Problem 28 - University Physics with Modern Physics

Question

(a) Show that, as \(n\) gets very large, the energy levels of the hydrogen atom get closer and closer together in energy. (b) Do the radii of these energy levels also get closer together?

Step-by-Step Solution

Answer

Energy levels become closer together, but radii differences increase as \( n \) grows.

Step 1: Understand Energy of Hydrogen Atom

The energy levels of a hydrogen atom are given by \[E_n = -\frac{13.6}{n^2} \text{ eV}\]where \( n \) is the principal quantum number. For large \( n \), this describes how the energy changes as \( n \) increases.

Step 2: Analyze Energy Level Differences

Calculate the difference in energy between successive energy levels:\[\Delta E = E_{n+1} - E_n = -\frac{13.6}{(n+1)^2} + \frac{13.6}{n^2}\]For large \( n \), approximate using the binomial expansion to simplify:\[\Delta E \approx \frac{27.2n}{n^4} = \frac{27.2}{n^3}\]This shows that as \( n \) increases, \( \Delta E \) decreases.

Step 3: Understand Radii of Hydrogen Atom

The radius of an electron in the hydrogen atom is given by \[r_n = n^2 \cdot a_0\]where \( a_0 \) is the Bohr radius. As \( n \) increases, the effect on \( r_n \) needs examination.

Step 4: Analyze Radii Differences

Calculate the difference in radii between successive levels:\[\Delta r = r_{n+1} - r_n = (n+1)^2 a_0 - n^2 a_0\]This simplifies to \[\Delta r = (2n + 1) a_0\]As \( n \) increases, \( \Delta r \) increases, showing that radii differences get larger.

Key Concepts

Principal Quantum Number

The principal quantum number, denoted as \( n \), is a crucial part of the quantum mechanical model of the atom. It identifies the energy level where an electron is most likely to be found around the nucleus. The integer values of \( n \) start from 1 and increase upwards. Each increase in \( n \) corresponds to an electron being in a higher energy level. The higher the energy level, the more energy the electron has and the farther it is from the nucleus.

This principle directly relates to the energy of an electron in a hydrogen atom, which is given by the formula \( E_n = -\frac{13.6}{n^2} \text{ eV} \). Here, \( E_n \) is the energy of the electron at level \( n \), and as \( n \) increases, the energy becomes less negative, indicating higher energy states. This foundational concept helps us understand the structure of atoms and why energy levels are quantized.

• Values of \( n \) start at 1: higher values imply higher energy and larger orbitals.
• Principal quantum number determines the electron's energy level.
• Electrons are more energetic and further from the nucleus as \( n \) increases.

Energy Level Differences

As the principal quantum number \( n \) grows, the spacing between energy levels in a hydrogen atom decreases. This is captured by examining the difference between successive energy levels, \( \Delta E = E_{n+1} - E_n \). Using quantum mechanics, it's shown that as \( n \) becomes very large, \( \Delta E \) approaches zero.

The formula for energy level differences simplifies for large \( n \) values using an approximation: \( \Delta E \approx \frac{27.2}{n^3} \). It reveals that even though energy levels become closer together, each additional level requires less energy to move to. This is why, at high energies, the energy states are almost continuous.

• Energy levels become closer as \( n \) increases.
• Difference in energy between levels is represented as \( \Delta E \).
• At high \( n \), energy differences approach zero, creating closely packed levels.

Bohr Radius

The Bohr radius, represented as \( a_0 \), is approximately 0.529 Angstroms. It defines the most probable distance of the electron from the nucleus in a hydrogen atom when the electron is in its ground state (\( n = 1 \)). For any energy level \( n \), the radius \( r_n \) of the electron's orbit is given by \( r_n = n^2 \cdot a_0 \). This means the radius grows with the square of the quantum number.

As \( n \) increases, not only does the energy level increase but so does the radius, leading to more spread-out energy levels spatially. However, the difference between successive radii, \( \Delta r = (2n + 1) a_0 \), indicates that spacing gets bigger. Thus, while the energy levels converge energy-wise, spatially they diverge more.

• Bohr radius \( a_0 \) is a key reference point in atomic physics.
• The radius grows with the square of the quantum number \( n \).
• Differences in radii increase, making orbits spatially farther as \( n \) rises.