Problem 45 - Chemistry: The Central Science

Question

Order the following transitions in the hydrogen atom from smallest to largest frequency of light absorbed: \(n=3\) to \(n=6, n=4\) to \(n=9, n=2\) to \(n=3\), and \(n=1\) to \(n=2\).

Step-by-Step Solution

Answer

The order of the transitions from the smallest to the largest frequency of light absorbed is: \(n=4\) to \(n=9\), \(n=3\) to \(n=6\), \(n=2\) to \(n=3\), and \(n=1\) to \(n=2\).

Step 1: Apply the Rydberg formula for hydrogen atoms

The formula that relates the energy difference between two energy levels in a hydrogen atom and the frequency of light absorbed is as follows: \(ν = R_H ( \frac{1}{n_1^2} - \frac{1}{n_2^2} )\) where \(ν\) is the frequency of light absorbed, \(R_H\) is the Rydberg constant for hydrogen, \(n_1\) and \(n_2\) are the principal quantum numbers of the initial and final states of the electron, respectively.

Step 2: Calculate the frequency of light absorbed for each transition

Using the given transitions and the Rydberg formula, we can calculate the frequency of light absorbed for each transition. 1. For \(n=3\) to \(n=6\): \(ν_1 = R_H ( \frac{1}{3^2} - \frac{1}{6^2} ) = \frac{5 R_H}{36}\) 2. For \(n=4\) to \(n=9\): \(ν_2 = R_H ( \frac{1}{4^2} - \frac{1}{9^2} ) = \frac{65 R_H}{324}\) 3. For \(n=2\) to \(n=3\): \(ν_3 = R_H ( \frac{1}{2^2} - \frac{1}{3^2} ) = \frac{5 R_H}{36}\) 4. For \(n=1\) to \(n=2\): \(ν_4 = R_H ( \frac{1}{1^2} - \frac{1}{2^2} ) = \frac{3 R_H}{4}\) Now, let's compare the frequencies.

Step 3: Order the transitions from smallest to largest frequency of light absorbed

We can order the frequencies (\(ν_1\), \(ν_2\), \(ν_3\), and \(ν_4\)) obtained in Step 2, from the smallest to the largest. Comparing the frequencies, we have: \(ν_2 < ν_1 = ν_3 < ν_4\) Therefore, the order of the transitions from the smallest to the largest frequency of light absorbed is: \(n=4\) to \(n=9\), \(n=3\) to \(n=6\), \(n=2\) to \(n=3\), and \(n=1\) to \(n=2\).

Key Concepts

Hydrogen Atom Transitions

In a hydrogen atom, electrons occupy specific energy levels, also known as orbits, which are defined by quantum numbers. These energy levels are not fixed; electrons can move, or "transition," from one energy level to another. Such transitions are crucial because they either absorb or emit light, which is how we can study these processes experimentally.
When moving from a lower energy level to a higher one, an electron absorbs energy. The difference in energy between these two levels will determine the frequency of light absorbed.
Some common types of transitions in a hydrogen atom are:

• *Lyman series:* Transitions where the electron ends at the n=1 level.
• *Balmer series:* Transitions where the electron ends at the n=2 level.
• *Paschen series:* Transitions where the electron ends at the n=3 level.

This energy absorption results in different wavelengths or colors of light, which are characteristic of the transition series.
Understanding these transitions helps us order the frequency of light absorbed in different circumstances.

Frequency of Light Absorbed

The frequency of light absorbed during a hydrogen atom transition is determined by the energy difference between initial and final quantum states. Frequency, denoted as \( ν \), is directly related to the amount of energy absorbed during this transition.
To calculate this frequency, we use the Rydberg formula:\[ ν = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]where \( R_H \) is the Rydberg constant, \( n_1 \) is the initial energy level, and \( n_2 \) is the final energy level. This formula expresses how frequency depends on the square of the inverse of these quantum numbers.
When comparing transitions, a larger energy gap (where \( n_2 \) is much larger than \( n_1 \)) will result in a higher frequency of light absorbed. This relationship confirms that larger transitions involve more energy absorption, resulting in higher frequencies.

Quantum Numbers

Quantum numbers are sets of numbers that describe various properties of an electron in an atom. For hydrogen atom transitions, the principal quantum number, often represented as \( n \), is the most relevant. It indicates the energy level or shell in which an electron resides. Lower values of \( n \) correspond to lower energy levels closer to the nucleus.
The principal quantum number can take positive integer values such as 1, 2, 3, etc. When an electron transitions between these levels, it either absorbs or emits energy. This movement is associated with distinct changes in the electron's quantum state.
The relationship between quantum numbers and electron transitions is fundamental in calculating the frequency of light absorbed during these transitions. The differences in the reciprocal of the squares of the initial and final quantum numbers are central to the Rydberg formula. In essence, quantum numbers are key to understanding atomic behavior, energy absorption, and consequent light frequencies observed in hydrogen atom transitions.