Problem 84 - Chemistry: The Central Science

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 jewelry is approximately 61.72% gold by mass and has a purity of 14.81 carats.

Step 1: Write down the given information

We are given the following values: Weight of the jewelry (m_total) = 9.85 g Volume of the jewelry (V_total) = 0.675 cm³ Density of gold (ρ_gold) = 19.3 g/cm³ Density of silver (ρ_silver) = 10.5 g/cm³ We are asked to find the percentage of gold by mass and the jewelry's purity in carats.

Step 2: Calculate the mass of gold and silver using their densities

We can calculate the mass of gold (m_gold) and silver (m_silver) in the jewelry using the densities: m_gold = ρ_gold × V_gold m_silver = ρ_silver × V_silver Since the total volume of the jewelry is the sum of the volumes of gold and silver: V_total = V_gold + V_silver Rearrange to find V_gold: V_gold = V_total - V_silver Substitute the expression for V_gold into the equation for m_gold: m_gold = ρ_gold × (V_total - V_silver)

Step 3: Solve for V_silver

We can use the mass conservation law to solve for V_silver. The total mass of the jewelry is equal to the sum of the masses of gold and silver: m_total = m_gold + m_silver Substitute the expressions for m_gold and m_silver from Step 2: 9.85 = (19.3 × (0.675 - V_silver)) + (10.5 × V_silver) Now, solve for V_silver: 9.85 = 13.005 - 19.3 V_silver + 10.5 V_silver 0 = 3.155 - 8.8 V_silver V_silver = \(\frac{3.155}{8.8}\) V_silver ≈ 0.358 cm³

Step 4: Calculate the mass of gold and silver

Now, we can use the values for V_silver and ρ_silver to calculate the mass of silver: m_silver = ρ_silver × V_silver m_silver = 10.5 × 0.358 m_silver ≈ 3.760 g Next, we can use the value for V_silver to find the volume of gold: V_gold = V_total - V_silver V_gold = 0.675 - 0.358 V_gold ≈ 0.317 cm³ Then, calculate the mass of gold: m_gold = ρ_gold × V_gold m_gold = 19.3 × 0.317 m_gold ≈ 6.090 g

Step 5: Calculate the percentage of gold by mass

Using the mass of gold and the total mass of the jewelry, we can find the percentage of gold by mass: Percentage of gold by mass = \(\frac{m_{gold}}{m_{total}}\) × 100 Percentage of gold by mass = \(\frac{6.090}{9.85}\) × 100 Percentage of gold by mass ≈ 61.72%

Step 6: Express the purity of gold in carats

Since pure gold is 24 carats, we can find the jewelry's purity in carats by multiplying the percentage of gold by a factor of 24: Purity in carats = 0.6172 × 24 Purity in carats ≈ 14.81 carats So, the jewelry is approximately 61.72% gold by mass and has a purity of 14.81 carats.

Key Concepts

Density Calculation

Calculating the density of a mixture, such as gold alloy, involves understanding that the proportions of each component contribute to overall density. Here, the jewelry piece has an overall density that is affected by both the gold and silver inside. To start, we were given the total mass and volume of the jewelry.

The formula for density is:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]In this exercise, we apply the density formula to the individual metals. For each metal, density is used to express the relationship between mass and volume:

• Density of gold = 19.3 g/cm³
• Density of silver = 10.5 g/cm³

These densities help us allocate parts of the combined volume and mass to each metal constituent. By solving for the volume and then the mass of each component, we determined how much of each metal is present in the piece.

Mass Percentage

Mass percentage is a way to convey the concentration of a component in a mixture. It tells us how much of a particular metal, like gold, is present in an alloy compared to the total mass.

In the exercise, the mass of gold is calculated to be approximately 6.090 grams, and the total mass of the jewelry is 9.85 grams. To find the mass percentage, the following formula is used:
\[\text{Mass Percentage} = \left(\frac{\text{Mass of Component}}{\text{Total Mass}}\right) \times 100\]For gold, this becomes:
\[\text{Mass Percentage of Gold} = \left(\frac{6.090}{9.85}\right) \times 100 \approx 61.72\%\]

• This means that 61.72% of the jewelry's weight is due to gold.
• Mass percentage provides a clear picture of gold's dominance in the alloy, essential for valuation or quality assessment.

Carats Purity

Carats are a traditional measure of gold purity. They range from 0 to 24, with 24 carats representing pure gold. The value in carats indicates how much gold is present compared to other metals.

To calculate the carats of the gold alloy:
First, determine the percentage of gold, which is approximately 61.72%. Then, use the carat scale, where 100% gold is 24 carats. The calculation formula is:
\[\text{Carat Purity} = \left(\frac{\text{Mass Percentage of Gold}}{100}\right) \times 24\]Applying this:
\[\text{Carat Purity} = 0.6172 \times 24 \approx 14.81 \, \text{carats}\]

• This shows the jewelry is approximately 14.81 carats.
• Carat purity helps customers and jewelers understand the value and purity level of the gold in any piece of jewelry.
• Higher carats mean higher purity and typically greater value.