Textbook Question
Silicon carbide, SiC, has the three-dimensional structure shown in the figure.
(b) Would you expect the bonding in SiC to be predominantly ionic, metallic, or covalent?
Ch.12 - Solids and Modern Materials
Chapter 12, Problem 114
(a) How many copper atoms are in a piece of copper metal in the shape of a cube with an edge length of 0.5 mm? The density of copper is 8.96 g/cm³. (c) Is this spacing larger, substantially smaller, or about the same as the 1 * 10⁻¹⁸ J separation between energy levels in a hydrogen atom?
Verified step by step guidance1
First, calculate the volume of the cube. The edge length is given as 0.5 mm, which needs to be converted to cm for consistency with the density units. Use the conversion factor: 1 mm = 0.1 cm. Therefore, the edge length in cm is 0.5 mm * 0.1 cm/mm = 0.05 cm. The volume of a cube is calculated using the formula: \( V = \text{edge length}^3 \). Substitute the edge length in cm to find the volume: \( V = (0.05 \text{ cm})^3 \).
Next, use the density of copper to find the mass of the copper cube. The density \( \rho \) is given as 8.96 g/cm³. The mass \( m \) can be calculated using the formula: \( m = \rho \times V \). Substitute the density and the volume calculated in the previous step to find the mass of the copper cube.
Now, convert the mass of copper to moles. Use the molar mass of copper, which is approximately 63.55 g/mol. The number of moles \( n \) is calculated using the formula: \( n = \frac{m}{\text{molar mass}} \). Substitute the mass of copper and the molar mass to find the number of moles.
Calculate the number of copper atoms using Avogadro's number, which is \( 6.022 \times 10^{23} \) atoms/mol. The number of atoms \( N \) is calculated using the formula: \( N = n \times \text{Avogadro's number} \). Substitute the number of moles calculated in the previous step to find the number of copper atoms.
For part (c), compare the calculated spacing between copper atoms with the given energy separation in a hydrogen atom, \( 1 \times 10^{-18} \) J. Discuss whether the spacing is larger, substantially smaller, or about the same, considering typical atomic and molecular dimensions and energy scales.
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Density and Volume Calculations
Density is defined as mass per unit volume, typically expressed in grams per cubic centimeter (g/cm³). To find the mass of a piece of copper, we first calculate its volume using the formula for the volume of a cube (V = edge length³). Once we have the volume, we can multiply it by the density of copper to find the mass, which is essential for determining the number of atoms present.
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Density Concepts
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 copper into the number of atoms. By calculating the number of moles of copper from its mass and then multiplying by Avogadro's number, we can determine how many individual copper atoms are present in the sample.
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Energy Levels in Atoms
Energy levels in atoms refer to the quantized states that electrons can occupy, with specific energy separations between them. In hydrogen, the energy separation between levels is on the order of 10⁻¹⁸ joules. Comparing this to the spacing derived from the dimensions of the copper cube helps to contextualize atomic scales versus macroscopic dimensions, illustrating the differences in scale between atomic structures and bulk materials.
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Related Practice
Textbook Question
Energy bands are considered continuous due to the large number of closely spaced energy levels. The range of energy levels in a crystal of copper is approximately 1 × 10–19 J. Assuming equal spacing between levels, the spacing between energy levels may be approximated by dividing the range of energies by the number of atoms in the crystal. (b) Determine the average spacing in J between energy levels in the copper metal in part (a).
Textbook Question
Sodium oxide (Na2O) adopts a cubic structure with Na atoms represented by green spheres and O atoms by red spheres.
(c) The unit cell edge length is 5.550 Å. Determine the density of Na2O.