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 \(1 \times 10^{-19} \mathrm{J}\) . 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. (a) How many copper atoms are in a piece of copper metal in the shape of a cube with edge length 0.5 \(\mathrm{mm} ?\) The density of copper is 8.96 \(\mathrm{g} / \mathrm{cm}^{3} .\) (b) Determine the average spacing in J between energy levels in the copper metal in part (a).(c) Is this spacing larger, substantially smaller, or about the same as the 1 \(\times 10^{-18}\) J separation between energy levels in a hydrogen atom?

Short answer

The number of copper atoms in a 0.5 mm cube is approximately \(1.06 \times 10^{19}\) atoms. The average spacing between energy levels in the copper metal is about \(9.4 \times 10^{-39}\) J, which is substantially smaller than the \(1 \times 10^{-18}\) J separation between energy levels in a hydrogen atom.

Step-by-step solution

Calculate the number of copper atoms in the cube

First, we need to find the mass of the copper cube. The volume of the cube is given by \(V = L^3\), where L is the edge length. Since we need to work with grams and cubic centimeters, we will convert the edge length from millimeters to centimeters: \(L_{cm} = 0.5\,mm × \frac{1\,cm}{10\,mm} = 0.05\,cm\). Now we can calculate the volume of the cube: \[V = L_{cm}^3 = (0.05\,cm)^3 = 1.25 \times 10^{-4}\,cm^3\] Next, using the density of copper (\(8.96\,g/cm^3\)), we can calculate the mass of the copper cube: \[Mass = Density \times Volume = 8.96\,g/cm^3 \times 1.25 \times 10^{-4}\,cm^3 = 1.12 \times 10^{-3}\,g\] The molar mass of copper is approximately \(63.5\,g/mol\). We'll use this to find the number of moles of copper atoms in the cube: \[Number\,of\,moles = \frac{Mass}{Molar\,mass} = \frac{1.12 \times 10^{-3}\,g}{63.5\,g/mol} = 1.76 \times 10^{-5}\,mol\] Finally, we can calculate the number of copper atoms using Avogadro's number (\(6.022 \times 10^{23}\,atoms/mol\)): \[Number\,of\,atoms = Number\,of\,moles \times Avogadro's\,number = 1.76 \times 10^{-5}\,mol \times 6.022 \times 10^{23}\,atoms/mol = 1.06 \times 10^{19}\,atoms\]

Determine the average spacing between energy levels

We are given that the range of energy levels in a crystal of copper is approximately \(1 \times 10^{-19}J\). The spacing between energy levels may be approximated by dividing the range of energies by the number of atoms in the crystal. So, the average spacing between energy levels is: \[Spacing = \frac{Range\, of\, energies}{Number\, of\, atoms} = \frac{1 \times 10^{-19}\,J}{1.06 \times 10^{19}\,atoms} = 9.4 \times 10^{-39}\,J\]

Compare this spacing to the 1 x 10^{-18} J separation in a hydrogen atom

Now that we know the spacing between energy levels in the copper cube, we can compare it to the separation between energy levels in a hydrogen atom, which is given as \(1 \times 10^{-18}J\). We can see that the spacing in the copper cube (\(9.4 \times 10^{-39}\,J\)) is substantially smaller than the separation in a hydrogen atom (\(1 \times 10^{-18}\,J\)). Therefore, the answer is that the spacing is substantially smaller.

Key concepts

Energy Bands in Solids

In the context of solids, particularly metals like copper, energy bands provide a striking illustration of quantum mechanics at play. These bands are formed due to the overlap of atomic orbitals when atoms come together to form a solid. Unlike isolated atoms that have discrete energy levels for their electrons, the interaction between adjacent atoms in a solid causes these levels to merge into continuous bands.

The energy bands are crucial for understanding how electrons move through a solid and are segmented into two main types: the valence band, which is filled with valence electrons, and the conduction band, where electrons gain energy and are free to conduct electricity. The gap between these two bands, known as the band gap, determines the electrical conductivity of the material; conductors have overlapping bands or a very small bandgap, while insulators have a large bandgap.

For copper, a good conductor, the energy range within its band is approximately 1 x 10^-19 J. This range is continuous and vast because of the huge number of atoms within the crystal, each contributing its discrete energy levels to the formation of the band.

Atomic Structure of Copper

To understand why copper has such favorable electrical properties, one must delve into its atomic structure. Copper is a transition metal with atomic number 29, meaning it has 29 protons and usually also 29 electrons. Its electrons are arranged in shells around the nucleus, with the configuration [Ar] 3d10 4s1.

The outermost electron in the 4s orbital is easily shared with neighboring atoms in the solid lattice, contributing to copper's excellent ability to conduct electricity. When copper atoms are packed together in a solid, these outer electrons delocalize and move freely, forming a 'sea' of electrons that make up the conduction band.

This structural trait is central to its electrical conductivity and the formation of a continuous range of energy levels, or bands, that we observe in metallic copper.

Quantum Physics in Solids

Quantum physics lays the groundwork for our understanding of energy bands in the realm of solids, and by extension, the entire field of solid-state physics. The principles of quantum mechanics dictate that electrons in atoms occupy specific energy levels. However, when atoms are brought close together to form a solid, these energy levels combine to form bands due to the wave-like nature of electrons and the Pauli exclusion principle which prevents them from occupying the same quantum state.

Within a solid like copper, the electrons are not confined to one atom but are distributed throughout the entire material, creating a collective behavior that quantum physics uniquely explains. The spacing between individual energy levels becomes so minute, as in the copper crystal, that it can no longer be distinguished, leading to the formation of bands with significantly smaller energy spacing than found in discrete atomic systems such as a hydrogen atom.

Avogadro's Number

One of the central concepts in chemistry and solid-state physics is Avogadro's number, designated as 6.022 x 10^23. This value represents the number of constituent particles, typically atoms or molecules, in one mole of a substance.

Avogadro's number is pivotal when calculating the number of atoms in a given sample, as illustrated in the problem's solution for the copper cube. By knowing the mass and the molar mass of copper, we can use Avogadro's number to convert the mass into atoms. This conversion is essential not only in theoretical calculations but also in practical applications that span from determining the amount of a substance needed in a chemical reaction to computing the number of charge carriers in a semiconductor.

Understanding Avogadro's number allows us to bridge the gap between the macroscopic world we can measure and the microscopic world of atoms and molecules, and it is a fundamental step in calculating the energy level spacing in materials.