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Chapter 41: Problem 30

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.)

Short Answer

Expert verified
Approximately 8.16 x 10^{21} occupied states are within the energy range.

Step by step solution

01

Calculate the Total Number of Electrons

First, we need to find out the total number of conduction electrons in the given volume of the metal. The volume is given as \(6.00 \times 10^{-6} \) m鲁. Multiply this by the electron density:\[\text{Total electrons} = 1.70 \times 10^{28} \text{ electrons/m}^3 \times 6.00 \times 10^{-6} \text{ m}^3\]Thus, the total number of electrons is \(1.02 \times 10^{23}\).
02

Determine the Fermi-Dirac Distribution Function

The number of occupied states within a small energy range is determined by the Fermi-Dirac distribution function:\[f(E) = \frac{1}{e^{(E - E_F) / kT} + 1}\]Where:\(E = 4.00 \times 10^{-19} \text{ J}\) is the center of our energy range, \(k = 1.38 \times 10^{-23} \text{ J/K}\) is the Boltzmann constant, and \(T = 200 \text{ K}\) is the temperature. We need an approximate value for the Fermi energy \(E_F\), which is typically a known constant for the specific metal.
03

Calculate the Number of Occupied States

Assuming the Fermi energy \(E_F\) and since the temperature is low, the approximation for \(f(E)\) should hold. We need to multiply the total number of electrons by the Fermi-Dirac distribution:\[N_{states} = \text{Total electrons} \times f(E) \]Assuming near-room temperature and that we have the correct Fermi energy \(E_F\):\[f(E) \approx 1.0 \text{ (since } E < E_F \text{ in low temperatures)}\]So, \(N_{states} \approx 1.02 \times 10^{23}\). Finally, adjust this value by considering only the small range around \(3.20 \times 10^{-20} \text{ J}\).
04

Refine for the Small Energy Range

This extra fine adjustment requires that we consider the difference due to a small change in energy. Given the symmetry and small energy width, the occupied states in this energy range (which is centered around the Fermi level and has a narrow range \(3.20 \times 10^{-20} \text{ J}\)) should about occupy similar probability as calculated under \(f(E)\). In practice, it is essentially this ratio of the small energy window (compared to total energies) which is:\[= \frac{3.20 \times 10^{-20}}{2} \text{ J}\]With \(N_{states} = \text{Total electrons} \times \frac{3.20 \times 10^{-20}}{kT}\approx 1\).
05

Calculate Final Number of Occupied States

The step must yield the equation factoring temperature and energy width:Since energy window \(\Delta E = 3.20 \times 10^{-20} \text{ J}\) is proportionately very small:\[ N_{occupied} = 1.02 \times 10^{23} \times \frac{3.20 \times 10^{-20}}{4 \times 10^{-19}} \]Yielding approximately \(N_{occupied} = 8.16 \times 10^{21}\) occupied states, just within the range centered at \(4.00 \times 10^{-19} \text{ J}\).

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Key Concepts

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

Conduction Electrons
Conduction electrons are the electrons that are free to move within a metal, allowing it to conduct electricity. Unlike electrons bound to atoms, conduction electrons can travel throughout the metal's lattice. In a conductor, these electrons can be thought of as a "sea" of electrons that freely move and facilitate the flow of electric current. In the context of the given exercise, we have been provided with an electron density of \(1.70 \times 10^{28}\) electrons/m鲁. This density indicates how many conduction electrons exist in a cubic meter of metal. With a given volume, we can then calculate the total number of conduction electrons using the formula:
  • Total electrons = Electron density \(\times\) Volume of the metal
  • Given an example with Volume = \(6.00 \times 10^{-6}\) m鲁, Total electrons = \(1.02 \times 10^{23}\)
Understanding how conduction electrons behave is crucial since their distribution among energy states determines how metals conduct electricity and respond to temperature changes.
Energy States
Energy states in a metal refer to the allowed energy levels that electrons can occupy. In the quantum world of conduction electrons, the Fermi-Dirac distribution is used to determine the likelihood that an energy state is occupied by an electron at a specific temperature.The exercise highlights an energy range centered around \(4.00 \times 10^{-19} \text{ J}\). This description implies a small energy window where various energy states can be either occupied or unoccupied, depending on the statistics of the electron distribution. Electrons usually occupy the lowest available energy states because of the Pauli exclusion principle, meaning no two electrons can occupy the same state simultaneously.To find the number of occupied states in this specific energy window, we need to apply the probability function from Fermi-Dirac statistics:\[f(E) = \frac{1}{e^{(E - E_F) / kT} + 1}\]This expression helps in evaluating how electrons populate energy states near the Fermi energy \(E_F\), especially at very low temperatures such as the 200 K in our problem.
Boltzmann Constant
The Boltzmann constant, denoted \(k\), appears in various important equations in statistical mechanics, including the Fermi-Dirac distribution function. It links the average kinetic energy of particles in a gas with the temperature of the gas. Its value is \(1.38 \times 10^{-23} \text{ J/K}\) and acts as a bridge in thermodynamics and the statistical view of physics.In this particular exercise, the Boltzmann constant plays a critical role in adjusting the energy scale used in the Fermi-Dirac distribution function. By influencing the denominator in the exponent of that distribution, Boltzmann's constant impacts how sharply the probability of an energy state being occupied by an electron changes with energy and temperature. Using the constant in evaluating conditions at specific temperatures ensures that the relationship between thermal energy and energy levels is properly accounted for. Without this constant, calculations regarding probability distributions of electrons among various energy states would not align correctly with empirical observations.

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Most popular questions from this chapter

(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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