Problem 63 - Chemistry The Central Science In Si Units

Question

Gold can be hammered into extremely thin sheets called gold leaf. An architect wants to cover a \(30 \mathrm{~m} \times 25 \mathrm{~m}\) ceiling with gold leaf that is twelve-millionths of a centimeter thick. The density of gold is \(19.32 \mathrm{~g} / \mathrm{cm}^{3},\) and gold costs \(\$ 1654\) per troy ounce ( 1 troy ounce \(=31.1034768 \mathrm{~g}\) ). How much will it cost the architect to buy the necessary gold?

Step-by-Step Solution

Answer

The total cost to buy the gold is approximately \$9245.86.

Step 1: Calculate the Volume of Gold Needed

First, we need to find the volume of gold required to cover the ceiling. The ceiling area is given by \( 30 \, \text{m} \times 25 \, \text{m} = 750 \, \text{m}^2 \). Convert this area to square centimeters: \( 750 \, \text{m}^2 = 7,500,000 \, \text{cm}^2 \). The thickness of the gold leaf is \( 0.0000012 \, \text{cm} \). The volume \( V \) is calculated by \( V = \text{Area} \times \text{Thickness} = 7,500,000 \, \text{cm}^2 \times 0.0000012 \, \text{cm} = 9 \, \text{cm}^3 \).

Step 2: Determine the Mass of Gold Required

Next, calculate the mass of gold using the density formula: \( \text{Mass} = \text{Density} \times \text{Volume} \). Given that the density of gold is \( 19.32 \, \text{g/cm}^3 \), the mass \( m \) is: \( m = 19.32 \, \text{g/cm}^3 \times 9 \, \text{cm}^3 = 173.88 \, \text{g} \).

Step 3: Convert Mass to Troy Ounces

We need to convert the mass from grams to troy ounces, where 1 troy ounce is \( 31.1034768 \, \text{g} \). The mass in troy ounces is \( \frac{173.88 \, \text{g}}{31.1034768 \, \text{g/troy ounce}} \approx 5.59 \, \text{troy ounces} \).

Step 4: Calculate the Total Cost

Finally, calculate the cost of the gold leaf. With a cost of \( \\(1654 \) per troy ounce, the total cost \( C \) is: \( C = 5.59 \, \text{troy ounces} \times \\)1654 \) per troy ounce = \( \$9245.86 \).

Key Concepts

Volume Calculation

When dealing with gold leaf, understanding how to calculate the volume of gold needed without making errors is crucial. Volume calculation involves determining how much space a substance occupies. This is usually found by multiplying the area of the surface you want to cover by the thickness of the material. In this particular exercise, the surface to be covered is a ceiling measuring 30 meters by 25 meters.

• First, calculate the total area in square meters: 30 m × 25 m = 750 m².
• Since volume uses cubic measurements, convert the area into a compatible unit: square centimeters. There are 10,000 square centimeters in one square meter, so 750 m² = 7,500,000 cm².
• Given the thickness of the gold leaf is only 0.0000012 cm, find the volume by multiplying the area by the thickness: 7,500,000 cm² × 0.0000012 cm = 9 cm³.

Understanding how to switch units and apply formulas in the right context ensures accurate calculations and prevents mistakes.

Density and Mass Conversion

Density is a fundamental concept used to relate the mass of a substance to its volume. The density tells us how much mass you'll find in a given volume. This relationship is expressed with the formula:\[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\]

In this case, you want to find the total mass of the gold leaf that can cover the ceiling, given the density of gold is 19.32 g/cm³. Here's how to use the information:

• First, note you already calculated the volume as 9 cm³.
• Apply the density formula in reverse to solve for mass: \[\text{Mass} = \text{Density} \times \text{Volume} = 19.32 \, \text{g/cm}^3 \times 9 \, \text{cm}^3 = 173.88 \, \text{g}\]

This step of the conversion helps you understand exactly how many grams of gold you need to produce the gold leaf and why understanding the meaning of density is foundational in these applications.

Unit Conversion

In many scientific and practical scenarios, converting the units of measurement is essential to understanding quantities and costs involved accurately. In this gold-leaf problem, conversion between different units is key to determining the cost.First, you need to convert the mass of gold from grams to troy ounces. Why troy ounces? Because the price of gold usually is quoted in troy ounces.

• The given mass is 173.88 grams, and we know that 1 troy ounce is equivalent to 31.1034768 grams.
• To convert to troy ounces, you divide the mass in grams by the conversion factor: \[\frac{173.88 \, \text{g}}{31.1034768 \, \text{g/troy ounce}} = 5.59 \, \text{troy ounces}\]

By converting units accurately, you can easily proceed to calculate the total cost. Using the given price per troy ounce \( \$1654 \), multiply by 5.59 troy ounces to derive the total expense. Such conversions arm you with the ability to translate scientific data into fiscal terms, adding practical meaning to the calculations.