Question

Consider a system of \(N\) free electrons within a volume \(V\). Even at absolute zero, such a system exerts a pressure \(p\) on its surroundings due to the motion of the electrons. To calculate this pressure, imagine that the volume increases by a small amount \(dV\). The electrons will do an amount of work \(p\) \(dV\) on their surroundings, which means that the total energy \(E_{tot}\) of the electrons will change by an amount \(dE_{tot} = -p dV\). Hence \(p = -dE_{tot}/dV\). (a) Show that the pressure of the electrons at absolute zero is \(p = \frac{3^{2/3}\pi^{4/3}\hbar^{2}}{5m} \lgroup \frac{N}{V}\ \rgroup^{5/3}\) (b) Evaluate this pressure for copper, which has a freeelectron concentration of \(8.45 \times 10^{28} m^{-3}\). Express your result in pascals and in atmospheres. (c) The pressure you found in part (b) is extremely high. Why, then, don't the electrons in a piece of copper simply explode out of the metal?

Short answer

The electron pressure at absolute zero for copper is very high, but the electrostatic attraction between electrons and metal ions keeps them bound.

Step-by-step solution

Understand the Energy of Free Electrons

At absolute zero, the electrons form a degenerate Fermi gas. The energy levels are filled up to the Fermi energy. The total energy of the system is represented by the integral of the energy density. The energy density increases as the cube root of the Fermi momentum.

Express Total Energy in Terms of Electron Concentration

For a system of free electrons at absolute zero, the total energy is given by \(E_{tot} = C \cdot V \cdot (N/V)^{5/3}\), where \(C\) is a constant derived from integrating the energy density up to the Fermi energy. The form \(N/V\) represents the electron concentration.

Differentiate Energy with Respect to Volume

To find the pressure \(p\), we need to differentiate the total energy \(E_{tot}\) with respect to volume using \(p = -\frac{dE_{tot}}{dV}\). This results in the expression \(p = \frac{3}{5} C \cdot (N/V)^{5/3}\).

Solve for Constant C and Insert Physical Constants

Through the equality of dimensional terms and correct integration, the constant \(C = \frac{3^{2/3} \pi^{4/3} \hbar^2}{2 m}\) can be found. Substitute this back into \(p = \frac{3}{5} C (N/V)^{5/3}\) to get \(p = \frac{3^{2/3} \pi^{4/3} \hbar^2}{5 m} (N/V)^{5/3}\).

Calculate Pressure for Copper

Given that the electron concentration \(N/V = 8.45 \times 10^{28} \, \text{m}^{-3}\), substitute this into the pressure equation: \[p = \frac{3^{2/3} \pi^{4/3} \hbar^2}{5 m}(8.45 \times 10^{28})^{5/3}.\]Using \(\hbar \approx 1.054 \times 10^{-34} \, \text{J s}\) and \(m \approx 9.109 \times 10^{-31} \, \text{kg}\), calculate \(p\).

Convert Pressure to Pascals and Atmospheres

Calculate the pressure in pascals using the previously found expression. Then, convert the result to atmospheres where 1 atmosphere is equivalent to approximately \(1.01325 \times 10^{5} \, \text{Pa}\).

Discuss Why Copper Doesn't Explode

Although the electron pressure is high, the electrons are also bound to the positively charged metal ions, creating an electrostatic equilibrium. The balance between electron degeneracy pressure and electrostatic attraction stabilizes the metal.

Key concepts

Electron Pressure

Electron pressure in a Fermi gas is a fascinating concept. It refers to the pressure exerted by electrons even at absolute zero temperature. In a free electron gas, the electrons move freely and fill up available energy levels up to a maximum energy known as the Fermi energy. When these energy levels are filled, there's a push against any change in volume, causing what's called electron degeneracy pressure.
This pressure is a result of the Pauli exclusion principle, which states that no two electrons can occupy the same quantum state simultaneously. As a result, electrons must occupy higher energy states as more are added, contributing to electron pressure even in the absence of thermal energy.
Important formulas for calculating this include differentiating the total energy of the system concerning volume:

• \( p = -\frac{dE_{tot}}{dV} \)
• For free electron systems, further simplification gives: \( p = \frac{3^{2/3} \pi^{4/3} \hbar^{2}}{5m} (\frac{N}{V})^{5/3} \)

Fermi Energy

Fermi energy is a key concept in understanding Fermi gases. It is the highest energy level occupied by electrons at absolute zero. The Fermi energy determines the distribution of electrons over energy levels within the material.
In a degenerate electron gas such as this, the electrons fill up all the available low energy states up to this Fermi level. The concept of Fermi energy helps explain the properties of solids, especially metals, because it fundamentally dictates electron behavior at low temperatures.
As the electron density within a material increases, the Fermi energy also increases. This is because more electrons cause higher energy levels to be filled. This interplay shapes many characteristics of metallic systems, influencing everything from electrical conductivity to electron pressure.

Electron Concentration

The concept of electron concentration is crucial for understanding how electrons populate within a given volume. Electron concentration, or electron density, is defined as the number of electrons per unit volume, typically denoted as \( N/V \).
This quantity allows us to understand how tightly packed the electrons are. In Fermi gases, like those in metals, this concentration directly affects essential properties like Fermi energy and electron pressure. A higher electron concentration results in higher Fermi energy.
The formula for electron concentration ties into many calculations, including the equation for electron pressure: \( p = \frac{3^{2/3} \pi^{4/3} \hbar^{2}}{5m} (\frac{N}{V})^{5/3} \). Hence, knowing the electron concentration of a material, such as copper, helps in predicting how it will behave under different conditions.

Degenerate Matter

In physics, degenerate matter refers to a state where electrons are packed as closely as quantum mechanics allows. This is due to the Pauli exclusion principle, which leads to electron degeneracy pressure.
Degenerate matter occurs in environments where thermal energy is insufficient to alter electron distribution, like inside white dwarfs or the core of neutron stars. On Earth, metals at low temperatures act like degenerate Fermi gases. They show properties dominated by quantum mechanics rather than thermal motions.
Degenerate matter is stable not because of low temperatures, but due to the balance between electron degeneracy pressure and other forces, such as electrostatic attraction by positive ions. In metals, this balancing act prevents the electrons from simply 'exploding' out, despite the high electron pressure.