Problem 95 - General Chemistry: Principles and Modern Applications

Question

The Pfund series of the hydrogen spectrum has as its longest wavelength component a line at \(7400 \mathrm{nm}\) Describe the electron transitions that produce this series. That is, give a Bohr quantum number that is common to this series.

Step-by-Step Solution

Answer

For the Pfund series of the hydrogen spectrum, the Bohr quantum number corresponding to the transition that produces the longest wavelength of 7400 nm is 6.

Step 1: Identify given values and unknowns

It is given that the Pfund series' longest wavelength is \( 7400 nm \). This series corresponds to the transitions where the electron ends in the \( n=5 \) energy level of the hydrogen atom. What's common with the Pfund series is the lower energy level, \( n_1 \), which is equal to 5. The higher energy level, \( n_2 \), will be different for each transition and is the one to solve.

Step 2: Solve for the higher quantum number

Substitute the given values into the formula: \[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \] Convert \( \lambda \) to meters and solve for \( n_2 \). Remember that \( R \) has a value of \( 1.097373 x 10^7 m^{-1} \).

Step 3: Calculation

After rearranging the formula and substituting the known values, calculate for \( n_2 \). The result should be rounded off to the nearest whole number as quantum number cannot be a fraction or decimal.

Step 4: Interpret the result

The quantum number attained indicates to which energy level the electron transitioned to emit the longest wavelength of light. This is the Bohr quantum number for the given Pfund series transition.

Key Concepts

Hydrogen Spectrum

The hydrogen spectrum is a fascinating aspect of atomic theory that helps us understand the behavior of electrons in a hydrogen atom. It arises when electrons transition between different energy levels, leading to the emission or absorption of light at specific wavelengths. These emissions or absorptions form a series of spectral lines, unique to hydrogen, known as its emission or absorption spectrum.

The visible portion of the hydrogen spectrum is known as the Balmer series. However, there are other series beyond the visible range, including the Lyman, Paschen, Brackett, and Pfund series. Each series corresponds to electron transitions that end at a specific energy level, denoted by the Bohr quantum number. These spectral lines are crucial for understanding quantum mechanics and exploring atomic structures.

In the context of the Pfund series, the transitions terminate at the fifth energy level ( = 5) in a hydrogen atom. Though these lines are not visible to the human eye, they occur in the infrared region of the electromagnetic spectrum and provide insight into the behavior of atomic electrons.

Electron Transitions

Electron transitions are central to the emission of light in the hydrogen spectrum. When an electron in a hydrogen atom absorbs energy, it moves to a higher energy level. Conversely, when it returns to a lower energy level, it releases energy in the form of a photon. The energy and wavelength of this photon are directly related to the difference in energy between the two levels involved in the transition.

Each transition is characterized by a decrease in the electron's energy, and the emitted photon's wavelength depends on this energy difference. In the Pfund series, the electron transitions involve electrons falling from higher energy levels to the n = 5 level. Each drop results in the emission of a photon, adding various spectral lines to this series.

The formula used to describe these transitions is: \[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \] where \( \lambda \) is the wavelength of the emitted light, \( R \) is the Rydberg constant, \( n_1 \) is the lower energy level (common to all transitions in the series), and \( n_2 \) is the initial higher energy level from which the electron falls.

Bohr Quantum Number

The Bohr quantum number, often denoted as \( n \), is an integer that represents the specific energy levels within an atom where electrons reside. This concept is integral to the Bohr model of the atom, which postulates that electrons orbit the nucleus in defined energy levels or shells and can move between them through absorption or emission of energy.

In the context of spectral series like the Pfund series, the Bohr quantum number becomes particularly useful. Each series in the hydrogen spectrum ends at a specific level associated with a particular Bohr quantum number. For the Pfund series, this number is 5. Therefore, every electron transition that results in a Pfund series line ends with the electron in the \( n = 5 \) level.

The Bohr quantum number helps categorize and predict the spectral lines observed in different series, making it easier to study and understand atomic behavior. It provides clarity on how and why certain wavelengths are emitted as electrons transition between various energy levels in the hydrogen atom.