Question

Calculate the wavelength and frequency of light emitted when an electron changes from \(n=4\) to \(n=3\) in the H atom. In what region of the spectrum is this radiation found?

Short answer

The emitted light has a wavelength of 193.7 nm and frequency of approximately \( 1.548 \times 10^{15} \) Hz, lying in the UV region.

Step-by-step solution

Identify the Rydberg Formula

The Rydberg formula is used to calculate the wavelength of light as electrons transition between energy levels in a hydrogen atom. The formula is given by \( \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \), where \( R_H \) is the Rydberg constant \( 1.097 \times 10^7 \text{ m}^{-1} \), \( n_1 \) and \( n_2 \) are the principal quantum numbers with \( n_2 > n_1 \).

Assign Quantum Numbers

Given that the electron transitions from the \( n=4 \) level to the \( n=3 \) level, assign \( n_1 = 3 \) and \( n_2 = 4 \).

Calculate the Wavelength

Substitute the values \( n_1 = 3 \) and \( n_2 = 4 \) into the Rydberg formula:\[ \frac{1}{\lambda} = 1.097 \times 10^7 \left( \frac{1}{3^2} - \frac{1}{4^2} \right) \]Calculate the expression:\[ \frac{1}{\lambda} = 1.097 \times 10^7 \left( \frac{1}{9} - \frac{1}{16} \right) = 1.097 \times 10^7 \left( \frac{7}{144} \right) \]\[ \frac{1}{\lambda} \approx 5.162 \times 10^6 \text{ m}^{-1} \]Now solve for \( \lambda \):\[ \lambda \approx \frac{1}{5.162 \times 10^6} \approx 1.937 \times 10^{-7} \text{ m} \] or approximately 193.7 nm.

Calculate Frequency

The frequency \( u \) can be found using the speed of light equation: \( c = \lambda u \). Solve for \( u \):\[ u = \frac{c}{\lambda} = \frac{3.00 \times 10^8 \text{ m/s}}{193.7 \times 10^{-9} \text{ m}} \]\[ u \approx 1.548 \times 10^{15} \text{ Hz} \]

Identify Spectral Region

The wavelength calculated, 193.7 nm, lies within the ultraviolet (UV) region of the electromagnetic spectrum.

Key concepts

Hydrogen Atom

The hydrogen atom is the simplest atom in the universe. It consists of just one proton in its nucleus and one electron orbiting this nucleus. Despite its simplicity, the hydrogen atom plays a substantial role in understanding atomic structure and quantum mechanics.

Hydrogen atoms are particularly important in the study of spectral lines, which are the unique fingerprints of atoms. When an electron in a hydrogen atom moves between energy levels, it either absorbs or emits light at specific wavelengths. These transitions can be calculated using the Rydberg formula and lead to the production of spectral lines that help scientists identify the types of atoms in distant stars or other astronomical phenomena.

In the realm of quantum mechanics, the hydrogen atom serves as a fundamental model. Its simplicity allows scientists to formulate and test quantum theory concepts like the quantization of energy levels, which was a significant leap in physics when first introduced.

Electron Transition

Electron transition refers to the movement of an electron between energy levels within an atom. In the context of the hydrogen atom, these transitions are important because they result in the absorption or emission of light.

When an electron drops from a higher energy level to a lower one, it emits light. This process is dictated by the difference in energy between the two levels. Electrons can only occupy specific energy levels, defined by quantum numbers, and any movement between these levels requires a precise amount of energy.

Such transitions are characterized by the quantum numbers involved. For instance, in our example, the electron is transitioning from the fourth energy level (\( n=4 \)) to the third (\( n=3 \)). The energy released during this transition is what produces light at a wavelength specific to the hydrogen atom, contributing to its unique spectrum.

Ultraviolet Spectrum

The ultraviolet (UV) spectrum comprises light waves with wavelengths shorter than visible light but longer than X-rays. These wavelengths are typically below 400 nm, placing the UV spectrum beyond the purple end of visible light.

In the specific case of a hydrogen atom, when an electron transitions between close energy levels such as from (\( n=4 \)) to (\( n=3 \)), the emitted light falls within this ultraviolet region. With a calculated wavelength of approximately 193.7 nm, the light emitted in this specific transition belongs to the UV spectrum.

The UV spectrum is of particular interest in many scientific fields, including astronomy, where it assists in the study of celestial phenomena. Observing the UV lines in a star's spectrum can offer insight into the star's composition, temperature, and the processes occurring within.

Quantum Numbers

Quantum numbers are integral to understanding the energy levels of electrons in atoms. They provide a set of numerical values that describe the unique quantum state of an electron.

• Principal Quantum Number (\( n \)): This number describes the size and energy level of the orbital an electron occupies. Larger values of \( n \) mean higher energy levels further from the nucleus.
• Angular Momentum Quantum Number (\( l \)): While not needed for the hydrogen spectral calculations, \( l \) defines the shape of the orbital.
• Magnetic Quantum Number (\( m_l \)): This number specifies the particular orbital within a subshell where an electron is likely to be found.
• Spin Quantum Number (\( m_s \)): Specifies the direction of the electron's spin, either +1/2 or -1/2.

When calculating electron transitions, we mainly focus on the principal quantum number, \( n \), as it determines the energy involved in the transition. In the example provided, \( n \) values of 4 to 3 highlight the specific energy state changes leading to light emission. Understanding these quantum numbers is key in predicting the behavior of electrons and the resulting spectral lines an atom produces.