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Chapter 18: Problem 42

Gold is found in seawater at very low levels, about 0.05 ppb by mass. Assuming that gold is worth about \(\$ 1300\) per troy ounce, how many liters of seawater would you have to process to obtain \(\$ 1,000,000\) worth of gold? Assume to prosity of seawater is 1.03 \(\mathrm{g} / \mathrm{mL}\) and that your gold recovery process is 50\(\%\) efficient.

Short Answer

Expert verified
You would need to process approximately 934,579,439 liters of seawater to obtain $1,000,000 worth of gold, considering the given conditions of 0.05 ppb gold concentration, $1300 per troy ounce gold worth, 1.03 g/mL seawater prosity, and 50% gold recovery process efficiency.

Step by step solution

01

Convert ppb to mass fraction

First, we need to convert the gold concentration in seawater from parts per billion (ppb) to mass fraction. 1 ppb means 1 µg gold per 1g of seawater, so 0.05 ppb = 0.05 µg of gold per 1g of seawater. Hence the mass fraction would be: \(0.05 \times 10^{-6}\)
02

Calculate the mass of gold in 1 liter of seawater

Next, we need to find the mass of gold in 1 liter of seawater. We know that the prosity of seawater is 1.03 g/mL, meaning 1 liter (1000 mL) of seawater has a mass of 1000 mL × 1.03 g/mL = 1030 g. Multiplying the mass of seawater by the mass fraction of gold: \(1030\,\text{g}\, \times 0.05 \times 10^{-6}\, = 5.15 \times 10^{-5}\,\text{g}\)
03

Convert grams to troy ounces

Since the worth of gold is given in USD per troy ounce, we have to convert the mass of gold from grams to troy ounces.1 troy ounce is approximately equal to 31.1 grams. So, we convert the gold mass as follows: \((5.15 \times 10^{-5} \text{ g}) \times \frac{1\, \text{troy oz}}{31.1\,\text{g}} = 1.65 \times 10^{-6}\,\text{troy oz}\)
04

Calculate the worth of gold in 1 liter of seawater

Now, we can find the worth of gold in 1 liter of seawater by multiplying the mass of gold in troy ounces by the worth of gold per troy ounce (1300 USD/troy oz): \(1.65 \times 10^{-6} \,\text{troy oz}\, \times 1300\,\text{USD/troy oz} = 0.00214\,\text{USD}\)
05

Adjust for gold recovery process efficiency

Since the gold recovery process is only 50% efficient, we have to adjust the worth of gold that we can obtain from 1 liter of seawater: \(0.00214\,\text{USD} \times 50\% = 0.00107\,\text{USD}\)
06

Calculate the number of liters to process

Finally, we can find the number of liters of seawater we need to process to obtain $1,000,000 worth of gold by dividing the required worth by the worth of gold in 1 liter of seawater, adjusted for recovery process efficiency: \(\frac{1,000,000\,\text{USD}}{0.00107\,\text{USD}} = 934,579,439\,\text{liters}\) So, you would need to process approximately 934,579,439 liters of seawater to obtain $1,000,000 worth of gold, considering the given conditions.

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