Question

An 18 -karat gold necklace is \(75 \%\) gold by mass, \(16 \%\) silver, and \(9.0 \%\) copper. a. What is the mass, in grams, of the necklace if it contains \(0.24\) oz of silver? b. How many grams of copper are in the necklace? c. If 18 -karat gold has a density of \(15.5 \mathrm{~g} / \mathrm{cm}^{3}\), what is the volume in cubic centimeters?

Short answer

a) 42.52 g, b) 3.83 g, c) 2.744 cm^3.

Step-by-step solution

Convert the mass of silver to grams

First, convert the given mass of silver from ounces to grams. Given 1 ounce (oz) equals 28.3495 grams (g), we convert 0.24 oz: \[ 0.24 \text{ oz} \times 28.3495 \text{ g/oz} = 6.80388 \text{ g} \]

Calculate the total mass of the necklace

Since we know the mass of silver is 16% of the total mass, we use the mass of silver to find the total mass of the necklace using the equation: \[ \frac{6.80388 \text{ g}}{0.16} = 42.52425 \text{ g} \] Therefore, the total mass of the necklace is approximately 42.52 grams.

Determine the mass of copper

Next, calculate the mass of copper using the percentage given. The necklace is 9.0% copper by mass: \[ 0.09 \times 42.52425 \text{ g} = 3.82718 \text{ g} \] The mass of copper in the necklace is approximately 3.827 grams.

Calculate the volume of the necklace

Use the density of 18-karat gold to find the volume. The formula for density is: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \] Rearrange to solve for volume: \[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} = \frac{42.52425 \text{ g}}{15.5 \text{ g/cm}^3} = 2.74382 \text{ cm}^3 \] The volume of the necklace is approximately 2.744 cubic centimeters.

Key concepts

mass conversion

Mass conversion is the process of changing the unit of an object's mass without altering the actual quantity. In chemistry problems, this often involves converting between common units like ounces (oz), grams (g), and kilograms (kg).

To convert between ounces and grams, use the conversion factor where 1 ounce equals 28.3495 grams. For example, if you have 0.24 oz of silver, you multiply by 28.3495 to get:

\(\text{0.24 oz} \times 28.3495 \text{ g/oz} = 6.80388 \text{ grams}\).

Understanding mass conversion helps ensure that measurements are consistent and accurate across different systems of measurement.

percentage by mass

Percentage by mass refers to the proportion of a particular component in a mixture, expressed as a percentage of the total mass. It is calculated by dividing the mass of the component by the total mass of the mixture and then multiplying by 100.

For instance, if a necklace is 16% silver by mass, this means that silver makes up 16% of the total mass of the necklace.

In problems where percentages are given, you can use these to find unknown quantities. For example:
If the mass of silver is 6.80388 grams and it represents 16% of the total mass, you can find the total mass by the equation:
\(\text{Total Mass} = \frac{6.80388 \text{ g}}{0.16} ≈ 42.52 \text{ g}\).

Mastering percentage by mass calculations ensures you can accurately determine the composition of any mixture.

density calculations

Density is a property of material that relates its mass to its volume, typically expressed in grams per cubic centimeter (g/cm³) or kilograms per liter (kg/L). The formula for density is:
\(\text{Density} = \frac{\text{Mass}}{\text{Volume}}\).

To find the volume based on mass and density, you rearrange the formula to:
\(\text{Volume} = \frac{\text{Mass}}{\text{Density}}\).

In our case of an 18-karat gold necklace, where the total mass is 42.52 grams and the density is 15.5 g/cm³:
\(\text{Volume} = \frac{42.52 \text{ g}}{15.5 \text{ g/cm}^3} = 2.74382 \text{ cm}^3\). This means the volume of the necklace is approximately 2.744 cm³.

Understanding density calculations allows you to determine the volume or mass based on the other two quantities, which is useful in many real-world applications.

unit conversions

Unit conversions are critical in chemistry for ensuring that all measurements are consistent and comparable. Conversion factors allow you to switch from one unit of measurement to another. These can be between the same dimension (like grams to kilograms) or different dimensions (like volume to mass using density).

For example, converting ounces to grams uses the factor 1 oz = 28.3495 g. In our problem, converting 0.24 oz of silver to grams involves:
\(\text{0.24 oz} \times 28.3495 \text{ g/oz} = 6.80388 \text{ g}\).

Converting volume to mass using density involves:
\(\text{Mass} = \text{Density} \times \text{Volume}\).

Practicing unit conversions develops the accuracy needed for precise measurements in scientific problems, ensuring correct and meaningful results.