Question

A jewelry maker sells lots of sterling silver hearts with a mass of \(35.0 \mathrm{~g}\). Sterling silver is composed of \(92.5 \%\) silver and \(7.50 \%\) copper. (a) Write out the dimensional analysis factors that can be derived from the two percentages given above. Be sure to include units. (b) If the jeweler buys \(50.0\) lb of silver to make the hearts, how many sterling silver hearts can the jeweler make?

Short answer

The jeweler can make 700 sterling silver hearts.

Step-by-step solution

- Determine the mass of silver in one sterling silver heart

Calculate the mass of silver in one sterling silver heart using the given mass and the percentage of silver. Percentage of silver: 92.5%Mass of silver in one heart: \[ 35.0 \text{ g} \times \frac{92.5}{100} = 32.375 \text{ g} \]

- Convert pounds of silver to grams

Convert the given mass of silver in pounds to grams. 1 pound (lb) = 453.592 grams (g) Mass of silver in grams:\[ 50.0 \text{ lb} \times 453.592 \text{ g/lb} = 22679.6 \text{ g} \]

- Calculate the number of sterling silver hearts that can be made

Divide the total mass of silver available by the mass of silver in one heart to find out how many hearts can be made.Number of hearts:\[ \frac{22679.6 \text{ g}}{32.375 \text{ g/heart}} \ \ \approx 700.95 \text{ hearts} \]Since we cannot have a fraction of a heart, we round down to the nearest whole number.Therefore, the jeweler can make 700 hearts.

Key concepts

Sterling Silver Composition

Sterling silver is a popular alloy used in jewelry making. It consists of two main elements: silver (Ag) and copper (Cu). Specifically, sterling silver is composed of 92.5% silver and 7.5% copper. This precise composition is essential for maintaining the alloy’s desirable properties, such as its appearance, strength, and resistance to tarnishing.
Understanding the percentages in alloys like sterling silver helps jewelers create consistent and high-quality products. Knowing the composition also aids in calculations when determining the amounts of each element in a piece of jewelry.
For example, in a sterling silver heart weighing 35 grams, you need to calculate how much of that weight is silver and how much is copper. This involves multiplying the total mass by the percentage of each metal:
\[ 35 \text{ g} \times \frac{92.5}{100} = 32.375 \text{ g of silver} \] \[ 35 \text{ g} \times \frac{7.5}{100} = 2.625 \text{ g of copper} \]
This provides an accurate breakdown of the material composition.

Unit Conversion

Unit conversion is a crucial skill in chemistry and everyday life, allowing you to switch between different units of measurement. In the context of our jewelry exercise, we need to convert pounds of silver into grams to align with the other given measurements. One pound (lb) is equivalent to 453.592 grams (g).
Here's how you convert 50 pounds of silver into grams:
\[ 50 \text{ lb} \times 453.592 \text{ g/lb} = 22679.6 \text{ g} \]
By converting these units, you ensure accuracy in further calculations. It's always essential to use the correct conversion factors and to keep your units consistent to avoid errors in your results.

Percentage Calculations

Percentage calculations help determine the proportion of each component in a mixture or alloy. They are especially vital in chemistry for understanding the composition of compounds and mixtures. In our exercise, we needed to find out how much silver from a sterling silver heart contributes to the total weight.
To calculate percentages, you use the formula:
\[ \text{Percentage of a component} = \frac{\text{Part}}{\text{Total}} \times 100 \]
For the sterling silver heart weighing 35 grams, you can calculate the mass of silver as follows:
\[ 35 \text{ g} \times \frac{92.5}{100} = 32.375 \text{ g} \]
This shows that 32.375 grams of silver are present in one heart. Then, you can use this information to determine how many hearts can be made from 50 pounds of silver by dividing the total available silver by the silver content in one heart.
\[ \frac{22679.6 \text{ g}}{32.375 \text{ g/heart}} \ \text{or} \ 700.95 \text{ hearts (rounding down to nearest whole number)} \]
Therefore, the jeweler can craft 700 sterling silver hearts. Accurate percentage calculations ensure precise and predictable results in such processes.