Problem 97 - Chemistry Principles and Practice

Question

A Gold leaf, which is used for many decorative purposes, is made by hammering pure gold into very thin sheets. Assuming that a sheet of gold leaf is \(1.27 \times 10^{-5} \mathrm{~cm}\) thick, how many square feet of gold leaf could be obtained from \(28.35 \mathrm{~g}\) gold? The density of gold is \(19.3 \mathrm{~g} / \mathrm{cm}^{3}\).

Step-by-Step Solution

Answer

The gold leaf can cover approximately 12.44 square feet.

Step 1: Find the Volume of Gold

First, we calculate the volume of the gold using its mass and density. The formula to find the volume (V) is: \[ V = \frac{\text{mass}}{\text{density}} \]Substitute the given values:\[ V = \frac{28.35 \text{ g}}{19.3 \text{ g/cm}^3} \]When you calculate this, \[ V \approx 1.468 \text{ cm}^3 \]

Step 2: Find the Area of the Gold Leaf

Now that we have the volume of gold, we can find the area (A) of the gold leaf. Volume can also be calculated by multiplying the area by the thickness of the leaf. Therefore,\[ V = A \times \text{thickness} \]Substitute the known values and solve for A:\[ 1.468 \text{ cm}^3 = A \times 1.27 \times 10^{-5} \text{ cm} \]\[ A \approx \frac{1.468}{1.27 \times 10^{-5}} \text{ cm}^2 \]\[ A \approx 115590.55 \text{ cm}^2 \]

Step 3: Convert Area from Square Centimeters to Square Feet

Now convert the area from square centimeters to square feet. We know that 1 inch equals 2.54 cm, and 1 foot equals 12 inches, so:\[ 1 \text{ cm}^2 = \left(\frac{1}{2.54}\right)^2 \text{ inch}^2 = \left(\frac{1}{2.54 \times 12}\right)^2 \text{ ft}^2 \]Therefore,\[ A \approx 115590.55 \times \left(\frac{1}{12 \times 2.54}\right)^2 \text{ ft}^2 \]\[ A \approx 12.44 \text{ ft}^2 \]

Step 4: Final Computation and Answer

Thus, having converted the area into square feet, the total area of the gold leaf is about 12.44 square feet.

Key Concepts

Volume calculation

To understand how to find the volume of gold in this problem, we rely on one basic principle: the formula for volume. When it comes to solid objects, and especially metals like gold, volume (\( V \)) can be calculated using the formula:\[ V = \frac{\text{mass}}{\text{density}} \]This formula indicates that if you know the mass and density of a substance, you can easily find its volume. In this case, we started with a mass of 28.35 grams and a known density for gold that is 19.3 grams per cubic centimeter. Plugging these values into the formula gives us the volume of gold in cubic centimeters. Through calculation, we determine the volume to be approximately 1.468 cm³.Understanding how to calculate volume is essential when working with materials and helps in determining how much space a substance occupies.

Unit conversion

The next step in solving the problem requires us to convert units from the metric system to the imperial system. This is vital because in our problem, we first calculate area in square centimeters, but the final answer needs to be in square feet. To perform this conversion, we utilize the relationship between centimeters and inches, knowing that 1 inch equals 2.54 centimeters.
From this, we can derive:

• 1 cm equals \(\frac{1}{2.54}\) inches
• Subsequently, 1 cm² equals \(\left(\frac{1}{2.54}\right)^2\) inch²

Next, since there are 12 inches in a foot, we further this conversion:

• 1 foot equals 12 inches, so 1 inch equals \(\frac{1}{12}\) feet
• Hence, 1 cm² equals \(\left(\frac{1}{(2.54 \times 12)}\right)^2\) ft²

This systematic conversion allows us to translate the gold leaf's calculated area from 115,590.55 cm² to approximately 12.44 ft², completing the unit conversion step efficiently.

Density formula

Density is a fundamental concept in the study of materials. It tells us how much mass is contained in a given volume. The formula for density (\( \rho \)) is:\[ \rho = \frac{\text{mass}}{\text{volume}} \]This is often rearranged to find mass or volume depending on what data you have and what you're solving for. In most problems similar to this one, we use:

• Volume (\( V \)): \( V = \frac{\text{mass}}{\rho} \)

Gold, being heavy and dense, has a density of 19.3 g/cm³ which clearly shows it packs a lot of mass into a relatively small space. Evaluating this parameter helps us viscerally understand the opulence and weight of gold compared to other materials.This density formula isn't just crucial for solving academic problems, but it's also vital in industries where material properties need to be tightly controlled.

Gold leaf

Gold leaf is a term used to describe gold that has been hammered into very thin sheets, which can be as thin as micrometers. This process is called goldbeating. Gold leaf is widely used in various applications, from art and decoration to electronics and traditional gilding.The exercise underlines how small quantities of gold, thanks to its malleability, can be transformed into large sheets. For instance, with a thickness of just \(1.27 \times 10^{-5}\) cm, incredibly thin gold sheets are possible. This remarkable thinness highlights gold's unique ability to be stretched and molded due to its atomic structure.Knowing how to convert these dimensions into practical metrics is crucial for any application involving these sheets, whether it's artistic projects or manufacturing goods that require gold's unique qualities.