Question

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

Short answer

The edge length of the box would be approximately \(8.47 \times 10^{7} cm.\)

Step-by-step solution

Calculate the Volume of 1 mole of Sugar Cubes

First, it is needed to calculate the volume that 1 mole of sugar cubes would occupy. This can be done by multiplying the volume of a single sugar cube by Avogadro's number, since 1 mole of sugar cubes equals Avogadro's number of sugar cubes. Now, the volume of a single sugar cube can be calculated using the formula for the volume of a cube, which is edge length cubed. Since it is given that the edge length of a typical sugar cube is 1 cm, the volume of a single sugar cube can be calculated as \(1 cm \times 1 cm \times 1 cm = 1 cm^{3}\). So, the volume occupied by 1 mole of sugar cubes is \((1 cm^{3}) \times (6.022 \times 10^{23}) = 6.022 \times 10^{23} cm^{3}\).

Calculate the Edge Length of the Box

Next, it is needed to find out the edge length of the cubical box that can contain 1 mole of sugar cubes. For this, we use the formula for the volume of a cube, which is edge length cubed, and solve for the edge length. This can be done by taking cubic root of the volume of the box. Hence, the edge length of the box can be calculated as \(cube~root~of~(6.022 \times 10^{23} cm^{3})\).

Use a Calculator to Compute the Cubic Root

Finally, use a calculator to compute the cubic root of \(6.022 \times 10^{23}\) to get the edge length of the box. It should be around \(8.47 \times 10^{7} cm\).

Key concepts

Avogadro's Number

Avogadro's Number is a fundamental concept in chemistry that relates to the quantity of entities in a mole. A mole is a standard unit of measurement in the chemical world that represents a specific amount of substance, often used to count atoms, molecules, or other particles. Avogadro's number is approximately \(6.022 \times 10^{23}\). Thus, when you say you have one mole of sugar cubes, it means you have \(6.022 \times 10^{23}\) sugar cubes. This vast number is crucial in calculations where we need to relate macroscopic quantities of material to atomic or molecular scales.
In our exercise, we need to think of sugar cubes much like molecules. Even though we can visualise a sugar cube and grasp its size, imagining \(6.022 \times 10^{23}\) of them highlights how huge this number really is.
Understanding Avogadro's number is key, as it helps us transition between counting individual particles and discussing amounts large enough to see and use in the real world.

Volume Calculation

Calculating volume is a critical step in understanding how much space an object or a collection of objects occupies. In our exercise, we deal with the volume of sugar cubes. A cube's volume can be easily calculated using the formula:
- Volume = edge length \(\times\) edge length \(\times\) edge length, or "edge length cubed".
Given each sugar cube has an edge length of \(1\) cm, its volume is \(1 \text{ cm}^3\). But when considering a mole of sugar cubes, we multiply this volume by Avogadro's number since a mole equals \(6.022 \times 10^{23}\) sugar cubes.
This yields a total volume for one mole of sugar cubes as \(6.022 \times 10^{23} \text{ cm}^3\).
Understanding volume calculation here ensures we grasp the enormity of space occupied by a mole of sugar cubes, transitioning the concept of volume from the level of a single cube to a massive virtually infinite number in a comprehensible way.

Cubic Measurements

Cubic measurements involve understanding the spatial dimensions and volume of 3D objects. The unit cubic centimeter \(\text{cm}^3\) is especially important here. When dealing with cubes, each side of the cube has the same length, which simplifies calculations.
For example, finding the edge length of a larger cube containing \(6.022 \times 10^{23} \text{ cm}^3\) is a reverse of initial calculations. By knowing the volume \( (6.022 \times 10^{23} \text{ cm}^3) \) of the larger cube, we can find the edge length using the cubic root:
- Edge length = cube root of \( 6.022 \times 10^{23} \).
Utilizing a calculator simplifies this, resulting in an edge length of approximately \( 8.47 \times 10^7 \text{ cm} \).
Recognizing the connection between volume and edge length in cubic measurements aids in comprehending how vast a structure would be needed to contain something as large-scale as a mole of small sugar cubes. Understanding cubic measurements is essential for converting between different dimensions of objects in real-world contexts.