Ch.1 - Introduction: Matter, Energy, and Measurement

Brown - Chemistry: The Central Science 14th Edition

Brown14th EditionChemistry: The Central ScienceISBN: 9780134414232Not the one you use?Change textbook

Brown 14th Edition

Ch.1 - Introduction: Matter, Energy, and Measurement

Problem 64

Chapter 1, Problem 64

A copper refinery produces a copper ingot weighing 70 kg. If the copper is drawn into wire whose diameter is 7.50 mm, how many meters of copper can be obtained from the ingot? The density of copper is 8.94 g/cm³. (Assume that the wire is a cylinder whose volume V = πr²h, where r is its radius and h is its height or length.)

Verified step by step guidance

Convert the mass of the copper ingot from kilograms to grams by multiplying by 1000, since 1 kg = 1000 g.

Use the density formula $ \text{Density} = \frac{\text{Mass}}{\text{Volume}} $ to find the volume of the copper ingot in cubic centimeters. Rearrange the formula to $ \text{Volume} = \frac{\text{Mass}}{\text{Density}} $.

Convert the diameter of the wire from millimeters to centimeters by dividing by 10, since 1 cm = 10 mm. Then, calculate the radius by dividing the diameter by 2.

Use the formula for the volume of a cylinder $ V = \pi r^2 h $ to express the volume of the wire in terms of its length $ h $. Rearrange the formula to solve for $ h $: $ h = \frac{V}{\pi r^2} $.

Substitute the volume of the copper ingot and the radius of the wire into the rearranged formula to calculate the length $ h $ of the wire in centimeters, and then convert the length from centimeters to meters by dividing by 100.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Density

Density is defined as mass per unit volume and is a crucial property of materials. In this problem, the density of copper (8.94 g/cm³) allows us to convert the mass of the copper ingot (70 kg) into volume. Understanding how to manipulate the density formula (Density = Mass/Volume) is essential for determining how much space the copper occupies.

Volume of a Cylinder

The volume of a cylinder can be calculated using the formula V = πr²h, where r is the radius and h is the height (or length) of the cylinder. In this scenario, the copper wire is modeled as a cylinder, and knowing this formula is necessary to relate the volume of the copper ingot to the length of wire that can be produced.

Unit Conversion

Unit conversion is the process of converting a quantity from one unit to another, which is often necessary in chemistry problems. In this case, converting the mass of copper from kilograms to grams (1 kg = 1000 g) is essential for using the density value correctly. Mastery of unit conversions ensures accurate calculations and results.

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Textbook Question

Gold can be hammered into extremely thin sheets called gold leaf. An architect wants to cover a 30 m x 25 m ceiling with gold leaf that is twelve-millionths of a centimeter thick. The density of gold is 19.32 g/cm³, and gold costs \$1654 per troy ounce (1 troy ounce = 31.1034768 g). How much will it cost the architect to buy the necessary gold?

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Textbook Question

Classify each of the following as a pure substance, a solution, or a heterogeneous mixture: (a) a leaf (b) a 999 gold bar

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Textbook Question

Classify each of the following as a pure substance, a solution, or a heterogeneous mixture: (c) stainless steel.

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Textbook Question

(a) Which is more likely to eventually be shown to be incorrect: an hypothesis or a theory?

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Textbook Question

The density of air at ordinary atmospheric pressure and25 C is 1.19 g/L. What is the mass, in kilograms, of the air ina room that measures 4.5 m * 5.0 m * 2.5 m?

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