Problem 16 - Chemistry and Chemical Reactivity

Question

What are the wavelength and frequency of the radiation involved in the least energetic emission line in the Lyman series? What are the values of \(n_{\text {initial }}\) and \(n_{\text {final }} ?\)

Step-by-Step Solution

Answer

The wavelength is 364.5 nm, frequency is \(8.228 \times 10^6 \text{ m}^{-1}\), with \(n_{\text{initial}} = 2\) and \(n_{\text{final}} = 1\).

Step 1: Understand the Lyman series

The Lyman series is a set of spectral lines in the ultraviolet region. These lines correspond to electron transitions in a hydrogen atom where the electron falls to the ground state, with the principal quantum number \( n_{\text{final}} = 1 \).

Step 2: Identify the least energetic transition in the Lyman series

The least energetic transition in any series occurs when the initial state \( n_{\text{initial}} \) is the smallest possible value greater than \( n_{\text{final}} \). For the Lyman series, this transition is from \( n_{\text{initial}} = 2 \) to \( n_{\text{final}} = 1 \).

Step 3: Calculate the frequency of the transition

Use the formula for the frequency \( u \) of a transition: \[ u = R_H \left( \frac{1}{n_{\text{final}}^2} - \frac{1}{n_{\text{initial}}^2} \right) \]where \( R_H \) is the Rydberg constant \( (1.097 \times 10^7 \text{ m}^{-1}) \). For the transition from 2 to 1:\[ u = 1.097 \times 10^7 \left( \frac{1}{1^2} - \frac{1}{2^2} \right) \] \[ u = 1.097 \times 10^7 \left( 1 - 0.25 \right) \]\[ u = 1.097 \times 10^7 \times 0.75 \]\[ u = 8.228 \times 10^6 \text{ m}^{-1} \].

Step 4: Calculate the wavelength of the transition

The wavelength \( \lambda \) is related to frequency \( u \) by the speed of light \( c = 3.00 \times 10^8 \text{ m/s} \):\[ \lambda = \frac{c}{u} \]\[ \lambda = \frac{3.00 \times 10^8}{8.228 \times 10^6} \]\[ \lambda = 3.645 \times 10^{-7} \text{ m} \] or 364.5 nm.

Key Concepts

Spectral Lines

Spectral lines are unique patterns of light emitted or absorbed by atoms and molecules. These lines occur when electrons jump between energy levels within an atom.
Each line represents a photon of a specific energy either being absorbed or emitted.
Spectral lines are used to identify the elemental composition of distant stars and galaxies since each element has a distinct pattern of spectral lines.

• Emission Lines: Bright lines on a dark background, created when electrons emit energy.
• Absorption Lines: Dark lines on a spectrum, created when electrons absorb energy.

Understanding spectral lines helps in studying electron transitions, which gives insights into atomic structures and properties.

Electron Transitions

Electron transitions refer to the movement of electrons between energy levels in an atom. This movement is associated with the absorption or emission of energy.

• If an electron absorbs energy, it moves to a higher energy level (excitation).
• If an electron releases energy, it descends to a lower energy level (emission).

In a hydrogen atom, the energy levels are quantized, meaning they have distinct values.
When an electron moves from one level to another, the difference in energy levels determines the energy of the photon that is absorbed or emitted.

Hydrogen Atom

The hydrogen atom, being the simplest atom with just one proton and one electron, plays a crucial role in our study of atomic physics. Its simplicity allows precise calculations of electron transitions and spectral lines.

• Atomic Structure: Consists of a single proton in the nucleus and one electron in orbit.
• Bohr's Model: Successfully explains observed spectra by quantizing angular momentum.

In the Lyman series, hydrogen atom electrons transition from higher energy levels to the ground state (n=1).
These transitions produce ultraviolet light.

Wavelength and Frequency

Wavelength and frequency are integral parts of understanding electromagnetic radiation. Wavelength (\( \lambda \)) is the distance between two peaks in a wave, while frequency (\( u \)) refers to the number of oscillations that occur in a specific time period.

Both are inversely related: as one increases, the other decreases.

The speed of light (\( c \)) connects them using the formula: \[ \lambda = \frac{c}{u} \]

In solving problems involving spectral lines, knowing these relationships helps us calculate one if we know the other.
Analyzing the wavelength and frequency allows determination of the photon's energy released in electron transitions.

Rydberg Constant

The Rydberg constant (\( R_H \)) is a fundamental physical constant used in atomic physics to describe the wavelengths of spectral lines of many chemical elements.

• It is represented numerically as \( 1.097 \times 10^7 \text{ m}^{-1} \).
• Commonly used to calculate energies of transitions in hydrogen atoms: \[ u = R_H \left( \frac{1}{n_{\text{final}}^2} - \frac{1}{n_{\text{initial}}^2} \right) \]

Understanding the Rydberg constant’s role is crucial for predicting the wavelengths of light emitted or absorbed by hydrogen.
It helps scientists explain and calculate specific electron transitions, such as those in the Lyman series.