Problem 90 - Chemistry: The Central Science in SI Units, Global Edition

Question

Gold is alloyed (mixed) with other metals to increase its hardness in making jewelry. (a) Consider a piece of gold jewelry that weighs \(9.85 \mathrm{~g}\) and has a volume of \(0.675 \mathrm{~cm}^{3}\). The jewelry contains only gold and silver, which have densities of 19.3 and \(10.5 \mathrm{~g} / \mathrm{cm}^{3}\), respectively. If the total volume of the jewelry is the sum of the volumes of the gold and silver that it contains, calculate the percentage of gold (by mass) in the jewelry. (b) The relative amount of gold in an alloy is commonly expressed in units of carats. Pure gold is 24 carat, and the percentage of gold in an alloy is given as a percentage of this value. For example, an alloy that is \(50 \%\) gold is 12 carat. State the purity of the gold jewelry in carats.

Step-by-Step Solution

Answer

The percentage of gold (by mass) in the jewelry is approximately \(83.86 \% \), and the purity of the gold jewelry is approximately 20.13 carats.

Step 1: Determine the volume of gold and silver in the jewelry.

We will be using the formula for density to find the volume of gold and silver in the jewelry. The density formula is: \(density = \frac{mass}{volume}\). Let the volume of gold in the jewelry be \(V_{gold}\) and the volume of silver be \(V_{silver}\). The total volume of the jewelry is the sum of the volumes of gold and silver: \(0.675 \mathrm{~cm}^{3} = V_{gold} + V_{silver}\).

Step 2: Express the density of gold and silver using their masses and volumes respectively.

We will now express the density of gold and silver using their masses and volumes respectively: \(density_{gold} = \frac{mass_{gold}}{V_{gold}}\) \(density_{silver} = \frac{mass_{silver}}{V_{silver}}\). Let the mass of gold in the jewelry be \(m_{gold}\) and the mass of silver be \(m_{silver}\). The total mass of the jewelry is the sum of the masses of gold and silver: \(9.85 \mathrm{~g} = m_{gold} + m_{silver}\).

Step 3: Set up a system of equations to solve for the mass of gold and silver in the jewelry.

Using the density formula and the total mass equation, we can set up a system of equations: \(\frac{m_{gold}}{V_{gold}} = 19.3 g/cm^{3}\), \(\frac{m_{silver}}{V_{silver}} = 10.5 g/cm^{3}\), \(m_{gold} + m_{silver} = 9.85 g\), \(V_{gold} + V_{silver} = 0.675 cm^3\). This system of equations contains four equations and four unknowns: \(m_{gold}, m_{silver}, V_{gold}\), and \(V_{silver}\). We will first solve for the masses of gold and silver using substitution method.

Step 4: Solve the system of equations for the mass of gold and silver in the jewelry.

Start by solving the first two equations for the volume of gold and silver: \(V_{gold} = \frac{m_{gold}}{19.3}\), \(V_{silver} = \frac{m_{silver}}{10.5}\). Substitute these into the volume equation: \(\frac{m_{gold}}{19.3} + \frac{m_{silver}}{10.5} = 0.675\). Now we can express the mass of silver in terms of the mass of gold using the total mass equation and substitute this into the volume equation: \(m_{silver} = 9.85 - m_{gold}\). Substitute this into the volume equation and solve for the mass of gold: \(\frac{m_{gold}}{19.3} + \frac{9.85 - m_{gold}}{10.5} = 0.675\). Solve for \(m_{gold}\): \(m_{gold} \approx 8.265 g\). Now substitute the mass of gold back into the mass equation to find the mass of silver: \(m_{silver} = 9.85 - 8.265 \approx 1.585 g\).

Step 5: Calculate the percentage of gold (by mass) in the jewelry.

Now that we have the masses of gold and silver, we can calculate the percentage of gold (by mass) in the jewelry using the following formula: \(\% \; gold = \frac{mass_{gold}}{total \; mass} \times 100\%\). Substitute the values into the formula and solve for the percentage of gold: \(\% \; gold = \frac{8.265 \mathrm{~g}}{9.85 \mathrm{~g}} \times 100\% \approx 83.86\% \).

Step 6: Calculate the purity of the gold jewelry in carats.

To convert the percentage of gold into carats, we simply multiply the percentage by the maximum carat value (24 carat for pure gold) and divide by 100: \(purity \;(in \; carats) = \frac{\% \; gold \times 24}{100}\). Substitute the percentage of gold and solve for the purity in carats: \(purity \;(in \; carats) = \frac{83.86 \times 24}{100} \approx 20.13 \; carats\). The purity of the gold jewelry is approximately 20.13 carats.

Key Concepts

Density

Density is a fundamental concept that allows us to calculate the mass of an object if we know its volume, and vice-versa. When creating gold alloys for jewelry, understanding density helps determine the proportion of each metal in the alloy.
Density is calculated using the formula: \[\text{Density} = \frac{\text{mass}}{\text{volume}}\] This formula is crucial when working with mixtures like gold and silver alloys, where knowing individual densities (\(19.3 \, \text{g/cm}^3\) for gold and \(10.5 \, \text{g/cm}^3\) for silver) aids in determining the specific contributions of each metal.
By using density and the total volume, jewelers can calculate both the mass and proportion of metals in a piece of jewelry, ensuring the desired physical and aesthetic properties are achieved.

Carats

Carats measure the purity of gold in alloys, with pure gold being defined as 24 carats. This is a useful metric for consumers and jewelers alike to understand the value and quality of a gold alloy.
To determine the carat value of an alloy:

• Calculate the gold percentage by mass in the alloy.
• Multiply by 24 (since pure gold is 24 carat).

For example, if a piece of jewelry is 83.86% gold, its carat value can be calculated: \[\text{Carats} = \frac{83.86 \times 24}{100} \approx 20.13\] Carat value not only reflects the gold content but also helps indicate an item's resilience and intended use. Lower carat values often result in harder, more durable adornments suitable for everyday wear.

Jewelry

Jewelry made from gold alloys combines beauty with practicality. Gold on its own is soft and not ideal for items that endure wear and tear. By alloying with other metals like silver, jewelers enhance the material's durability.
In jewelry making, the choice of metal and its proportion impacts both the appearance and cost of the final product. High-gold content alloys maintain a classic look and precious value, while lower-gold content can offer intriguing variations in color and increased hardness.
Understanding the composition of jewelry is essential for both the purchase and crafting of these pieces, ensuring they meet the standards and expectations of artistry and longevity.