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 \(19.3 \mathrm{~g} / \mathrm{cm}^{3}\) 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.

Short answer

The percentage of gold in the jewelry is approximately 86.9%, and its purity is approximately 20.9 carats.

Step-by-step solution

Calculate the mass of gold and silver

Let's denote the mass of gold in the jewelry as m_g, and the mass of silver as m_s. We know that the total mass of the jewelry is 9.85 g, so the sum of the gold and silver mass should be equal to the total mass: m_g + m_s = 9.85

Calculate the volume of gold and silver

Let the volume of gold in the jewelry be V_g, and the volume of silver be V_s. We are given the total volume of the jewelry, which is equal to the sum of the volumes of gold and silver: V_g + V_s = 0.675

Express volumes in terms of mass using densities

Recall that the density of gold is 19.3 g/cm³ and the density of silver is 10.5 g/cm³. Density is defined as mass/volume, so we can express the volume of gold and silver like this: V_g = m_g / 19.3 V_s = m_s / 10.5 Now, substitute these expressions for V_g and V_s into the volume equation: (m_g / 19.3) + (m_s / 10.5) = 0.675

Solve the system of equations

We need to solve the following system of linear equations for m_g and m_s: m_g + m_s = 9.85 (m_g / 19.3) + (m_s / 10.5) = 0.675 We can solve it by substitution or elimination methods. Let's use the substitution method. From the first equation, express m_s as follows: m_s = 9.85 - m_g Now substitute m_s in the second equation with this expression: (m_g / 19.3) + ((9.85 - m_g) / 10.5) = 0.675 Solve for m_g: m_g ≈ 8.566 g Now substitute the value of m_g back into m_s = 9.85 - m_g to find m_s: m_s ≈ 1.284 g

Calculate the percentage of gold

Now that we know the mass of the gold (m_g) and the total mass, we can find the percentage of gold by dividing the mass of gold by the total mass, and multiplying the result by 100: Percentage of gold = (m_g / total mass) × 100 = (8.566 / 9.85) × 100 ≈ 86.9%

Calculate the purity of gold in carats

Recall that pure gold is 24 carat, and the percentage of gold in an alloy is given as a percentage of this value. We have found the percentage of gold in the jewelry (86.9%). To find the purity in carats, simply multiply this percentage by 24: Purity of gold in carats = 0.869 × 24 ≈ 20.9 carats Thus, the purity of the gold jewelry is approximately 20.9 carats.

Key concepts

Density

Density is an important concept when working with materials such as metals and alloys. Density is defined as the mass of a substance per unit volume, which mathematically can be expressed as \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \).
This property is crucial when determining the composition of objects like gold jewelry. By understanding the density of gold (19.3 g/cm³) and silver (10.5 g/cm³), we can use these values to identify the different amounts of each metal present in a mixture.

• If you know the density and volume of an object, you can easily find the mass by rearranging the formula: \( \text{Mass} = \text{Density} \times \text{Volume} \).
• Conversely, if you know the mass and density, you can find the volume: \( \text{Volume} = \frac{\text{Mass}}{\text{Density}} \).

Understanding density allows us to break down complex mixtures and accurately measure components of an alloy, as illustrated in the gold-silver jewelry example.

Mass and Volume Calculations

In the context of the gold alloy problem, calculating mass and volume is key to determining the amounts of gold and silver within the jewelry. Given a total mass and volume, we often need to solve a series of equations.
The first step involves setting up equations based on the information we have:

• The combined mass of gold and silver is equal to the total mass, so: \( m_g + m_s = 9.85 \text{ g} \).
• The volumes also add up to the total volume, using the given densities to express mass as a function of volume: \( \frac{m_g}{19.3} + \frac{m_s}{10.5} = 0.675 \text{ cm}^3 \).

With these equations, you can solve for the individual mass of each metal, either by substitution or elimination methods. This process is foundational in material science and quality control, ensuring we can confidently determine the composition of alloys.

Purity Measurement

Purity measurement in metals, especially in gold, is a standard practice to evaluate the quality of an alloy. In the exercise, we are asked to express the purity of the gold in carats.
Pure gold is considered 24 carats, representing 100% pure gold. Thus, any alloy composition is a percentage of this figure:

• If an alloy has 50% gold by mass, it is termed 12 carats.
• To find the carat value, calculate the percentage purity by mass and then translate it into carats by multiplying by 24.

In the case of the gold jewelry with approximately 86.9% purity, the calculation is \( 0.869 \times 24 \approx 20.9 \) carats. This measurement helps to determine the quality and value of the gold piece accurately, making it essential for both jewelers and consumers.