Problem 104 - Chemistry: The Central Science in SI Units, Global Edition

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

To find the value of the silver in the American Silver Eagle coin, we first calculate its volume using the formula \[V = \pi r^2 h \approx \pi (20.5 \mathrm{~mm})^2 (2.5 \mathrm{~mm})\]. Next, we convert the volume from cubic millimeters to cubic centimeters: \[V_\mathrm{cm^3} = V_\mathrm{mm^3} \times \frac{1 \mathrm{~cm}^3}{1000 \mathrm{~mm}^3}\]. Then, we calculate the mass of the coin using its volume and silver's density: \[M = V_\mathrm{cm^3} \times 10.5 \mathrm{~g/cm}^3\]. Finally, we find the value of the silver in the coin by multiplying the mass by the price per gram: \[Value = M \times \$ 0.51\].

Step 1: Calculate the volume of the coin

The American Silver Eagle coin is cylindrical in shape. The formula to calculate the volume of a cylinder is: \[V = \pi r^2 h\] where: - \(V\) is the volume of the cylinder, - \(r\) is the radius (which is half of the diameter), - \(h\) is the height (or thickness in this case), - and \(\pi\) (pi) is a mathematical constant approximately equal to \(3.14159\). Given the diameter of the coin is \(41 \mathrm{~mm}\), the radius can be determined as: \[r = \frac{41 \mathrm{~mm}}{2} = 20.5 \mathrm{~mm}\] Moreover, the thickness of the coin is \(2.5 \mathrm{~mm}\). On substituting the values in the formula, we will get the volume of the coin in cubic millimeters: \[V = \pi (20.5 \mathrm{~mm})^2 (2.5 \mathrm{~mm})\]

Step 2: Convert the volume to cubic centimeters

Since the given density and price are in terms of cubic centimeters and grams, respectively, we should convert the volume from cubic millimeters to cubic centimeters. To do this conversion, we use the following formula: \[1 \mathrm{~cm}^3 = 1000 \mathrm{~mm}^3\] So, to convert the volume from cubic millimeters to cubic centimeters: \[V_\mathrm{cm^3} = V_\mathrm{mm^3} \times \frac{1 \mathrm{~cm}^3}{1000 \mathrm{~mm}^3}\]

Step 3: Calculate the mass of the coin

Now that we have the volume in cubic centimeters, we can use the density of silver \(10.5 \mathrm{~g/cm}^3\) to obtain the mass of the coin. The following formula can be used: \[M = V_\mathrm{cm^3} \times D\] where: - \(M\) is the mass of the coin, - \(V_\mathrm{cm^3}\) is the volume of the coin in cubic centimeters, - and \(D\) is the density of silver (\(10.5 \mathrm{~g/cm}^3\)).

Step 4: Calculate the value of the silver in the coin

Finally, we can calculate the value of the silver in the coin using the mass obtained in step 3 and the given price of silver (\(\$ 0.51\) per gram). Using the following formula, we can find the value of the silver: \[Value = M \times P\] where: - \(Value\) is the value of the silver in the coin, - \(M\) is the mass of the coin, - and \(P\) is the price per gram of silver (\(\$ 0.51\)).

Key Concepts

Cylindrical Volume Formula

To determine the volume of the American Silver Eagle coin, we need to treat it as a cylinder. The cylindrical volume formula is essential here:

• \(V = \pi r^2 h\)

Let's decode this a bit:

• \(V\) represents the volume of the cylinder.
• \(r\) is the radius, which is exactly half of the diameter. For our coin, the diameter is given as 41 mm, so \(r = 20.5\) mm.
• \(h\) is the height — or thickness — of the coin, noted as 2.5 mm in this case.
• \(\pi\) (pi) is a well-known constant valued approximately at 3.14159.

By plugging these numbers into the formula, we can calculate the volume of the coin in cubic millimeters, a key first step before moving on to further calculations.

Unit Conversion

Converting units is crucial when dealing with measurements in different systems. For the American Silver Eagle coin, we're given dimensions in millimeters, but we need the volume in cubic centimeters to proceed with our other calculations.

• It's important to know that \(1 \text{ cm}^3 = 1000 \text{ mm}^3\).

This means that when we compute the volume in cubic millimeters, we need to divide by 1000 to get our desired cubic centimeters measure.

• This conversion allows us to align our volume measurement with other data provided, such as the density of silver, which is given in grams per cubic centimeter.

Always remember, being precise with unit conversions makes our final calculations accurate.

Silver Density

Understanding the concept of density helps us find the mass of our coin once we have its volume. Silver, like many materials, has a specific density, which in this case is provided as \(10.5 \text{ g/cm}^3\).

• Density is essentially a measure of how much mass fits into a particular volume — here, representing how heavy silver is per unit of space.

With the volume of our coin (in cubic centimeters) determined and the known density of silver, we can calculate its mass easily:

• The formula is simple: \(M = V_{\text{cm}^3} \times D\) where \(M\) is the mass.
• This calculation provides us with an understanding of how much silver is present in the coin, expressed in grams.

Grasping this relationship is pivotal in finding the value of the silver in our coin.

Silver Pricing

The final step is to determine the monetary value of the silver contained in our coin. Given that the price of silver is \$0.51 per gram, we utilize the mass previously calculated.

• The value of the silver can be found using: \(\text{Value} = M \times P\)

Here, \(M\) is the mass of the silver in grams, and \(P\) is the price per gram.

• This multiplication yields the total cost of the silver based on the current market price.
• It’s a straightforward way to connect the physical properties of the coin to real-world financial terms.

By following these steps, we can transform a simple piece of metal into an economic value, illustrating the practical application of basic math and science principles.