Problem 52 - Chemistry: An Atoms-Focused Approach

Question

Without calculating any frequency values, rank the following transitions in the hydrogen atom in order of increasing frequency of the electromagnetic radiation that could produce them. a. \(n=4 \rightarrow n=6\) b. \(n=6 \rightarrow n=8\) c. \(n=9 \rightarrow n=11\) d. \(n=11 \rightarrow n=13\)

Step-by-Step Solution

Answer

Question: Rank the following transitions in the hydrogen atom in order of increasing frequency of the electromagnetic radiation that can cause them: a) n=4 to n=6, b) n=6 to n=8, c) n=9 to n=11, d) n=11 to n=13. Answer: The transitions in order of increasing frequency are: d) n=11 to n=13, c) n=9 to n=11, b) n=6 to n=8, and, a) n=4 to n=6.

Step 1: 1. Recalling the formula for energy difference and frequency

The energy difference between two energy levels in a hydrogen atom is given by the Rydberg formula: $$\Delta E = -13.6\ \textit{eV}\left(\frac{1}{n_{f}²} - \frac{1}{n_{i}²}\right)$$ where \(n_{i}\) is the initial energy level, \(n_{f}\) is the final energy level, and \(\Delta E\) is the energy difference. The energy difference is related to the frequency of the electromagnetic radiation by Planck's equation: $$\Delta E = h\cdot f$$ where \(h\) is the Planck's constant, and \(f\) is the frequency. Since we don't want to calculate the frequency values, we'll focus on comparing the magnitudes of \(\Delta E\) for each transition.

Step 2: 2. Finding the energy differences for the given transitions

We will now find the energy differences for each of the given transitions: a. \(n=4 \rightarrow n=6\) $$\Delta E_a = -13.6\ \textit{eV}\left(\frac{1}{6^2} - \frac{1}{4^2}\right)$$ b. \(n=6 \rightarrow n=8\) $$\Delta E_b = -13.6\ \textit{eV}\left(\frac{1}{8^2} - \frac{1}{6^2}\right)$$ c. \(n=9 \rightarrow n=11\) $$\Delta E_c = -13.6\ \textit{eV}\left(\frac{1}{11^2} - \frac{1}{9^2}\right)$$ d. \(n=11 \rightarrow n=13\) $$\Delta E_d = -13.6\ \textit{eV}\left(\frac{1}{13^2} - \frac{1}{11^2}\right)$$

Step 3: 3. Comparing the energy differences to rank the transitions

Comparing the expressions for the energy differences, we find that: $$\Delta E_a > \Delta E_b > \Delta E_c > \Delta E_d$$ Since the energy difference is directly proportional to the frequency (\(\Delta E = h\cdot f\)), we can conclude that the transitions have the following order of increasing frequency: $$n=11 \rightarrow n=13 < n=9 \rightarrow n=11 < n=6 \rightarrow n=8 < n=4 \rightarrow n=6$$

Key Concepts

Rydberg formula

The Rydberg formula is a crucial mathematical tool for describing electronic transitions in a hydrogen atom. It specifically calculates the energy difference between two distinct energy levels. The formula is expressed as:\[ \Delta E = -13.6\ \text{eV} \left(\frac{1}{n_{f}^2} - \frac{1}{n_{i}^2}\right) \]In this equation:

• \(n_i\) is the initial energy level of the electron.
• \(n_f\) is the final energy level.
• \(\Delta E\) denotes the change in energy, measured in electron volts (eV).
• The constant, \(-13.6\ eV\), is known as the Rydberg constant for hydrogen.

The formula demonstrates how energy transitions are quantized, meaning electrons can only exist at distinct energy levels. When an electron transitions from a higher energy level \((n_i)\) to a lower energy level \((n_f)\), it releases energy, causing the emission of electromagnetic radiation. This energy difference can be positive or negative based on the direction of the transition, but is always a measure of how much energy is involved in moving between two distinct levels.

energy levels

In atomic physics, energy levels are the fixed distances from the nucleus of an atom where electrons can be found. These levels are quantized, meaning electrons can reside at certain specific intervals. Electrons in an atom occupy the lowest available energy level unless energy is added. Each energy level corresponds to a specific amount of energy. For hydrogen, the energy level expression is often represented in terms of the principal quantum number \(n\). The larger the value of \(n\), the higher is its energy, and electrons at this level are further from the atomic nucleus. When an electron transitions between these levels:

• Energy is absorbed if the electron moves to a higher energy level.
• Energy is released when it moves down to a lower energy level.

This absorption or emission of energy manifests as electromagnetic radiation. It's this radiation that can be measured and calculated using formulas like Planck's equation and the Rydberg formula. Understanding these stages helps in predicting the likelihood of electron transitions and their resulting spectral lines.

Planck's equation

Planck's equation is pivotal in connecting the energy of electrons transitioning between energy levels to the electromagnetic waves they emit or absorb. This equation establishes a relationship between energy changes and wave frequency:\[ \Delta E = h \cdot f \]Where:

• \(\Delta E\) represents the energy change between two levels.
• \(h\) is Planck's constant, valued at \(6.626 \times 10^{-34}\ \text{Js}\).
• \(f\) is the frequency of the electromagnetic radiation.

This formula shows that the energy emitted or absorbed during an electron transition is directly proportional to the frequency \(f\) of the radiation involved. When electrons move to higher energy levels, energy is absorbed; when they fall to lower levels, energy is emitted, creating radiation of a specific frequency measurable in experiments.By using Planck's equation in conjunction with the Rydberg formula, one can rank the transitions according to their frequencies, as demonstrated in the original exercise. This understanding not only aids in solving problems but also provides insight into the fundamental nature of atomic spectra.