Problem 58 - Chemistry: The Central Science

Question

A copper refinery produces a copper ingot weighing \(150 \mathrm{lb}\). If the copper is drawn into wire whose diameter is \(7.50 \mathrm{~mm}\), how many feet of copper can be obtained from the ingot? The density of copper is \(8.94 \mathrm{~g} / \mathrm{cm}^{3}\). (Assume that the wire is a cylinder whose volume \(V=\pi r^{2} h\), where \(r\) is its radius and \(h\) is its height or length.)

Step-by-Step Solution

Answer

Approximately \(709.04 \mathrm{~ft}\) of copper wire can be obtained from the ingot.

Step 1: Convert the weight of ingot to grams

Given that the weight of the copper ingot is \(150 \mathrm{lb}\), we need to convert this weight into grams. Use the conversion factor \(453.592 \mathrm{~g} / \mathrm{lb}\): Weight (in grams) \(= 150 \mathrm{lb} × 453.592 \mathrm{~g} / \mathrm{lb} = 68038.8 \mathrm{~g}\).

Step 2: Calculate the volume of the ingot

To find the volume of the ingot, we will use the density formula and the given density of copper: Density \(= \frac{\text{Mass}}{\text{Volume}}\) Rearranging for the Volume and applying the given density, we get: Volume of ingot \(= \frac{\text{Mass}}{\text{Density}} = \frac{68038.8 \mathrm{~g}}{8.94 \mathrm{~g} / \mathrm{cm}^{3}} = 7609.81 \mathrm{~cm}^{3}\)

Step 3: Calculate the radius of the wire

The diameter of the wire is given as \(7.50 \mathrm{~mm}\). To get the radius, divide the diameter by 2, and then convert from millimeters to centimeters: Radius (in cm) \(= \frac{7.50 \mathrm{~mm}}{2} × \frac{1 \mathrm{~cm}}{10 \mathrm{~mm}} = 0.375 \mathrm{~cm}\)

Step 4: Calculate the length of the wire using the volume of the ingot

We know that the volume of a cylinder is \(V = \pi r^{2}h\). We can rearrange this formula to solve for the height (length) \(h\), given the volume and radius: Length \(h = \frac{V}{\pi r^{2}}\) Now substitute the known values for the volume of the ingot and the radius of the wire: Length \(h = \frac{7609.81 \mathrm{~cm}^3}{\pi (0.375 \mathrm{~cm})^{2}} \approx 21619.8 \mathrm{~cm}\)

Step 5: Convert the length of the wire to feet

To convert the length of the wire from centimeters to feet, use the conversion factor \(1 \mathrm{~ft} = 30.48 \mathrm{~cm}\): Length (in feet) \(= \frac{21619.8 \mathrm{~cm}}{30.48 \mathrm{~cm} / \mathrm{ft}} \approx 709.04 \mathrm{~ft}\)

Step 6: Final answer

Therefore, approximately \(709.04 \mathrm{~ft}\) of copper wire can be obtained from the ingot.

Key Concepts

Cylinder Volume

When working with objects such as copper wires, it's helpful to imagine them as cylinders. A cylinder's volume is calculated using the formula \( V = \pi r^{2} h \), where \( r \) is the radius and \( h \) is the height or length of the cylinder. This formula helps in determining how much space a material occupies.

In the case of copper wire, once we know the volume of the copper available, we can find out how long the wire can be by calculating its cylindrical volume. Here, the volume obtained from the copper ingot is used to find how much of this volume can be shaped into a wire, given a specific radius. This concept brings together geometry and practical applications, such as shaping metal ingots into usable forms like wires.

Unit Conversion

Unit conversion is essential when working with measurements. It helps ensure that all quantities are in the same units for calculation and comparison.

In this exercise, we encounter several conversions:

• Weight conversion from pounds to grams.
• Length conversion from millimeters to centimeters, and finally to feet.

For example, since the initial mass of the copper ingot is given in pounds, it's crucial to convert it to grams. This is because the density of copper is given in grams per cubic centimeter. Converting correctly ensures that calculations like determining the volume, which relies on density, are accurate.

Density

Density is a property that describes how much mass is contained within a specific volume. The formula for density is \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). Knowing the density of a material, like copper, allows us to connect mass with volume.

In our problem, the density of copper is given as \(8.94 \text{ g/cm}^3\). This information is crucial because it allows us to determine the volume of the copper ingot from its mass. The calculated volume can then be used to find out how much wire can be made from this volume. Understanding density helps us bridge the gap between mass and three-dimensional space.

Copper Refinery

A copper refinery converts raw copper into products like ingots, wires, and sheets that can be used for various applications.

By refining copper, refineries ensure that the copper is pure and in a form that can be molded into objects, such as thin wires in this exercise. The process involves several steps like melting, purification, and casting the metal into a desired shape.

• Copper is often processed into ingots for ease of transportation and subsequent use.
• The refining process enhances attributes such as conductivity, making copper ideal for electrical wiring.

This exercise illustrates the practical application of copper refinement: turning an ingot into a useful length of copper wire.