefore, there are approximately 7.09848 * 10^2 cubic meters of gold in the oceans.
To solve the second question, we need to find the number of meters of 34-gauge wire that can be produced from the given amount of copper.
- Determine the mass of copper in the ore: 5.09 lb = 5.09 lb * (453.592 g/lb) = 2,313.09 g.
- Calculate the mass of copper in the ore using the percentage composition: 2,313.09 g * (66/100) = 1,525.77 g.
- Convert the mass of copper to volume using the density of copper: volume = mass / density = 1,525.77 g / 8.95 g/cm^3 = 170.36 cm^3.
- Convert the diameter of the wire from inches to cm: 6.304 * 10^-3 in * 2.54 cm/in = 0.160256 cm.
- Calculate the radius of the wire: radius = diameter / 2 = 0.160256 cm / 2 = 0.080128 cm.
- Calculate the volume of the wire using the formula for the volume of a cylinder: volume = π * r^2 * h = π * (0.080128 cm)^2 * h, where h is the length of the wire we want to find.
- Now, equate the volume of the wire to the volume of copper: π * (0.080128 cm)^2 * h = 170.36 cm^3.
- Solve for h: h = 170.36 cm^3 / (π * (0.080128 cm)^2) ≈ 6798.58 cm.
Therefore, approximately 6798.58 meters of 34-gauge wire can be produced from 5.09 lb of covellite ore.