Question

The average concentration of gold in seawater is \(100 \mathrm{fM}\) (femtomolar). Given that the price of gold is \(\$ 1764.20\) per troy ounce ( 1 troy ounce \(=31.103 \mathrm{~g}\) ), how many liters of seawater would you need to process to collect \(\$ 5000\) worth of gold, assuming your processing technique captures only \(50 \%\) of the gold present in the samples?

Short answer

To obtain $5000 worth of gold with a 50% recovery rate, you would need to process approximately \(4.86 \times 10^{13}\) liters of seawater.

Step-by-step solution

Calculate the mass of gold worth $5000

First, we need to determine the mass of gold equivalent to 5000 dollars. We are given the price of gold as $1764.20 per troy ounce 1 troy ounce \(=31.103 \mathrm{~g}\) Price of gold $\$ 1764.20 = 31.103 \mathrm{~g}\) Now, we'll find the grams of gold in 5000 dollars: \[\frac{5000}{1764.20} = \frac{5000}{1764.20}\times 31.103 \mathrm{~g}\]

Account for the 50% recovery rate

Considering 50% recovery rate, we need to process seawater with double the gold mass: \[\frac{5000}{1764.20}\times 31.103 \times 2 \mathrm{~g}\]

Convert concentration of gold in seawater to mass per liter

Now, we need to convert the concentration of gold in seawater from femtomolar (fM) into mass per liter: \(1 M (molar) = 1 \frac{mol}{L} = 6.022 \times 10^{23} \frac{atoms}{L} \) \(1 fM = 10^{-15} M\) The atomic weight of gold is approximately 197 g/mol, so: \(100 fM = 100 \times 10^{-15} \times 6.022 \times 10^{23} \times 197 \mathrm{~g / L} \)

Calculate the volume of seawater required

Finally, we can calculate the volume of seawater in liters: \[\frac{\frac{5000}{1764.20}\times 31.103 \times 2 \mathrm{~g}}{100 \times 10^{-15} \times 6.022 \times 10^{23} \times 197 \mathrm{~g / L}}\] Upon finding the result, we get the amount of seawater needed to process in order to collect $5000 worth of gold with the mentioned efficiency.

Key concepts

Concentration Measurement

Understanding concentration is key when dealing with chemistry calculations. Concentration refers to the amount of a substance present in a certain volume of a solution. In our problem, we are dealing with a very low concentration of gold in seawater, measured in the unit called femtomolar (fM). This is an incredibly tiny unit, where 1 femtomolar equals \(10^{-15}\) molar.

To make sense of this, consider that a molar concentration means there is 1 mole of a substance in 1 liter of solution. Avogadro's number tells us that one mole equals \(6.022 \times 10^{23}\) particles, which includes atoms or molecules. So, for gold in seawater, a concentration measurement of \(100 fM\) indicates there are \(100 \times 10^{-15} \) moles of gold per liter. This allows us to convert this concentration into grams, since we know that one mole of gold weighs about 197 grams.

Ultimately, these calculations help us understand how much gold is actually present in a vast amount of seawater by providing a mass measurement per liter.

Gold Recovery

Gold recovery refers to the process of retrieving gold from a mixture or solution. In various industries, especially the economic and mining sectors, understanding how to effectively recover gold can be quite crucial.

In this context, the challenge is to extract enough gold from seawater to equal a certain financial value, namely $5000 worth, while considering a recovery rate. Recovery rate indicates the efficiency of a process in capturing what is desired. In this case, we assume a 50% recovery rate, meaning that out of all the gold available in the seawater, only half can actually be recovered and utilized.

This requires doubling the amount of gold we need to gather from the seawater, since only half will end up being usable. So, when calculating how much seawater needs to be processed, the required amount of gold is twice as much as the pure monetary equivalent.

Understanding gold recovery is not just about the recovery rate but includes processes that impact the efficiency of capturing this precious metal from a solution.

Seawater Composition

Seawater is a complex solution, made up of a variety of substances. Predominantly, it's a mix of water and salts, but it also contains trace amounts of precious metals like gold. Interestingly, while the concentration of gold is minuscule within seawater, it represents a vast resource given the sheer volume of oceans.

On average, seawater contains about \(100 fM\) of gold, showcasing how distributed and diluted such metals are. Analyzing the composition of seawater becomes important when calculating how to extract gold efficiently. It's essential to understand not just the gold concentration but also the scale of processing required to yield a significant, financially viable amount of gold.

Given the vastness and variability of seawater, processes for gold extraction must account for the challenges posed by its composition. The task isn't merely chemically challenging but also logistically complex, making the precise calculation of things like volume and concentration pivotal to success.