Problem 104 - Chemistry The Central Science

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

\( radius = \frac{41 \text{ mm}}{2} = 20.5 \text{ mm} \) #tag_title#Step 2: Calculate the volume of the coin#tag_content#Using the radius and thickness of the coin, we can find the volume using the formula for the volume of a cylinder: \( V = \pi r^2h \) where V is the volume, r is the radius, and h is the thickness. The height in this case is the thickness of the coin, which is 2.5 mm. \( V = \pi (20.5 \text{ mm})^2 (2.5 \text{ mm}) = 3330.15 \text{ mm}^3 \) Now convert the volume to cm³: \( 1 \text{ cm}^3 = 1000 \text{ mm}^3 \) \( V = 3330.15 \text{ mm}^3 \times \frac{1 \text{ cm}^3}{1000 \text{ mm}^3} = 3.33015 \text{ cm}^3 \) #tag_title#Step 3: Calculate the mass of the coin#tag_content#Using the density and the volume of the coin, we can calculate its mass: \( mass = density \times volume \) The density of silver is given as 10.5 g/cm³: \( mass = (10.5 \, \text{g}/\text{cm}^3) \times 3.33015 \, \text{cm}^3 = 34.966575 \, \text{g} \) #tag_title#Step 4: Calculate the value of the silver in the coin#tag_content#Finally, we multiply the mass of the coin by the market price of silver per gram to find the value of the silver in the coin: \( Value = mass \times price \) The market price of silver is given as $0.51 per gram: \( Value = 34.966575 \, \text{g} \times \$0.51/\text{g} = \$17.83317375 \)

Step 1: Calculate the radius of the coin

Given the diameter of the coin, 41 mm, we can calculate its radius by dividing the diameter by 2: radius = diameter/2

Key Concepts

American Silver Eagle

The American Silver Eagle is a standout coin produced by the United States Mint. It is highly regarded for its close to pure silver content, making it both a valuable financial asset and a beloved collector's item. First introduced in 1986, this coin is part of the "Silver Bullion Coin" series. Measuring 41 mm in diameter and 2.5 mm in thickness, the American Silver Eagle boasts a significant presence. Its detailed design features Lady Liberty on one side and a heraldic eagle on the other. For calculations involving volume and value, these dimensions are essential since the coin's uniformity is considered in computations. Additionally, knowing the coin's pure silver content is crucial for understanding its intrinsic value tied to silver's market price.

Silver Coin

Silver coins like the American Silver Eagle aren't just currency; they are also a way to invest in silver. The purity of the silver content in such coins is a key factor in determining their worth. Silver coins serve multiple purposes:

• Investment in precious metals: Provides a hedge against inflation.
• Collectible appeal: With historical or aesthetic value.
• Liquidity: Easily traded or sold in the marketplace.

Given that silver has a density of 10.5 g/cm³, calculations related to the coin's volume help determine its mass. This requires understanding the basic geometry of the coin to calculate its volume as a cylinder. The formula for volume is crucial when figuring out how much pure silver is in the coin, providing insight into its market value.

Market Price of Silver

When analyzing the American Silver Eagle, understanding the market price of silver is fundamental. This price reflects the current value of silver per gram and can fluctuate based on supply, demand, and market conditions. As of the given context, the market price was set at $0.51 per gram. Knowing this price allows investors and collectors to gauge the real-time value of the silver content within the coin. If, for instance, the price of silver rises, the value of the silver in the coin increases as well. Thus, to find the coin's value from just its silver, one would multiply its silver content in grams by this market price. This relationship demonstrates how the market dynamics of silver impact the financial value of silver coins like the American Silver Eagle.