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Chapter 1: Problem 27

A typical sugar cube has an edge length of \(1 \mathrm{~cm}\). If you had a cubical box that contained a mole of sugar cubes, what would its edge length be? (One mole \(=6.02 \times 10^{23}\) units.)

Short Answer

Expert verified
The edge length of the cubical box is approximately \(8.43 \times 10^7\) cm.

Step by step solution

01

Calculate the volume of one sugar cube

The volume of one sugar cube can be found using the formula for the volume of a cube: \[ V = a^3 \]where \( a \) is the edge length of the cube. For one sugar cube, \( a = 1 \, \text{cm} \).So,\[ V = 1^3 = 1 \, \text{cm}^3 \]
02

Calculate the total volume for a mole of sugar cubes

Since one mole of sugar cubes contains \( 6.02 \times 10^{23} \) cubes, the total volume is given by multiplying the volume of one sugar cube by the number of cubes:\[ V_{\text{total}} = 6.02 \times 10^{23} \, \text{cm}^3 \]
03

Determine the edge length of the cubical box

To find the edge length of the cubical box that contains \( 6.02 \times 10^{23} \) sugar cubes, we use the volume of a cube formula in reverse. We need to find \( a \) such that:\[ a^3 = V_{\text{total}} \]Thus,\[ a = \sqrt[3]{V_{\text{total}}} = \sqrt[3]{6.02 \times 10^{23}} \]Using a calculator,\[ a \approx 8.43 \times 10^7 \, \text{cm} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mole Concept
The mole concept is a fundamental principle in chemistry. It's a way to quantify large numbers of tiny entities like atoms, molecules, or sugar cubes. One mole represents exactly 6.02 x 10^23 particles, an incredibly large number known as Avogadro's number.
Think of it like this: if you had a mole of anything, you'd have 6.02 x 10^23 of that thing. For example, in our exercise, we are dealing with a mole of sugar cubes. This helps chemists to describe and use quantities of substances in practical and manageable terms. Imagine counting each sugar cube individually鈥攊t would be impossible! The mole concept is like a helpful shortcut to understand the vast quantities in chemical calculations.
Cubic Volume
Determining the volume of a cube is quite straightforward. A cube is a three-dimensional shape with all equal sides. Thus, the volume of a cube can be calculated using the formula: \( V = a^3 \) where \( a \) is the edge length of the cube.
In our step-by-step solution, we started by calculating the volume of one sugar cube with an edge length of 1 cm. Plugging 1 cm into the formula, we get: \[ V = 1^3 = 1 \, \text{cm}^3 \]
This basic understanding is crucial because it allows you to move on to larger calculations, like finding the total volume for a mole of sugar cubes, which is simply the volume of one sugar cube multiplied by the number of sugar cubes (6.02 x 10^23).
Cube Root Calculation
Cube root calculations are useful when you need to determine the edge length of a cube from its volume. In our exercise, we were tasked with finding the edge length of a cubical box containing a mole of sugar cubes. Once we calculated the total volume of these sugar cubes (\[V_{\text{total}} = 6.02 \times 10^{23} \text{cm}^3\]), we needed the cube root to find the edge length \( a \).
This involves reversing the volume formula: \[ a = \sqrt[3]{V_{\text{total}}} \approx \sqrt[3]{6.02 \times 10^{23}} \approx 8.43 \times 10^7 \text{cm}\]
Cube roots may seem tricky, but tools like calculators can make them much easier to handle. In summary, cube root calculations are indispensable for translating volume back into a more tangible measurement, like edge length.

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Most popular questions from this chapter

Historically the English had a doubling system when measuring volumes; 2 mouthfuls equal 1 jigger, 2 jiggers equal 1 jack (also called a jackpot); 2 jacks equal 1 jill; 2 jills \(=\) 1 cup; 2 cups \(=1\) pint; 2 pints \(=1\) quart; 2 quarts \(=1\) pottle; 2 pottles \(=1\) gallon; 2 gallons \(=1\) pail. (The nursery rhyme "Jack and Jill" refers to these units and was a protest against King Charles I of England for his taxes on the jacks of liquor sold in the tavern. (See A. Kline, The World of Measurement, New York: Simon and Schuster, 1975, pp. \(32-39 .\) American and British cooks today use teaspoons, tablespoons, and cups; 3 teaspoons \(=1\) tablespoon; 4 tablespoons \(=1 / 4\) cup. Assume that you find an old English recipe requiring 3 jiggers of milk. How many cups does this represent? How many tablespoons? You can assume that the cups in the two systems represent the same volume.

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