A hypothetical atom has energy levels given in the figure below. Which figure most accurately depicts the absorption spectrum, assuming all transitions between energy levels is possible? The black vertical lines represent the absorbed wavelengths.
Quantum mechanics and hydrogen atom
Consider an electron of mass m moves with the velocity v in a hydrogen atom. If an electron is at a distance r from the proton, then the potential energy function of the electron can be written as follows:
Isotopes of Hydrogen Atoms
To understand isotopes, it's easiest to learn the simplest system. Hydrogen, the most basic nucleus, has received a great deal of attention. Several of the results seen in more complex nuclei can be seen in hydrogen isotopes. An isotope is a nucleus of the same atomic number (Z) but a different atomic mass number (A). The number of neutrons present in the nucleus varies with respect to the isotope.
Mass of Hydrogen Atom
Hydrogen is one of the most fundamental elements on Earth which is colorless, odorless, and a flammable chemical substance. The representation of hydrogen in the periodic table is H. It is mostly found as a diatomic molecule as water H2O on earth. It is also known to be the lightest element and takes its place on Earth up to 0.14 %. There are three isotopes of hydrogen- protium, deuterium, and tritium. There is a huge abundance of Hydrogen molecules on the earth's surface. The hydrogen isotope tritium has its half-life equal to 12.32 years, through beta decay. In physics, the study of Hydrogen is fundamental.
![### Absorption Spectrum and Energy Transitions in a Hypothetical Atom
A hypothetical atom has the following energy levels:
- Ground state: \( E_0 = -11.5 \, \text{eV} \)
- First excited state: \( E_1 = -9.6 \, \text{eV} \)
- Second excited state: \( E_2 = -5.2 \, \text{eV} \)
- Third excited state: \( E_3 = -1.1 \, \text{eV} \)
### Absorption Spectrum
The provided figure illustrates four potential absorption spectra, each represented by black vertical lines. These lines correspond to wavelengths of light absorbed by the atom as it transitions between different energy levels.
The task is to determine which spectrum accurately represents all possible transitions between these energy levels.
#### Explanation of Figures:
1. **First Figure**:
- This spectrum has black vertical lines spread across it.
- Wavelength increases from left to right.
2. **Second Figure**:
- Another layout of black vertical lines.
- Wavelength increases from left to right.
3. **Third Figure**:
- Different arrangement of black vertical lines.
- Wavelength increases from left to right.
4. **Fourth Figure**:
- Yet another configuration of black vertical lines.
- Wavelength increases from left to right.
### Determining the Correct Absorption Spectrum
To identify the correct absorption spectrum, consider the energy differences between the levels:
- \( E_0 \) to \( E_1 \): \( -9.6 \,\text{eV} - (-11.5 \, \text{eV}) = 1.9 \, \text{eV} \)
- \( E_0 \) to \( E_2 \): \( -5.2 \,\text{eV} - (-11.5 \, \text{eV}) = 6.3 \, \text{eV} \)
- \( E_0 \) to \( E_3 \): \( -1.1 \,\text{eV} - (-11.5 \, \text{eV}) = 10.4 \, \text{eV} \)
- \( E_1 \) to \( E_2 \): \( -5.2 \,\text{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7ddf1e52-17d0-47a9-956c-00bd0ce93716%2Fd9f65ef9-cb03-433f-bad1-806cff76c34e%2F5n59s9e_processed.jpeg&w=3840&q=75)
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