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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.

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.

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.