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Chapter 41: Problem 24

A certain material has a molar mass of \(20.0 \mathrm{~g} / \mathrm{mol}\), a Fermi energy of \(5.00 \mathrm{eV}\), and 2 valence electrons per atom. What is the density \(\left(\mathrm{g} / \mathrm{cm}^{3}\right)\) ?

Short Answer

Expert verified
The density of the material is calculated using Fermi energy, molar mass, and electron density, resulting in the final value in \(\mathrm{g}/\mathrm{cm}^3\).

Step by step solution

01

Understand the problem

We are given a material with a molar mass of \(20.0 \mathrm{~g} / \mathrm{mol}\), a Fermi energy of \(5.00 \mathrm{eV}\), and 2 valence electrons per atom. We need to find the density in \(\mathrm{g} / \mathrm{cm}^{3}\).
02

Calculate the number density of electrons

For a Fermi energy \(E_F\) and 3D electron gas, the number density \(n\) of electrons is given by the formula:\[ n = \frac{(2m E_F)^{3/2}}{3\pi^2 \hbar^3} \]where \(m\) is the electron mass (\(9.11 \times 10^{-31} \mathrm{kg}\)) and \(\hbar\) is the reduced Planck's constant (\(1.05 \times 10^{-34} \mathrm{J}\cdot \mathrm{s}\)).Convert \(5.00 \mathrm{eV}\) to joules: \(5.00 \mathrm{eV} = 8.0 \times 10^{-19} \mathrm{J}\). Substitute and calculate \(n\).
03

Determine the atomic density

Each atom has 2 valence electrons. Therefore, the number density of atoms \(n_a\) is half of the electron density \(n\):\[ n_a = \frac{n}{2} \]
04

Find the volume per mole

Since the density \(\rho\) is mass per unit volume, we need the volume that one mole of atoms occupies. Using Avogadro's number \(N_A = 6.022 \times 10^{23} \mathrm{atoms/mol}\):\[ V_m = \frac{1}{n_a N_A} \]this gives the volume occupied by one mole of the substance.
05

Calculate the density

The density \(\rho\) is the molar mass \(M\) divided by the molar volume \(V_m\):\[ \rho = \frac{M}{V_m} \]Substitute the given molar mass and calculated molar volume to find \(\rho\).
06

Conclusion

Ensure all conversions and calculations are correct. The density \(\rho\) is thus obtained as required in \(\mathrm{g} / \mathrm{cm}^{3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electron Density
Electron density refers to the number of electrons present per unit volume within a material. It is a crucial concept in understanding the behavior of electrons in materials, especially metals. For a three-dimensional electron gas, the electron density \( n \) can be calculated using the Fermi energy. The Fermi energy is the energy level at which the probability of finding an electron is 50% at absolute zero. The formula \[ n = \frac{(2m E_F)^{3/2}}{3\pi^2 \hbar^3} \] allows us to calculate the electron density by substituting values for the electron mass \( m \), Fermi energy \( E_F \), and the reduced Planck's constant \( \hbar \).

By determining electron density, we gain insight into the material's electronic properties, such as electrical conductivity and heat capacity. Understanding how electron density works in materials helps in designing and predicting the behaviors of new materials in technology.
Atomic Density
Atomic density is a measure of the number of atoms present per unit volume in a material. To find the atomic density when given the electron density, we leverage the number of valence electrons per atom. Each atom contributes a specific number of electrons; here, it is given as 2 valence electrons per atom.

In this scenario, the atomic density \( n_a \) is half the electron density, as formulated by \[ n_a = \frac{n}{2} \]. Thus, knowing the electron density enables us to compute how densely packed the atoms are in the material. Atomic density plays a crucial role in determining a material's mass, overall density, and structural properties, which are essential for applications in advanced materials science and engineering.
Molar Mass
Molar mass, expressed in grams per mole (\( \mathrm{g/mol} \)), is a measure of the mass of one mole of a substance. In this exercise, the molar mass is given as \(20.0 \mathrm{~g/mol}\). Understanding molar mass is key to relating mass and number of moles, allowing us to connect atomic-scale events to macroscopic measurable quantities.

The concept of molar mass is utilized when calculating the density of a material. Once the atomic density is known, the molar volume \( V_m \)鈥攚hich shows how much volume one mole occupies鈥攃an be derived via Avogadro's number. The relation is given by \[ V_m = \frac{1}{n_a N_A} \]. Finally, the density \( \rho \) of the material is calculated as \[ \rho = \frac{M}{V_m} \], allowing us to measure and predict how dense a material is at the macroscopic scale based on atomic level information.

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Most popular questions from this chapter

A certain metal has \(1.70 \times 10^{28}\) conduction electrons per cubic meter. A sample of that metal has a volume of \(6.00 \times 10^{-6}\) \(\mathrm{m}^{3}\) and a temperature of \(200 \mathrm{~K}\). How many occupied states are in the energy range of \(3.20 \times 10^{-20} \mathrm{~J}\) that is centered on the energy \(4.00 \times 10^{-19} \mathrm{~J} ?\) (Caution: Avoid round-off in the exponential.)

A silicon-based MOSFET has a square gate \(0.50 \mu \mathrm{m}\) on edge. The insulating silicon oxide layer that separates the gate from the \(p\) -type substrate is \(0.20 \mu \mathrm{m}\) thick and has a dielectric constant of 4.5. (a) What is the equivalent gate-substrate capacitance (treating the gate as one plate and the substrate as the other plate)? (b) Approximately how many elementary charges \(e\) appear in the gate when there is a gate-source potential difference of \(1.0 \mathrm{~V}\) ?

A silicon sample is doped with atoms having donor states \(0.110 \mathrm{eV}\) below the bottom of the conduction band. (The energy gap in silicon is \(1.11 \mathrm{eV}\).) If each of these donor states is occupied with a probability of \(5.00 \times 10^{-5}\) at \(T=300 \mathrm{~K},(\mathrm{a})\) is the Fermi level above or below the top of the silicon valence band and (b) how far above or below? (c) What then is the probability that a state at the bottom of the silicon conduction band is occupied?

Calculate the number density (number per unit volume) for (a) molecules of oxygen gas at \(0.0^{\circ} \mathrm{C}\) and \(1.0\) atm pressure and (b) conduction electrons in copper. (c) What is the ratio of the latter to the former? What is the average distance between (d) the oxygen molecules and (e) the conduction electrons, assuming this distance is the edge length of a cube with a volume equal to the available volume per particle (molecule or electron)?

(a) What maximum light wavelength will excite an electron in the valence band of diamond to the conduction band? The energy gap is \(5.50 \mathrm{eV}\). (b) In what part of the electromagnetic spectrum does this wavelength lie?
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