Understanding the Problem
To find the length of the side of a cube formed by one mole of copper, we need to bridge the gap between microscopic properties (moles) and macroscopic properties (volume and dimensions).
We are given:
- Density of Copper ($\\rho$): $8.92 \text{ g/mL}$ (or $8.92 \text{ g/cm}^3$)
- Amount of substance: $1 \text{ mole}$
- Atomic weight of Copper ($M$): $63.5 \text{ g/mol}$
Step-by-Step Calculation
1. Calculate the Mass of 1 Mole of Copper
Using the atomic weight, we know that 1 mole of copper has a mass ($m$) of: $$m = 1 \text{ mol} \times 63.5 \text{ g/mol} = 63.5 \text{ g}$$
2. Calculate the Volume of the Copper
We use the density formula: $\rho = \frac{m}{V}$. Rearranging for volume ($V$): $$V = \frac{m}{\rho}$$ $$V = \frac{63.5 \text{ g}}{8.92 \text{ g/cm}^3}$$ $$V \approx 7.1188 \text{ cm}^3$$
3. Calculate the Side Length of the Cube
For a cube with side length $s$, the volume is $V = s^3$. To find $s$, we take the cube root of the volume: $$s = \sqrt[3]{V}$$ $$s = \sqrt[3]{7.1188 \text{ cm}^3}$$ $$s \approx 1.924 \text{ cm}$$
Summary of Results
The length of each side of the cube formed by one mole of copper is approximately 1.92 cm.
Intuition
It is fascinating to realize that one mole of copper—which contains roughly $6.022 \times 10^{23}$ individual atoms—is only about 2 centimeters wide. This highlights the incredible density and small size of atoms, showing how a vast number of particles can fit into a small, tangible object.