How many atoms are in a 2.0 cm×2.0 cm×2.0 cm cube of aluminum?

Verified step by step guidance

1. Step 1: Calculate the volume of the aluminum cube. Use the formula for the volume of a cube, which is \( V = a^3 \), where \( a \) is the length of one side of the cube. Here, \( a = 2.0 \, \text{cm} \).
2. Step 2: Convert the volume from cubic centimeters (\( \text{cm}^3 \)) to cubic meters (\( \text{m}^3 \)) because the density of aluminum is typically given in \( \text{kg/m}^3 \). Use the conversion factor \( 1 \, \text{cm}^3 = 10^{-6} \, \text{m}^3 \).
3. Step 3: Use the density of aluminum, \( \rho = 2700 \, \text{kg/m}^3 \), to calculate the mass of the aluminum cube. The formula is \( m = \rho \cdot V \), where \( \rho \) is the density and \( V \) is the volume in \( \text{m}^3 \).
4. Step 4: Determine the number of moles of aluminum in the cube. Use the molar mass of aluminum, \( M = 26.98 \, \text{g/mol} \), and the formula \( n = \frac{m}{M} \), where \( m \) is the mass in grams (convert \( \text{kg} \) to \( \text{g} \) by multiplying by 1000).
5. Step 5: Calculate the number of atoms in the cube using Avogadro's number, \( N_A = 6.022 \times 10^{23} \, \text{atoms/mol} \). Multiply the number of moles \( n \) by \( N_A \) using the formula \( N = n \cdot N_A \).

Key Concepts

Atomic Structure

Atoms are the basic building blocks of matter, consisting of protons, neutrons, and electrons. In the context of aluminum, each atom has a specific arrangement of these subatomic particles, which determines its properties. Understanding atomic structure is essential for calculating the number of atoms in a given volume of a material.

Density

Density is defined as mass per unit volume and is a critical property for determining how many atoms are present in a specific volume of a substance. For aluminum, the density is approximately 2.7 g/cm³. By knowing the density, one can calculate the mass of the aluminum cube and subsequently the number of atoms it contains.

Avogadro's Number

Avogadro's number, approximately 6.022 x 10²³, is the number of atoms or molecules in one mole of a substance. This concept is crucial for converting the mass of aluminum into the number of atoms. By using the molar mass of aluminum (about 27 g/mol), one can determine how many moles are in the cube and then multiply by Avogadro's number to find the total number of atoms.