Problem 49 - Chemistry: An Atoms-Focused Approach

Question

Does the clectromagnetic energy emitted by an excited-state H atom depend on the individual values of \(n_{1}\) and \(n_{2}\) or only on the difference between them \(\left(n_{1}-n_{2}\right) ?\)

Step-by-Step Solution

Answer

Answer: The electromagnetic energy emitted by an excited-state hydrogen atom depends on the individual values of the quantum numbers \(n_1\) and \(n_2\).

Step 1: Recall the energy of hydrogen energy levels

In hydrogen-like atoms, the energy of an electron in energy level \(n\) is given by: \(E_n = -\dfrac{13.6 \text{ eV}}{n^2}\) where \(n\) represents the principal quantum number of an energy level.

Step 2: Calculate the energy change during a transition

During a transition from an initial energy level \(n_1\) to a final energy level \(n_2\), the energy change will be: \(\Delta E = E_{n_2} - E_{n_1}\) Now substitute the energy level formula into the equation: \(\Delta E = \left[-\dfrac{13.6 \text{ eV}}{(n_2)^2}\right] - \left[-\dfrac{13.6 \text{ eV}}{(n_1)^2}\right]\)

Step 3: Simplify the equation

Distribute the negative sign to both terms in the equation and factor out the constant value \(13.6 \text{ eV}\): \(\Delta E = 13.6 \text{ eV} \left[\dfrac{1}{(n_2)^2}-\dfrac{1}{(n_1)^2}\right]\)

Step 4: Analyze the dependence on \(n_1\) and \(n_2\)

From the equation for the energy change during a transition, we can conclude that the energy depends on the inverse square of the quantum numbers \(n_1\) and \(n_2\). It does not explicitly depend on their difference \((n_1 - n_2)\), but rather on the individual values of \(n_1\) and \(n_2\). Therefore, the electromagnetic energy emitted by an excited-state hydrogen atom does depend on the individual values of \(n_1\) and \(n_2\).

Key Concepts

Quantum Numbers

In the magical world of quantum mechanics, quantum numbers are crucial for understanding the behavior of electrons in an atom. They're like the seat numbers for electrons, helping us pinpoint their position within an atom. For the hydrogen atom, the principal quantum number, denoted by \( n \), is particularly important. It tells us about the energy level or shell in which the electron is residing. Some important points regarding quantum numbers include:

• They are integers (1, 2, 3, ...), reflecting the different energy levels.
• The bigger the number, the higher the energy level and the farther the electron from the nucleus.
• This quantum number determines the energy of the electron in the hydrogen atom as per the formula \( E_n = -\dfrac{13.6 \text{ eV}}{n^2} \).

These numbers enable us to calculate the change in energy when an electron transitions from one level to another. In summary, to understand the behavior and energy of electrons in hydrogen atoms, quantum numbers are a fundamental concept.

Energy Transition

When an electron in a hydrogen atom moves between energy levels, we call it an energy transition. This is like an electron taking an elevator to a different floor in a building. During this move, an electron jumps from an initial energy level, \( n_1 \), to a final energy level, \( n_2 \). This can either release or absorb energy. The change in energy, \( \Delta E \), can be calculated using the formula:\[ \Delta E = 13.6 \text{ eV} \left[\dfrac{1}{(n_2)^2}-\dfrac{1}{(n_1)^2}\right] \]A couple of key things about energy transitions:

• Energy is released when \( n_2 < n_1 \), meaning the electron falls to a lower energy level.
• Energy is absorbed when \( n_2 > n_1 \), indicating the electron goes to a higher energy level.

It's fascinating that the energy transition is based on these quantum numbers. The key point here is: The change in energy depends on the difference in the inverse squares of those numbers (\( \frac{1}{(n_2)^2} - \frac{1}{(n_1)^2} \)), not the actual numeric difference \( n_1 - n_2 \).

Electromagnetic Energy Emission

Every time an electron in a hydrogen atom transitions to a lower energy level, there's a beautiful spectacle of electromagnetic energy emission. This is like a little fireworks show at the atomic level.Here's what happens:

• The electron releases the energy difference as photons (light particles).
• The energy of these emitted photons corresponds exactly to the energy change \( \Delta E \) of the electron.
• This emitted light contributes to what scientists call the atomic spectrum of hydrogen.

This process doesn't require \( n_1 \) and \( n_2 \) to be directly subtracted from each other. Instead, it's the respective energy levels they represent that determine the energy of the emitted light.So, in the grand scheme of things, each transition gives us a different wavelength of light, creating the unique spectral lines seen in hydrogen. It's like painting with numbers, where each set of quantum numbers gives us a unique color!