51 lines
4.2 KiB
Markdown
51 lines
4.2 KiB
Markdown
The Rydberg formula (or Rydberg equation) is a mathematical formula used to predict the [wavelength](https://www.thoughtco.com/energy-from-wavelength-example-problem-609479) of light resulting from an electron moving between energy levels of an atom.
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When an electron changes from one atomic orbital to another, the electron's energy changes. When the electron changes from an orbital with high energy to a lower energy state, a [photon of light](https://www.thoughtco.com/definition-of-photon-605908) is created. When the electron moves from low energy to a higher energy state, a photon of light is absorbed by the atom.
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Each element has a distinct spectral fingerprint. When an element's gaseous state is heated, it will give off light. When this light is passed through a prism or diffraction grating, bright lines of different colors can be distinguished. Each element is slightly different from other elements. This discovery was the beginning of the study of spectroscopy.
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![[Pasted image 20241021220213.png]]
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---
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## Rydberg Equation
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Johannes Rydberg was a Swedish physicist who attempted to find a mathematical relationship between one spectral line and the next of certain elements. He eventually discovered there was an integer relationship between the wavenumbers of successive lines.
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His findings were combined with Bohr's model of the atom to create this formula:
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> 1/λ = RZ2(1/n12 - 1/n22)
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where
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> λ is the wavelength of the photon (wavenumber = 1/wavelength)
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> R = Rydberg's constant (1.0973731568539(55) x 107 m-1)
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> Z = [atomic number](https://www.thoughtco.com/definition-of-atomic-number-604376) of the atom
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> n1 and n2 are integers where n2 > n1.
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It was later found that n2 and n1 were related to the principal quantum number or energy quantum number. This formula works very well for transitions between energy levels of a hydrogen atom with only one electron. For atoms with multiple electrons, this formula begins to break down and give incorrect results. The reason for the inaccuracy is that the amount of screening for inner [electrons](https://www.thoughtco.com/definition-of-electron-chemistry-604447) or outer electron transitions varies. The equation is too simplistic to compensate for the differences.
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The Rydberg constant, typically denoted as RHR_HRH, is associated with the energy levels of electrons in hydrogen-like atoms. The formula for the Rydberg constant includes the term for the reduced mass of the electron-nucleus system, which is crucial for accurate calculations of spectral lines.
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The reduced mass (μ\muμ) is defined as:
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μ=memNme+mN\mu = \frac{m_e m_N}{m_e + m_N}μ=me+mNmemN
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where:
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- mem_eme is the mass of the electron,
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- mNm_NmN is the mass of the nucleus (like the proton in hydrogen).
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When dealing with the Rydberg formula, the use of reduced mass accounts for the fact that both the electron and nucleus move, rather than assuming the nucleus is fixed.
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The term −1-1−1 in the Rydberg formula arises from the quantum mechanical treatment of the system, specifically when calculating the energy levels. It reflects the energy state adjustments due to the interaction between the electron and nucleus, leading to the overall expression for the energy levels being proportional to −RHμn2-\frac{R_H \mu}{n^2}−n2RHμ, where nnn is the principal quantum number.
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In summary, the inclusion of −1-1−1 relates to the energy being negative, indicating that the electron is in a bound state within the atom. The use of reduced mass further ensures that this model accurately describes the behavior of the electron in relation to the nucleus.
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[](https://light-measurement.com/images/wavelength-range.jpg) |