Problem 42 - Chemistry The Central Science In Si Units

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 ocean contains about 9.9 million troy ounces of platinum, valued at approximately $15.84 billion.

Step 1: Convert ocean volume to mass

First, convert the ocean's total volume from liters to milliliters, knowing that 1 liter is equal to 1000 milliliters. This gives us \(1.3 \times 10^{21} \text{ mL}\). Next, use the density of seawater \(1.03 \text{ g/mL}\) to find the mass of the ocean in grams by multiplying the volume by the density:\[\text{Mass} = 1.3 \times 10^{21} \text{ mL} \times 1.03 \text{ g/mL} = 1.339 \times 10^{21} \text{ g}\]

Step 2: Calculate mass of platinum in the ocean

Using the concentration of platinum in seawater, \(0.23 \text{ ppt}\), which means 0.23 grams of platinum per \(10^{12}\) grams of seawater:\[\text{Mass of Pt} = \frac{0.23 \text{ g Pt}}{10^{12} \text{ g seawater}} \times 1.339 \times 10^{21} \text{ g seawater}\]\[= 0.23 \times 1.339 \times 10^9 \text{ g Pt} \]\[≈ 3.08 \times 10^8 \text{ g Pt}\]

Step 3: Convert grams to troy ounces

There are approximately 31.1035 grams in a troy ounce. To convert the mass of platinum from grams to troy ounces:\[\text{Troy ounces of Pt} = \frac{3.08 \times 10^8 \text{ g}}{31.1035 \text{ g/troy oz}}\]\[\approx 9.9 \times 10^6 \text{ troy oz}\]

Step 4: Estimate the price of platinum

Multiply the total troy ounces of platinum by the price per troy ounce to find the total value:\[\text{Value} = 9.9 \times 10^6 \text{ troy oz} \times \\(1,600 \/\text{troy oz}\]\[\approx \\)15.84 \times 10^9\]

Step 5: Conclusion

The calculation shows that the entire ocean contains approximately 9.9 million troy ounces of platinum, valued at approximately $15.84 billion given the current price.

Key Concepts

Platinum Concentration

When we talk about platinum concentration in seawater, we often refer to terms like 'parts per trillion' (ppt). This implies that for every trillion particles in seawater, 0.23 of them are platinum. It's an extremely small fraction, but considering the vastness of the ocean, these tiny amounts add up significantly.
Let's break down what this means. A concentration of 0.23 ppt implies that in a mass of a trillion grams of seawater, there are 0.23 grams of platinum. This tiny concentration requires precise calculations to determine how much platinum is present across the entire volume of the ocean. By understanding this concentration, we're able to scale the findings to immense quantities like the ocean as a whole.

Density of Seawater

The density of seawater plays a crucial role in our calculations. Density is the measure of how much mass exists in a given volume, typically expressed in grams per milliliter (g/mL). For seawater, the density is approximately 1.03 g/mL. This means that each milliliter of seawater weighs 1.03 grams.
In practical terms, this density value lets us convert a known volume of seawater into a mass. This conversion is essential because concentration values (like ppt) are mass-based, requiring the mass of seawater to calculate the mass of substances dissolved in it, like platinum. By multiplying the ocean's total volume in milliliters by the density, we calculate the total mass of seawater, facilitating further calculations.

Unit Conversion

Unit conversion is a fundamental part of carrying out precise chemical calculations. In this exercise, we primarily conducted conversions from volume to mass and from mass to different units.

• First, we converted the volume of the entire ocean from liters to milliliters by recognizing that there are 1,000 milliliters in a liter. This was necessary to proceed with multiplying by the density to get mass in grams.
• Next, for platinum's worth in the ocean, we converted its mass from grams to troy ounces. Knowing that about 31.1035 grams make up one troy ounce, this conversion allowed us to calculate the economic value since platinum prices are typically quoted per troy ounce.

By using proper unit conversion, we ensured accuracy in comparing scientific data to real-world values like monetary worth.

Price Calculation

After determining the mass of platinum in the ocean expressed in troy ounces, we need to assign monetary value to these calculations. The standard method involves multiplying the number of troy ounces by the current market price per troy ounce.
In this case, we assume a market price of $1,600 per troy ounce of platinum. By calculating the product of the total number of troy ounces and price, we get the estimated total value of the platinum present in all the world's oceans. The final estimation showed a staggering value due to the vast oceanic volume, illustrating the potential economic importance of even minute concentrations of valuable metals like platinum in sea water.
Price calculation is the final step that translates physical quantities into practical financial insights, bridging chemistry with economics.