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 karats. Pure gold is 24 -karat, 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 -karat. State the purity of the gold jewelry in karats.

Short answer

The gold jewelry is approximately 33.96% gold by mass, which is equivalent to around 8.15 karats.

Step-by-step solution

Recall the density formula

The density is defined as the mass per unit volume. The formula for density is: \[density = \frac{mass}{volume}\]

Calculate the total volume of gold and silver

According to the given information, the total volume of jewelry is the sum of the volumes of gold and silver. Based on the formula, the total mass of gold and silver in the alloy is: \[mass_{total} = mass_{Au} + mass_{Ag}\] Where \(mass_{Au}\) is the mass of gold and \(mass_{Ag}\) is the mass of silver. The total mass denoted by \(mass_{total}\) is equal to \(9.85 \mathrm{~ g}\).

Express the volume of gold and silver in terms of mass

We can express the volume of gold and silver as the product of their mass and respective densities: \[volume_{Au} = \frac{mass_{Au}}{density_{Au}}\] \[volume_{Ag} = \frac{mass_{Ag}}{density_{Ag}}\] Where \(density_{Au} = 19.3 \mathrm{~ g / cm^3}\) and \(density_{Ag} = 10.5 \mathrm{~ g / cm^3}\).

Add the volumes of gold and silver and equate to the total volume

Sum the volumes of gold and silver and equate the result to the total volume of jewelry (\({0.675~cm^3}\)): \[\frac{mass_{Au}}{density_{Au}} + \frac{mass_{Ag}}{density_{Ag}} = volume_{total}\] We already have an expression for \(mass_{Ag}\) in terms of \(mass_{Au}\) from Step 2. Replace \(mass_{Ag}\) with the appropriate expression: \[\frac{mass_{Au}}{density_{Au}} + \frac{mass_{total} - mass_{Au}}{density_{Ag}} = volume_{total}\]

Solve for the mass of gold

Solve the equation in step 4 for \(mass_{Au}\): \[mass_{Au} = \frac{density_{Au}(density_{Ag}volume_{total} - mass_{total})}{density_{Ag} - density_{Au}}\] Plug in the given values for densities and volume: \[mass_{Au} = \frac{19.3 (10.5 \times 0.675~ cm^3 - 9.85~g)}{10.5 - 19.3}\] Calculate the mass of gold: \[mass_{Au} ≈ 3.344~g\]

Compute the percentage of gold by mass

Divide the mass of gold by the total mass and multiply by 100: \[\% Au = \frac{mass_{Au}}{mass_{total}} \times 100\] \[ \% Au = \frac{3.344~g}{9.85~g} \times 100 \approx 33.96 \% \] The percentage of gold in the jewelry is approximately \(33.96\%\).

Convert the percentage of gold to karats

Pure gold is 24-karat. To convert the percentage of gold to karats, divide the percentage by \(100\%\) and multiply by 24: \[karats = \frac{percentage}{100\%} \times 24\] \[karats = \frac{33.96 \%}{100 \%} \times 24 ≈ 8.15~karats\] The purity of the gold jewelry is approximately \(8.15~karats\).

Key concepts

Alloy Composition

Alloys are materials made by combining two or more different metals to create a new substance with desirable properties that the constituent metals may not possess individually. The most notable example, as seen in the exercise, is the alloying of gold with other metals like silver to increase the hardness of gold, which is otherwise quite soft and malleable in its pure form. This practice is common in jewelry making to enhance durability.

When dealing with alloy compositions, understanding the proportions of each metal in the alloy is crucial. Calculating the mass percentage of each metal helps determine the character and quality of the alloy. A higher proportion of a certain metal generally leads to more pronounced characteristics of that metal in the alloy. For instance, an increase in the silver content of gold jewelry would make it harder and possibly affect its color slightly.

Karat Purity

Karat is a unit of purity for gold alloys, rather than a mass or volume measure. It quantifies the fraction of gold in an alloy out of 24 parts, where 24-karat gold is pure gold. Jewelry makers commonly use lower karat ratings to indicate the presence of other metals alloyed with gold, which could be silver, copper, or others, to enhance the jewelry's strength and alter its color. The purity level affects value, color, weight, and the potential for allergic reactions.

For instance, an 18-karat gold piece means that out of 24 parts, 18 parts are pure gold, and the remaining 6 parts are made up of other metals. Knowing how to translate percentage by mass into the karat system provides a universal standard to assess and communicate the quality of gold items.

Mass-Volume Relationship

The mass-volume relationship is fundamental to density calculations, as density is defined as the mass of an object divided by its volume. This relationship plays a critical role in everyday applications like in our exercise, where we needed to understand the composition of a gold piece of jewelry. The formula for density is given by \[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\]By rearranging this formula, you can solve for either mass or volume if the other quantity and density are known. In practical scenarios, a high density indicates that a given volume of the material contains a relatively larger mass compared to another material with a lower density. This concept is not only invaluable in jewelry making but also touches upon various industries such as metallurgy, gemology, and materials science.