Problem 47 - Chemistry and Chemical Reactivity

Question

A distant acquaintance has offered to sell you a necklace, said to be pure ( 24 -carat) gold, for \(\$ 300 .\) You have some doubts, however; perhaps it is gold plated. You decide to run a test. You have a graduated cylinder and a small balance. You partially fill the cylinder with water and immerse the necklace; the height of water rises from \(22.5 \mathrm{mL}\) to \(26.0 \mathrm{mL} .\) Then you determine the mass to be \(67 \mathrm{g} .\) You recall that the density of gold is \(19.3 \mathrm{g} / \mathrm{cm}^{3},\) and that no other element has a density near this value. (Silver has a density of \(\left.11.5 \mathrm{g} / \mathrm{cm}^{3} .\right)\) The price of gold on the open market is \(\$ 380\) per troy ounce ( 1 troy ounce \(=31.1 \mathrm{g}\) ). Is the necklace gold? Explain your conclusion. Is \(\$ 300\) a good price?

Step-by-Step Solution

Answer

The necklace's density suggests it is pure gold, making the $300 price very low compared to its $817 gold value.

Step 1: Calculate the Volume of the Necklace

The volume of the necklace is the difference in the water level before and after immersing it. Use the formula:\[ \text{Volume of Necklace} = \text{Final Volume} - \text{Initial Volume} = 26.0 \, \text{mL} - 22.5 \, \text{mL} = 3.5 \, \text{mL}\]Since 1 mL is equivalent to 1 cm³, the volume of the necklace is 3.5 cm³.

Step 2: Calculate the Density of the Necklace

Density is calculated using the formula \(\text{Density} = \frac{\text{Mass}}{\text{Volume}}\). Substitute the known values:\[\text{Density of Necklace} = \frac{67 \, \text{g}}{3.5 \, \text{cm}^3} = 19.14 \, \text{g/cm}^3\]Compare this to the density of pure gold which is 19.3 g/cm³.

Step 3: Determine the Metal Composition

The calculated density of the necklace is 19.14 g/cm³, which is very close to the density of pure gold (19.3 g/cm³). This suggests the necklace is likely made mostly of gold and is not gold-plated.

Step 4: Calculate the Market Value of the Necklace

First, convert the mass of the necklace from grams to troy ounces:\[\text{Mass in troy ounces} = \frac{67 \, \text{g}}{31.1 \, \text{g/troy ounce}} \approx 2.15 \, \text{troy ounces}\]Then, calculate the market value of the gold based on the gold price of \(380 per troy ounce:\[\text{Market Value} = 2.15 \, \text{troy ounces} \times \\)380/\text{troy ounce} = \$817\]

Step 5: Analyze the Offer Price

Given that the calculated market value of the gold content in the necklace is \\(817, the offer price of \\)300 is a very low price. This discrepancy further supports the idea that the necklace is most likely pure or almost pure gold.

Key Concepts

Graduated Cylinder

A graduated cylinder is an essential tool in chemistry used to measure the volume of liquids. It's a transparent container marked with a series of lines (graduations) to indicate volume measurements.
These lines let you precision-measure the volume of the liquid inside. Graduated cylinders come in various sizes, allowing you to measure volumes ranging from a few milliliters to several liters. They're commonly made of glass or plastic.
When using a graduated cylinder, it's essential to read the volume at the bottom of the meniscus, the curved surface of the liquid. This ensures accurate measurements.

• **Before each measurement, ensure the graduated cylinder is clean and free from previous residues.**
• **Always place the cylinder on a flat surface and view at eye level for an accurate reading.**

In the exercise, a graduated cylinder measures the volume difference when the necklace is submerged, determining its volume swiftly and accurately. This change in water level reflects the volume of the necklace via water displacement, a classic Archimedes principle.

Density of Gold

The density of a substance is its mass per unit volume, offering insight into how heavy or light a material is relative to its size. For this exercise, determining the density of gold helps ascertain whether the necklace is genuine.
Gold is a dense, heavy element, with a density of 19.3 g/cm³. This characteristic makes it stand out since few other substances have similar densities.
Calculating density is straightforward:

• **Use the formula:** \[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\]
• **Compare calculated density with known values**: If the outcome aligns closely with gold’s density, as in this situation (19.14 g/cm³), there's a high chance the necklace is indeed primarily composed of gold.

Therefore, this calculated density offers critical clues, suggesting the necklace's material composition without specialized equipment, only basic calculation and measurement skills.

Troy Ounce

The troy ounce is a unit of measure used primarily for precious metals. It's slightly heavier than the standard ounce, which is often used for other goods.
To put this into perspective:

• **1 troy ounce is equal to 31.1 grams**, while a regular ounce is approximately 28.35 grams.
• **This measurement standard is crucial when dealing with metals like gold, silver, and platinum, ensuring precision in trading and valuations.**

In our exercise, converting the necklace's weight from grams to troy ounces is essential for assessing its market value. The mass of the necklace when translated into troy ounces is evaluated against current market prices for precious metals.
This conversion is pivotal in determining the necklace's worth and assessing whether the offered sale price is fair. The exercise highlights the role of accurate conversions between measurement units in financial calculations, emphasizing their importance in providing clarity and fairness in trade.