Problem 103 - Chemistry: The Central Science in SI Units, Global Edition

Question

A U.S. 1 -cent coin (a penny) has a diameter of \(19 \mathrm{~mm}\) and a thickness of \(1.5 \mathrm{~mm}\). Assume the coin is made of pure copper, whose density and approximate market price are \(8.9 \mathrm{~g} / \mathrm{cm}^{3}\) and \(\$ 2.40\) per pound, respectively. Calculate the value of the copper in the coin, assuming its thickness is uniform.

Step-by-Step Solution

Answer

Based on the given information, we can find the value of the copper in the penny by following these steps: 1. Calculate the volume of the coin: \(V = π(9.5)^2(1.5)\) 2. Convert volume to mass: \(m = 8.9~g/cm^3 × V~cm^3\) 3. Convert mass to weight in pounds: \(mass~(lbs) = \frac{m~(g)}{453.592}\) 4. Calculate the value of the copper: \( Value = \$ 2.40/lb × mass~(lbs)\) By performing these calculations, we can determine the value of the copper in the penny.

Step 1: Calculate the volume of the coin

First, we need to find the volume of the copper used in the penny. Since the penny is assumed to be a flat cylinder in shape, we can use the formula for the volume of a cylinder: \(V = πr^2h\), where V is the volume, r is the radius, and h is the height (thickness) of the coin. Given the diameter of the penny is 19 mm, the radius is \(r = \frac{19}{2}\) mm = 9.5 mm. The thickness (height) of the coin is given as 1.5 mm. Now, calculate the volume: \(V = π(9.5)^2(1.5)\)

Step 2: Convert volume to mass

We have the volume of the copper used in the penny measured in mm^3. Convert the volume from mm^3 to cm^3: \(1~cm^3 = 1000~mm^3\). In order to find the mass of copper used in the penny, we need the given density of copper which is 8.9 g/cm^3. The mass (m) of copper in the coin can be obtained by multiplying the density by volume: m = density × volume.

Step 3: Convert mass to weight in pounds

To find the value of the copper in the penny, we need first to convert the mass of the copper (in grams) to pounds (lbs). Convert the mass of copper from grams to pounds, using the conversion factor: \(1 lb = 453.592 g\).

Step 4: Calculate the value of the copper in the penny

Now that we have the mass of copper in pounds, we can use the given market price of copper, which is $2.40 per pound, to calculate the value of the copper in the penny: Value = Market price × Mass of copper in pounds. Once you find the value of the copper in the penny, you can compare it to the actual value of the coin (1 cent).

Key Concepts

Density

Density is a measure of how much mass is contained in a given volume. Essentially, it tells us how compact or crowded a substance is. The formula for density is given by:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]In this exercise, we're dealing with the density of copper, which is specified as 8.9 g/cm³. This value indicates that every cubic centimeter (cm³) of copper weighs 8.9 grams. Density is a crucial factor when converting volumes to masses, as seen in the penny exercise. Knowing the density allows us to determine the mass of a material from its volume, providing a direct link between these two properties.

Mass Conversion

Mass conversion involves changing the measurement of an object's mass from one unit to another. In many calculations, especially involving currency or comparison with market prices, using a standard unit like pounds is necessary. Here are some key conversions to remember:

• 1 gram (g) = 0.0022 pounds (lbs)
• 1 pound (lb) = 453.592 grams (g)

In our scenario, after calculating the copper mass in grams, we must convert that mass to pounds to engage with the market price. This conversion is essential since the market price for copper is given in dollars per pound, and without it, the monetary value calculation wouldn't align properly.

Volume to Mass Conversion

To convert volume to mass, we apply the concept of density. Once you know the volume of an object and its density, you can easily find its mass. This conversion is key in the copper penny exercise.The steps are:

1. Calculate the volume of the object. For cylinders, use the formula: \[ V = \pi r^2 h \]
2. Convert the volume to the unit associated with the density if necessary (e.g., from mm³ to cm³).
3. Multiply the volume by the material's density to find the mass:

\[ \text{Mass} = \text{Density} \times \text{Volume} \]In this exercise, we first find the volume of the penny considering it as a cylinder. Then by utilizing the density of copper, we successfully convert this volume into the mass of copper.

Market Price

Market price is the current price at which a particular commodity can be bought or sold. It plays a crucial role when determining the value of raw materials like copper in economic terms.For copper, the market price is given as $2.40 per pound. Once the mass of copper is converted to pounds, we can easily compute its value by multiplying the mass with the market price:\[ \text{Value} = \text{Market Price} \times \text{Mass in Pounds} \]This calculation directly translates the physical measurement of materials into a monetary figure, allowing comparisons with costs and values of currency items, such as determining if the raw material value of a penny exceeds its face value of one cent.