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?

Step by step solution

01

Calculate the mass of the copper cube

First, convert the edge length of the copper cube from mm to cm: \[ 0.5 \, \mathrm{mm} = 0.05 \,\mathrm{cm}. \] Calculate the volume (V) of the cube: \[ V = (\mathrm{edge \, length})^3 = (0.05 \, \mathrm{cm})^3 = 1.25 \times 10^{-4} \, \mathrm{cm^3}. \] Calculate the mass (m) of the copper cube using the volume (V) and the density (ρ): \[ m = ρV = (8.96 \,\mathrm{g/cm^3})(1.25 \times 10^{-4} \, \mathrm{cm^3}) = 1.12 \, \mathrm{g}. \]

02

Find the number of copper atoms in the cube

Determine the moles (n) of copper in the cube: \[ n = \frac{\mathrm{mass \, of \, copper}}{\mathrm{molar \, mass \, of \, copper}} = \frac{1.12 \, \mathrm{g}}{63.5 \, \mathrm{g/mol}} = 1.76 \times 10^{-2} \, \mathrm{mol}. \] Calculate the number of copper atoms (N) using Avogadro's number: \[ N = n \times \mathrm{Avogadro's \, number} = (1.76 \times 10^{-2} \, \mathrm{mol})(6.022 \times 10^{23} \, \mathrm{atoms/mol}) \approx 1.06 \times 10^{22} \, \mathrm{atoms}. \]

03

Calculate the average energy spacing

Divide the range of energies by the number of atoms in the copper crystal to find the average energy spacing: \[ \mathrm{Average\, spacing} = \frac{1 \times 10^{-19} \, \mathrm{J}}{1.06 \times 10^{22} \, \mathrm{atoms}} \approx 9.43 \times 10^{-42} \, \mathrm{J/atom}. \]

04

Compare the energy spacing with hydrogen atom energy levels

The given energy spacing for hydrogen is \(1 \times 10^{-18}\) J. Comparing the two energy spacings: \[ \frac{9.43\times 10^{-42} \, \mathrm{J/atom}}{1 \times 10^{-18} \, \mathrm{J}} \approx 9.43 \times 10^{-24} . \] Since the ratio is extremely small, the energy spacing in the copper crystal is substantially smaller than that of the hydrogen atom.

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