Problem 104 - Chemistry The Central Science In Si Units

Question

The U.S. Mint produces a dollar coin called the American Silver Eagle that is made of nearly pure silver. This coin has a diameter of \(41 \mathrm{~mm}\) and a thickness of \(2.5 \mathrm{~mm} .\) The density and approximate market price of silver are \(10.5 \mathrm{~g} / \mathrm{cm}^{3}\) and \(\$ 0.51\) per gram, respectively. Calculate the value of the silver in the coin, assuming its thickness is uniform.

Step-by-Step Solution

Answer

The value of the silver in the coin is approximately \( \$17.68 \).

Step 1: Calculate the Volume of the Coin

The coin is a cylinder, thus its volume is given by the formula \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height (thickness) of the cylinder. The diameter is \( 41 \mathrm{~mm} \), thus the radius \( r = 20.5 \mathrm{~mm} \). Convert \( r \) and \( h = 2.5 \mathrm{~mm} \) to centimeters (since density is given in \( \mathrm{g/cm}^3 \)): \( r = 2.05 \mathrm{~cm} \) and \( h = 0.25 \mathrm{~cm} \). Then compute the volume: \[ V = \pi (2.05)^2 \times 0.25 \approx 3.302 \mathrm{~cm}^3 \].

Step 2: Calculate the Mass of the Coin

Using the density formula \( \text{density} = \frac{\text{mass}}{\text{volume}} \), rearrange it to find mass as \( \text{mass} = \text{density} \times \text{volume} \). With a density of silver \( 10.5 \mathrm{~g/cm}^3 \), compute the mass: \[ \text{mass} = 10.5 \times 3.302 \approx 34.671 \mathrm{~g} \].

Step 3: Calculate the Value of the Silver in the Coin

The value of the silver in the coin is found by multiplying the price per gram by the mass of the silver. With the price being \( \\(0.51 \) per gram: \[ \text{value} = 0.51 \times 34.671 \approx \\)17.68 \].

Key Concepts

Volume of a Cylinder

To find the volume of a cylinder, you can use the formula for volume, which is \( V = \pi r^2 h \). Here, \( r \) stands for the radius of the base, and \( h \) means the height or thickness of the cylinder.
For a cylinder, the base is typically a circle. Hence, you need the radius, which is half of the diameter. In this example, given a diameter of 41 mm, the radius \( r = 20.5 \) mm or 2.05 cm when converted to centimeters for accuracy in density calculations. The height or thickness \( h \) is 2.5 mm, which converts to 0.25 cm. Always keep units consistent; centimeters in this case.
Once you have converted the measurements, insert them into the formula:

• First, square the radius: \( r^2 = (2.05)^2 \).
• Multiply the result by \( \pi \) (approximately 3.14159): \( \pi r^2 \).
• Finally, multiply by the height \( h = 0.25 \) cm to find the volume \( V \approx 3.302 \text{ cm}^3 \).

Understanding this process helps ensure accurate input into subsequent calculations.

Mass from Density

Density tells us how much mass is packed into a certain volume. It's important in identifying how much a specific amount of substance weighs. The formula for density is \( \text{density} = \frac{\text{mass}}{\text{volume}} \), which can be rearranged to find mass: \( \text{mass} = \text{density} \times \text{volume} \).
Given the density of silver is 10.5 g/cm³, and knowing the volume from the previous step is \( 3.302 \text{ cm}^3 \), you can calculate the mass of the coin:

• Multiply the density by the volume: \( 10.5 \times 3.302 \).
• This results in a mass of approximately 34.671 g.

The mass calculation is crucial because it directly affects the amount of silver present in the coin, which you need in order to evaluate the coin's market value.

Market Value Calculation

After calculating the mass of silver in the coin, you need to find out its market value. This is done using the formula \( \text{value} = \text{price per gram} \times \text{mass} \).
The price of silver is given as \( 0.51 \) per gram. This means that for each gram of silver, it costs 51 cents. Multiply this price by the mass previously calculated: 34.671 grams.

• Perform the multiplication: \( 0.51 \times 34.671 \).
• The calculation results in an approximate value of the coin being \( 17.68 \).

This demonstrates how to compute the worth of this silver coin based on its intrinsic silver content, aligning physical measurements with economic value.