Problem 170 - General Chemistry

Question

You have a piece of gold jewelry weighing \(9.35 \mathrm{~g}\). Its volume is \(0.654 \mathrm{~cm}^{3}\). Assume that the metal is an alloy (mixture) of gold and silver, which have densities of \(19.3 \mathrm{~g} / \mathrm{cm}^{3}\) and \(10.5 \mathrm{~g} / \mathrm{cm}^{3}\), respectively. Also assume that there is no change in volume when the pure metals are mixed. Calculate the percentage of gold (by mass) in the alloy. The relative amount of gold in an alloy is measured in karats. Pure gold is 24 karats; an alloy of \(50 \%\) gold is 12 karats. State the proportion of gold in the jewelry in karats.

Step-by-Step Solution

Answer

The jewelry is approximately 74.2% gold, making it about 17.81 karats.

Step 1: Define the variables

Let the mass of gold in the jewelry be denoted by \( m_g \) and the mass of silver be \( m_s \). The total mass is given by \( m_g + m_s = 9.35 \).

Step 2: Use volume and density to express masses

Using the density relation \( \text{density} = \frac{\text{mass}}{\text{volume}} \), express the masses: \[ \frac{m_g}{19.3} + \frac{m_s}{10.5} = 0.654 \].

Step 3: Solve the system of equations

We have two equations from the previous steps: \( m_g + m_s = 9.35 \) and \( \frac{m_g}{19.3} + \frac{m_s}{10.5} = 0.654 \). Solve this system of equations to find \( m_g \) and \( m_s \).

Step 4: Set up the equations

The first equation is \( m_s = 9.35 - m_g \). Substitute this in the second equation: \[ \frac{m_g}{19.3} + \frac{9.35 - m_g}{10.5} = 0.654 \].

Step 5: Simplify and solve for \( m_g \)

Simplify the equation and solve for \( m_g \): \[ \frac{m_g}{19.3} + \frac{9.35}{10.5} - \frac{m_g}{10.5} = 0.654 \], which simplifies to \( m_g \approx 6.94 \text{ g} \).

Step 6: Calculate the percentage of gold

The percentage of gold by mass is \( \frac{m_g}{9.35} \times 100 \% \approx 74.2 \% \).

Step 7: Determine the karats of the alloy

Since 24-karat gold is pure, 12-karat gold is 50%, and so on, the equation for karats is \( \frac{74.2}{100} \times 24 \approx 17.81 \text{ karats} \).

Key Concepts

Density

Density is an essential concept that helps us determine how much mass is contained within a given volume. It's a measure of how tightly matter is packed together in an object. In mathematical terms, density is illustrated as \(\text{density} = \frac{\text{mass}}{\text{volume}}.\)
In the context of our gold-silver alloy, knowing the densities of pure gold and silver allows us to figure out the composition of the alloy. Gold has a density of \(19.3 \ \text{g/cm}^3\) while silver has a density of \(10.5 \ \text{g/cm}^3\).
By utilizing these values in the formula, we can derive valuable information about how much of each metal is in our jewelry. When the alloy's overall density is given, and the individual metal densities are known, they can help determine their proportions in the mixture.

Mass and Volume Relationship

The relationship between mass and volume is fundamental in solving problems relating to mixtures like alloys. Mass tells us how much matter is in a substance, while volume measures the space that this matter occupies.
In our exercise, we have a total mass and a fixed volume for the piece of jewelry. These aspects are tied directly by the densities of the metals being mixed. For each metal:

• The mass can be determined by multiplying density by volume.
• The volume can then be used to break down how much of the total mass comes from each component of the alloy.

In practice, once the mass of the jewelry is spread across the known volume, we utilize specific densities to partition this mass between the gold and silver components. This relationship makes it possible to decipher the exact makeup of an alloy, such as our gold-silver blend.

Karats Involved in Gold Jewelry

Understanding karats is crucial for anyone interested in gold jewelry as it indicates the purity of the gold component within an alloy. The term "karat" quantifies the fineness of gold. Pure gold is marked 24 karats, representing 100% purity.
In alloys, each karat represents approximately \(\frac{1}{24}\) of the whole. Hence, a 12-karat gold piece signifies that the metal is \(50\%\) gold, effectively making it half pure.
In the exercise, the jewelry's karats are calculated based on how much gold, by mass, is present in the alloy. If knowledge about the percentage composition of gold is available, one can calculate the equivalent karat value by connecting this percentage to a full karat scale: \(\frac{\text{gold percentage}}{100} \times 24\), providing a straightforward means of conveying the purity level.

Percentage Composition in a Mixture

The percentage composition in a mixture indicates how much of each component exists in relation to the entire mixture by mass. It is expressed as a percentage of the total mass.
To determine the composition of our gold-silver alloy:

• Calculate the mass of each metal in the mixture using the given densities and total mass.
• Once the mass for each component is identified, find the percentage by dividing the mass of each metal by the total mass, then multiplying by 100%.

For instance, in the solved exercise, a proportion of the jewelry's mass can be attributed to gold. Calculating its percentage in relation to the total mass of \(9.35\) grams provides a clear picture of how much of the alloy is actual gold, solidifying the final composition of the piece.