Problem 74 - University Physics with Modern Physics

Question

CP Consider a simple model of the helium atom in which two electrons, each with mass \(m,\) move around the nucleus (charge \(+2 e\) ) in the same circular orbit. Each electron has orbital angular momentum \(\hbar\) (that is, the orbit is the smallest-radius Bohr orbit), and the two electrons are always on opposite sides of the nucleus. Ignore the effects of spin. (a) Determine the radius of the orbit and the orbital speed of each electron. [Hint: Follow the procedure used in Section 39.3 to derive Eqs. \((39.8)\) and \((39.9) .\) Each electron experiences an attractive force from the nucleus and a repulsive force from the other electron. \(J\) (b) What is the total kinetic energy of the electrons? (c) What is the potential energy of the system (the nucleus and the two electrons)? (d) In this model, how much energy is required to remove both electrons to infinity? How does this compare to the experimental value of 79.0 \(\mathrm{eV}\) ?

Step-by-Step Solution

Answer

The calculated energy to remove both electrons differs from the experimental value, reflecting model limits.

Step 1: Understanding the Problem

We have a helium atom model with 2 electrons (mass \( m \)) in a circular orbit around a nucleus with charge \(+2e\). Each electron has an orbital angular momentum of \( \hbar \). Our task is to calculate (a) the orbit's radius and velocity, (b) the total kinetic energy of the electrons, (c) the system's potential energy, and (d) the energy to remove the electrons.

Step 2: Applying Angular Momentum Quantization

Given that each electron has angular momentum \( \hbar \), and using the quantization condition for angular momentum \( mvr = n\hbar \) with \( n = 1 \), we set \( m v r = \hbar \). By solving for \( v \), we find: \[ v = \frac{\hbar}{mr} \].

Step 3: Using Bohr's Model Equations

To find the radius \( r \), we consider the forces acting on each electron: the electrostatic attraction from the nucleus and repulsion from the other electron. The effective nuclear charge is reduced by the repulsion to \( +1e \), leading to: \[ k_e \frac{e^2}{r^2} = \frac{mv^2}{r} \]. Using \( v = \frac{\hbar}{mr} \) and solving \( r \), we find: \[ r = \frac{\hbar^2}{mek_e e^2} \].

Step 4: Calculating Orbital Speed

Using the formula \( v = \frac{\hbar}{mr} \) and substituting \( r \) from the previous step, calculate \( v \). Thus, \( v = \frac{k_e e^2}{\hbar} \).

Step 5: Finding Total Kinetic Energy

Each electron has kinetic energy \( KE = \frac{1}{2} mv^2 \). For two electrons, the total kinetic energy \( KE_{total} = mv^2 \). Substituting \( v \) gives \( KE_{total} = \frac{k_e^2 e^4}{2\hbar^2} \).

Step 6: Determining Total Potential Energy

The potential energy of interaction between one electron and the nucleus is \( -k_e \frac{2e^2}{r} \), and between electrons is \( k_e \frac{e^2}{r} \). Thus, total potential energy \( U_{total} = -3k_e \frac{e^2}{r} \).

Step 7: Calculating Total Energy to Remove Electrons

The total energy required to remove both electrons is given by \( E = KE_{total} + U_{total} = \frac{k_e^2 e^4}{2\hbar^2} - 3k_e \frac{e^2}{r} \). Substitute \( r \) and \( v \) to express in eV.

Step 8: Comparison with Experimental Values

Calculate the value from previous results and compare it with the experimental value of 79.0 eV. The difference provides insight into model accuracy limitations.

Key Concepts

Angular Momentum Quantization

The concept of angular momentum quantization is a fundamental part of quantum mechanics. In the Bohr model, electrons orbit around the nucleus in fixed paths called orbits. Each orbit corresponds to a specific quantized angular momentum value. The angular momentum, denoted by "L", for each electron is given by the equation:

• \[ L = mvr = n\hbar \]

where "m" is the mass of the electron, "v" its velocity, "r" the radius of the orbit, "n" a positive integer representing the quantum number, and \(\hbar\) is the reduced Planck's constant.
Angular momentum quantization implies that only certain discrete values of "L" are possible, depending on the quantum number "n". In the special helium atom model under consideration, each electron follows the smallest possible orbit, where \(n = 1\), so the angular momentum is just \(\hbar\). This fixes the allowed motion of the electrons so that they remain in stable, well-defined orbits around the nucleus.
This quantization of angular momentum helps in determining other properties of the electron orbit, such as its radius and velocity. These quantized levels affect how the electrons interact with each other and the nucleus.

Helium Atom

The helium atom is the second simplest atom and consists of two protons, two neutrons in the nucleus, and two electrons orbiting around. In this model, it is assumed that the electrons orbit the nucleus in a circular path, and the helium nucleus has a charge of "+2e" due to the protons. For simplicity, we neglect the effect of electron spin and focus only on the orbital motion.
In a real helium atom, complex interactions exist due to the presence of two electrons, which complicate calculations. However, in our simplified model:

• Electrons experience an attractive force from the nucleus and a repulsive force from each other.
• The effective nuclear charge becomes \(+1e\) due to these interactions, simplifying calculations using Bohr's model.

It's important to recognize that while this model allows us to compute key values, it doesn't account for everything seen experimentally. Despite this, it lays down a strong foundation for understanding more intricate quantum mechanics involved in real helium atoms.

Electron Orbitals

Electron orbitals describe the regions around the nucleus where electrons are likely to be found. They are not like planetary orbits but are more about probability distribution in shapes defined by quantum mechanics.
In the Bohr model, which simplifies electron behavior, orbitals are considered as fixed paths. However, quantum mechanics reveals a more complex picture:

• Electrons don't follow precise paths but rather occupy regions called orbitals.
• Different types of orbitals (s, p, d, f) have different shapes, like spheres or dumbbells, affecting electron distribution.
• In multi-electron atoms like helium, electrons fill these orbitals in a specific sequence (known as the Aufbau principle).

For our simplified helium model, each electron is viewed as if in a fixed path with a specific angular momentum due to quantization. Although this doesn't perfectly match quantum mechanical orbitals, it serves as a useful approximation for calculations, allowing us to estimate atomic behavior and energy levels.

Kinetic and Potential Energy Calculation

In atomic physics, understanding the energies present in the atom is crucial.
Kinetic Energy (KE) is given by the motion of the electrons. For each electron, the kinetic energy is computed using:

• \[ KE = \frac{1}{2}mv^2 \]

where "m" is the electron's mass and "v" its speed.
For two electrons, the total kinetic energy is \[ KE_{total} = mv^2 \]. This takes into account both electrons revolving in opposite directions at the same speed.
Potential Energy (U) in the system arises due to electrostatic interactions. The attractive potential between the electron and nucleus is:

• \[ U_{nucleus} = -k_e \frac{2e^2}{r} \]

The repulsion between the two electrons adds additional potential energy:

• \[ U_{electrons} = k_e \frac{e^2}{r} \]

Summing these gives the total potential energy as \[ U_{total} = -3k_e \frac{e^2}{r} \]. Both energies are essential to determine the total energy required to remove electrons, and this concept forms the basis for understanding various atomic energy exchanges.