Question

Scientists have found interstellar hydrogen atoms with quantum number \(n\) in the hundreds. Calculate the wavelength of light emitted when a hydrogen atom undergoes a transition from \(n=236\) to \(n=235 .\) In what region of the electromagnetic spectrum does this wavelength fall?

Short answer

The wavelength is approximately \(2.70 \times 10^{-3} \text{ m}\), falling in the microwave region.

Step-by-step solution

Understand the Rydberg Formula

The wavelength of light emitted during a hydrogen atom's electron transition between two energy levels can be calculated using the Rydberg formula: \[\frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)\] where \( \lambda \) is the wavelength, \( R_H \) is the Rydberg constant \((1.097 \times 10^7 \text{ m}^{-1})\), and \( n_1 \) and \( n_2 \) are the principal quantum numbers of the two levels (\( n_1 < n_2 \)).

Substitute Values

For the given transition, we have \( n_1 = 235 \) and \( n_2 = 236 \). Substitute these values into the Rydberg formula:\[ \frac{1}{\lambda} = R_H \left( \frac{1}{235^2} - \frac{1}{236^2} \right) \] Calculating the terms:\[ \frac{1}{235^2} = \frac{1}{55225} \] \[ \frac{1}{236^2} = \frac{1}{55696} \]

Calculate the Difference

Calculate the difference between the two fractions:\[ \Delta = \frac{1}{235^2} - \frac{1}{236^2} \approx 3.37 \times 10^{-8} \text{ m}^{-1} \]

Use the Rydberg Constant

Multiply the difference by the Rydberg constant:\[ \frac{1}{\lambda} = 1.097 \times 10^7 \times 3.37 \times 10^{-8} \] \[ \frac{1}{\lambda} \approx 369.729 \text{ m}^{-1} \]

Calculate the Wavelength

Find the wavelength by taking the reciprocal of the previous result:\[ \lambda \approx \frac{1}{369.729} \approx 2.70 \times 10^{-3} \text{ m} \]

Determine the Electromagnetic Spectrum Region

A wavelength of approximately \(2.70 \times 10^{-3} \text{ m}\) falls within the microwave region of the electromagnetic spectrum.

Key concepts

Wavelength Calculation

The wavelength calculation for hydrogen atom transitions revolves around the Rydberg Formula. This formula is a critical tool in quantum physics used to calculate the wavelength of light emitted when electrons in a hydrogen atom shift between energy levels. Given the Rydberg constant, denoted as \( R_H = 1.097 \times 10^7 \text{ m}^{-1}\), this formula provides insights into the energy released as electromagnetic radiation during electron transitions.

Here's how it works: the Rydberg formula is expressed as:\[\frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)\]Where:

• \( \lambda \) is the wavelength
• \( n_1 \) and \( n_2 \) are the principal quantum numbers of the initial and final states

By substituting the values, such as \( n_1 = 235 \) and \( n_2 = 236 \) into the formula, one can calculate the resulting wavelength. This gives a specific picture of the light emitted and the kind of energy transfer happening within the atom.

Quantum Numbers

Quantum numbers are fundamental aspects of quantum mechanics that define the unique state of an electron in an atom. The principal quantum number, represented by \( n \), essentially describes the size of the electron's orbit and the energy level of the electron within an atom.

Here’s why they matter:

• They are principal identifiers of electron shells within an atom.
• The larger the quantum number, the higher the energy level, and the greater distance from the nucleus.

In the problem, the electrons transition between energy levels \( n=236 \) and \( n=235 \), showcasing high-energy states rarely observed in typical terrestrial conditions. Such transitions involve large quantum numbers because interstellar conditions can excite electrons to very high levels, owing mainly to low density and high energy environments. Understanding these transitions is crucial in interpreting the spectral lines emitted by distant cosmic objects.

Electromagnetic Spectrum

The electromagnetic spectrum includes all types of electromagnetic radiation, which are differentiated by their wavelengths. This spectrum stretches from gamma rays, with very short wavelengths, to radio waves, which have very long wavelengths. Knowing where a particular wavelength falls on this spectrum tells us about the type of radiation and its source.

Specifically:

• A wavelength of about \(2.70 \times 10^{-3}\) meters is located in the microwave region.
• This part of the spectrum is associated with lower energy emissions compared to visible light or ultraviolet rays.

In the context of hydrogen transitions, the microwave region signifies that the energy differences between very high quantum numbers like those in the transition between \( n=236 \) and \( n=235 \) are small, resulting in longer wavelengths typical of microwaves. This knowledge aids in applications such as astrophysical observations, where detecting such transition wavelengths can reveal conditions of celestial hydrogen.

Hydrogen Atom Transition

Hydrogen atom transitions occur when the electron within the atom jumps between different energy levels. This quantum leap results in the emission or absorption of light at specific wavelengths determined by the energy difference between those levels.

For instance:

• In a hydrogen atom, the transition from one high energy level, like \( n=236 \) to a slightly lower energy level, \( n=235 \), involves minimal energy differences.
• This process emits light in the microwave spectrum given the small energy difference and subsequently the long wavelength calculated.

These transitions are significant in astrophysical studies since they reveal the presence and characteristics of hydrogen in outer space. By studying the emitted wavelengths, scientists can infer the abundance, location, and energy states of hydrogen atoms in interstellar regions, enhancing our understanding of the universe's composition.