Question

In Bohr’s atomic model, what electron-orbit transition produces the blue-green line in hydrogen’s atomic emission spectrum?

Short answer

The electron-orbit transition that produces the blue-green line in hydrogen's atomic emission spectrum according to Bohr's atomic model occurs when an electron jumps from the $n_2=4$ energy level to the $n_1=2$ energy level.

Step-by-step solution

Recall Bohr's Atomic Model

Bohr's atomic model is based on the concept that electrons revolve around the nucleus in fixed orbits or energy levels (like planets revolve around the sun in solar system). When an electron jumps from a higher energy level to a lower energy level, it releases energy in the form of electromagnetic radiation (light). The energy and wavelength of the emitted light are related to the difference in energy between the two orbits.

Understand Rydberg Formula

The wavelength of the light emitted during an electron-orbit transition can be calculated using the Rydberg formula: $${\displaystyle \frac{1}{\lambda }}=R_{\mathrm{H}}({\displaystyle \frac{1}{n_1^2}}-{\displaystyle \frac{1}{n_2^2}})$$ where: - $\lambda$ is the wavelength of the emitted light - $R_{\mathrm{H}}$ is the Rydberg constant specific to hydrogen ($R_{\mathrm{H}}=1.097\times {10}^7{\mathrm{m}}^{-1}$) - $n_1$ is the principal quantum number of the lower energy level - $n_2$ is the principal quantum number of the higher energy levels ($n_2>n_1$)

Identify Blue-Green Wavelength

Blue-green light falls within the range of 490 - 520 nm. For this problem, we will take the middle value of this range, which is 505 nm.

Use Rydberg Formula to Find Electron-Orbit Transition

Now, we will use the Rydberg formula and plug in the wavelength of blue-green light to find the values of $n_1$ and $n_2$ that correspond to this wavelength. As the constant ($R_{\mathrm{H}}$) and the wavelength ($\lambda$) are given, we only need to try different combinations of $n_1$ and $n_2$ until we find the correct transition. We will try different values for $n_1$ starting from 1 and try different corresponding values for $n_2(>n_1)$ until we find the right combination that gives us a wavelength close to 505 nm. By trying different combinations, we find that: For $n_1=2$ and $n_2=4$: $${\displaystyle \frac{1}{\lambda }}=R_{\mathrm{H}}({\displaystyle \frac{1}{2^2}}-{\displaystyle \frac{1}{4^2}})$$ $$\lambda ={\displaystyle \frac{1}{R_{\mathrm{H}}\left({\displaystyle \frac{3}{16}}\right)}}={\displaystyle \frac{1}{(1.097\times {10}^7{\mathrm{m}}^{-1})\times {\displaystyle \frac{3}{16}}}}$$ $$\lambda \approx 486.1\mathrm{nm}$$ The calculated wavelength $486.1\mathrm{nm}$ falls within the blue-green range, so the electron-orbit transition that produces the blue-green line in hydrogen's atomic emission spectrum is when an electron jumps from the $n_2=4$ energy level to the $n_1=2$ energy level.

Key concepts

Electron-Orbit Transition

In Bohr's atomic model, an electron-orbit transition is a fascinating process that explains the behavior of electrons within an atom. According to the model, electrons revolve around the nucleus in specific paths or 'orbits' at fixed distances called energy levels. When an electron transitions between these levels, it either absorbs or emits energy.

This energy emission or absorption occurs because electrons need to move from one orbit to another. The change in energy corresponds to the difference between the higher and lower energy levels involved in the transition.

• If the electron moves to a higher energy level (further from the nucleus), it absorbs energy.
• If it moves to a lower energy level (closer to the nucleus), it releases energy in the form of electromagnetic radiation.

Understanding these transitions is essential for explaining the colors we see in atomic emission spectra, such as the distinct blue-green line in the hydrogen spectrum when electrons descend from the fourth to the second energy level.

Rydberg Formula

The Rydberg formula is a fundamental equation used to calculate the wavelengths of light emitted or absorbed during an electron-orbit transition in a hydrogen atom. This formula gives us a mathematical way to understand the exact energy changes happening during these transitions.

The formula is expressed as:
$${\displaystyle \frac{1}{\lambda }}=R_{\mathrm{H}}({\displaystyle \frac{1}{n_1^2}}-{\displaystyle \frac{1}{n_2^2}})$$
Where:

• $\lambda$ is the wavelength of the light emitted.
• $R_{\mathrm{H}}$ is the Rydberg constant for hydrogen, approximately $1.097\times {10}^7{\mathrm{m}}^{-1}$.
• $n_1$ is the lower principal quantum number, and $n_2$ is the higher principal quantum number.

The Rydberg formula enables scientists and students to predict the wavelengths of spectral lines, crucial for identifying elements and their electron transitions.

Hydrogen Atomic Emission Spectrum

The hydrogen atomic emission spectrum is a series of discrete spectral lines associated with transitions of electrons among energy levels in a hydrogen atom. Antoine's light bulb moment came when he noticed these lines had distinct colors, each corresponding to a particular wavelength.

When an electron drops from a higher to a lower energy level, such as from the fourth to second level, it emits light. The specific wavelengths of light correspond to the differences in energy between these levels, which is beautifully represented by the Rydberg formula.

This spectrum is unique to hydrogen, making it a fingerprint for identifying hydrogen atoms. The visible part of the spectrum includes the prominent blue-green line, explaining why hydrogen emits a pale greenish hue under certain conditions.

Energy Levels

Energy levels in Bohr's atomic model refer to the fixed orbits where electrons reside around a nucleus. Each level is designated by a principal quantum number $n$ — the lower the number, the closer the level is to the nucleus, and the lower the energy.

Energy levels are like shelves in a cupboard where electrons sit. An electron can only jump from one shelf to another if it can absorb or release the exact amount of energy difference between these shelves.

• Lower energy levels have less energy and are nearer to the nucleus.
• Higher energy levels require more energy and are further from the nucleus.

This concept is critical for predicting the electron-orbit transitions that produce various spectral lines in the hydrogen emission spectrum. It emphasizes the quantized nature of electrons in atoms, meaning they can't exist between defined energy levels.

Electromagnetic Radiation

Electromagnetic radiation is the form of energy released or absorbed when electrons transition between energy levels in an atom. This radiation encompasses a broad spectrum, including visible light, ultraviolet, infrared, and more.

In Bohr's model, when an electron moves to a lower energy level, it emits electromagnetic radiation, which may be visible or invisible to the naked eye depending on its wavelength. Conversely, when an electron absorbs electromagnetic radiation, it jumps to a higher energy level.

• The blue-green line in hydrogen's emission spectrum is an example of visible electromagnetic radiation emitted when electrons drop energy levels within the atom.
• These emissions provide crucial insights into the electronic structure of atoms and are fundamental in fields like astronomy, where they help discern the composition of distant stars.

Understanding electromagnetic radiation is vital in appreciating how and why atoms emit light and other forms of energy, a cornerstone of atomic physics.