Problem 64 - Chemistry The Central Science In Si Units

Question

A copper refinery produces a copper ingot weighing \(70 \mathrm{~kg}\). If the copper is drawn into wire whose diameter is \(7.50 \mathrm{~mm}\), how many meters 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

The wire can be extended to approximately 177.15 meters.

Step 1: Convert Ingot Mass to Volume

First, convert the mass of the copper ingot from kilograms to grams, since the density is given in grams per cubic centimeter. The mass is \(70 \, \text{kg} = 70000 \, \text{g}\). Next, use the density to find the volume using the formula: \(V = \frac{\text{mass}}{\text{density}}\). Substitute:\[ V = \frac{70000 \, \text{g}}{8.94 \, \text{g/cm}^3} \approx 7827.4 \, \text{cm}^3 \].

Step 2: Convert Wire Diameter to Radius in Centimeters

Convert the diameter of the wire from millimeters to centimeters and then find the radius. Since \(7.50 \, \text{mm} = 0.75 \, \text{cm}\), the radius \(r\) is \(\frac{0.75 \, \text{cm}}{2} = 0.375 \, \text{cm}\).

Step 3: Calculate Wire Volume as a Cylinder Using Formula

We know that the volume of a cylinder is given by the formula \(V = \pi r^2 h\), where \(r\) is the radius and \(h\) is the height (or length of the wire). Replace \(V\) with \(7827.4 \, \text{cm}^3\) from Step 1 and \(r\) with \(0.375 \, \text{cm}\):\[ 7827.4 = \pi (0.375)^2 h \].

Step 4: Solve for Wire Length

Rearrange the cylinder volume equation to solve for \(h\):\[ h = \frac{7827.4}{\pi (0.375)^2} \]. Simplify the calculations:\[ h \approx \frac{7827.4}{\pi \times 0.140625} \approx \frac{7827.4}{0.441787} \approx 17715 \]. Thus, the length of the wire is approximately \(17715 \, \text{cm}\).

Step 5: Convert Wire Length to Meters

Finally, convert the length of the wire from centimeters to meters. Since \(1 \, \text{m} = 100 \, \text{cm}\), the wire length in meters is \(\frac{17715}{100} \approx 177.15 \, \text{m}\).

Key Concepts

Volume Conversion

Volume conversion is an essential skill in science and engineering, as it allows us to change measurements from one unit to another. In the copper wire calculation, we start by converting the mass of the copper ingot into volume. The mass is given in kilograms, so we first convert it to grams to match the density unit. Since 1 kilogram equals 1000 grams, a simple multiplication helps us do this. We then use the formula:

• \( V = \frac{\text{mass}}{\text{density}} \)

This formula helps us calculate the volume in cubic centimeters (\( cm^3 \)) given the density in grams per cubic centimeter (\( g/cm^3 \)). Volume conversion helps us find the physical quantity needed to proceed with further calculations, especially when working with different units of measurement.
It's crucial for students to be comfortable with such conversions, as they form the foundation for problem-solving in physics and engineering.

Cylinder Volume Formula

The cylinder volume formula is instrumental when you need to calculate the volume or any dimension of a cylinder, like the height or radius. This formula is written as:

• \( V = \pi r^2 h \)

Here, \( V \) is the volume, \( r \) is the radius of the base, and \( h \) is the height (or length) of the cylinder. In our context, the copper wire is shaped as a long cylinder, and the formula allows us to express the volume in terms of the wire's length and radius.
By rearranging this formula, we can solve for any one of the quantities if the other two are known, which makes it incredibly versatile. For example, knowing the volume and radius can help us calculate the height, which corresponds to the length of the wire in this problem. Understanding how to apply this formula is key in tasks involving cylindrical objects, such as pipes or wires.

Density of Copper

The density of copper, which is given as \(8.94 \, \text{g/cm}^3\), helps us relate the mass of copper to its volume. Density is defined as the mass per unit volume and is crucial in determining the amount of material required for a given volume, or conversely, the volume that a given mass will occupy. In the case of the copper wire calculation, we use density to determine how much volume the copper ingot occupies. Understanding density is important for

• calculating mass, volume, or density itself when the other two are known,
• solving problems involving buoyancy or material selection,
• and considering real-world applications like metalworking and thus selecting materials based on specific weight criteria.

Knowing the density of materials like copper aids in making informed decisions in manufacturing and engineering contexts.

Unit Conversion in Measurements

Unit conversion in measurements is a fundamental aspect of solving mathematical and scientific problems. In the context of the copper wire exercise, we start by converting the diameter from millimeters to centimeters. Since 1 centimeter equals 10 millimeters, dividing the diameter in millimeters by 10 gives us its equivalent in centimeters. Further, we find the radius by dividing the diameter by two. Another conversion happens at the end when we find the wire's length in centimeters and need it in meters. Since 1 meter equals 100 centimeters, dividing the length in centimeters by 100 provides the measurement in meters. These unit conversions are crucial for consistency in calculations and avoiding errors. It's important to:

• use the correct conversion factors,
• ensure that all measurements are in the same unit system,
• and apply units consistently to make the calculations meaningful and correct.

By mastering unit conversion, students ensure precision and accuracy in scientific calculations.