Problem 42 - Chemistry: The Central Science in SI Units, Global Edition

Question

Platinum is found in seawater at very low levels, about 0.23 ppt (parts per trillion) by mass. How much platinum can be found in the entire ocean \(\left(1.3 \times 10^{21} \mathrm{~L}\right)\) ? Assume the density of seawater is \(1.03 \mathrm{~g} / \mathrm{mL}\). Estimate the price of the following amount of platinum: \(\$ 1,600\) per troy ounce.

Step-by-Step Solution

Answer

The estimated price of the platinum in the entire ocean is approximately \( \$ 15.843\) trillion.

Step 1: Calculate the mass of the seawater

To find the mass of the seawater, we need to use the formula mass = volume * density. We are given the volume of the ocean as \(1.3\times10^{21}\mathrm{~L}\) and the density of seawater as \(1.03\mathrm{~g/mL}\). First, convert the volume of the ocean to milliliters (1 L = 1000 mL): \(1.3\times10^{21}\mathrm{~L} \times 1000 \mathrm{~mL/L} = 1.3\times10^{24}\mathrm{~mL}\) Now we can find the mass of the seawater by multiplying the volume by the density: \(1.3\times10^{24}\mathrm{~mL} \times 1.03 \mathrm{~g/mL} = 1.339\times10^{24} \mathrm{~g}\)

Step 2: Calculate the mass of platinum in the entire ocean

We are given the concentration of platinum in seawater as 0.23 ppt (parts per trillion). To find the mass of platinum, we need to multiply the mass of the seawater by the ratio of platinum to seawater: \((1.339\times10^{24}\,\mathrm{g})\times(0.23 \,\mathrm{ppt}) = (1.339\times10^{24}\,\mathrm{g})\times(0.23\times 10^{-12})\) \(= 3.0797\times10^{11}\,\mathrm{g\, of\, platinum}\)

Step 3: Convert the mass of platinum to troy ounces

Now, we need to convert the mass of platinum from grams to troy ounces. There are approximately 31.1035 grams in a troy ounce, so: \((3.0797\times10^{11}\,\mathrm{g})\times\dfrac{1\,\mathrm{troy\, ounce}}{31.1035\,\mathrm{g}} = 9.902\times10^{9}\,\mathrm{troy\,ounces\,of\,platinum}\)

Step 4: Estimate the price of the platinum

To estimate the price of the platinum, we need to multiply the amount of platinum in troy ounces by the price per troy ounce. We are given the price per troy ounce as \( \$ 1,600\), so: \((9.902\times10^{9}\,\mathrm{troy\, ounces})\times(\$1,600/\mathrm{troy\,ounce}) = \$15.843\times10^{12}\) So, the estimated price of the platinum in the entire ocean is approximately \( \$ 15.843\) trillion.

Key Concepts

Seawater Density

The density of seawater is a crucial factor when calculating the mass of water in the ocean. Density is a measure of how much mass is contained in a given volume. The formula to find mass using density is:

• \(\text{mass} = \text{volume} \times \text{density}\)

In this exercise, the density of seawater is given as \(1.03 \mathrm{~g/mL}\). When dealing with large volumes, such as the volume of the entire ocean, every small variation in density can account for significant changes in the calculated mass.
The ocean's volume is provided as \(1.3 \times 10^{21} \text{ liters}.\) To use the formula properly, convert this volume into milliliters, because density is given in grams per milliliter:

• \(1.3 \times 10^{21} \text{ liters} \times 1000 \text{ mL/L} = 1.3 \times 10^{24} \text{ mL}\)

Then, multiply by the density to find the mass:

• \(1.3 \times 10^{24} \text{ mL} \times 1.03 \text{ g/mL} = 1.339 \times 10^{24} \text{ grams of seawater}\)

This mass is the basis for calculating how much platinum can be found in the sea.

Platinum Concentration

The concentration of platinum in seawater is expressed in parts per trillion (ppt), which is a tiny fraction. One part per trillion means one unit of platinum for every trillion units of seawater. In this problem, the platinum concentration is given as 0.23 ppt.
To estimate the mass of platinum in the entire ocean, multiply the total mass of seawater by this concentration. Given the small size of ppt, you need to convert it to a straightforward decimal form for calculation:

• \(0.23 \text{ ppt} = 0.23 \times 10^{-12}\)

The mass of platinum thus is:

• \((1.339 \times 10^{24} \mathrm{~grams}) \times (0.23 \times 10^{-12}) = 3.0797 \times 10^{11} \text{ grams of platinum}\)

Understanding this concept helps appreciate how trace elements are measured and calculated in a massive body like the ocean.

Troy Ounce Conversion

In the world of precious metals, the troy ounce is a standard unit of weight different from the regular ounce. One troy ounce equals approximately 31.1035 grams. Hence, converting grams of platinum into troy ounces is essential for determining its value in financial terms.
To perform the conversion:

• Take the calculated grams of platinum: \(3.0797 \times 10^{11}\)
• Divide by the grams in one troy ounce: \(31.1035 \text{ g/troy ounce}\)

This results in:

• \((3.0797 \times 10^{11} \mathrm{~grams}) \times \dfrac{1 \text{ troy ounce}}{31.1035 \text{ grams}} = 9.902 \times 10^{9} \text{ troy ounces}\)

Finally, to find the value, multiply the troy ounces by the price per troy ounce. This conversion is widely used when calculating costs and investments related to precious metals, making it an essential concept to grasp in economics and material science.