(a) Show that the various lines in the hydrogen spectrum can...

(a) Show that the various lines in the hydrogen spectrum can be expressed in angstroms as $$ \lambda(\dot{A})=\frac{911 \mathbf{n}_{1}^{2} \mathbf{n}^{2}}{\mathbf{n}^{2}-\mathbf{n}_{1}^{2}} $$ where $\mathbf{n}_{1}=1$ for the Lyman series, 2 for the Balmer series, and 3 for the Paschen series. The integer $\mathbf{n}$ is larger than $\mathbf{n}_{1}$. (b) Calculate $\lambda$ for the Lyman series to $\mathbf{n}=5$, the Balmer series to $\mathbf{n}=7$, and the Paschen series to $\mathbf{n}=10 .$ Plot the results as in Fig. $2-2 .$ What are the wavelength limits for each of the three series?

(a) Show that the various lines in the hydrogen spectrum can be expressed in angstroms as
$$
\lambda(\dot{A})=\frac{911 \mathbf{n}_{1}^{2} \mathbf{n}^{2}}{\mathbf{n}^{2}-\mathbf{n}_{1}^{2}}
$$
where $\mathbf{n}_{1}=1$ for the Lyman series, 2 for the Balmer series, and 3 for the Paschen series. The integer $\mathbf{n}$ is larger than $\mathbf{n}_{1}$.
(b) Calculate $\lambda$ for the Lyman series to $\mathbf{n}=5$, the Balmer series to $\mathbf{n}=7$, and the Paschen series to $\mathbf{n}=10 .$ Plot the results as in Fig. $2-2 .$ What are the wavelength limits for each of the three series?

Key Concepts

Hydrogen Spectrum

The hydrogen spectrum is the set of discrete wavelengths of light emitted by hydrogen atoms when their electrons make transitions between quantized energy levels. This phenomenon was one of the experimental cornerstones that led to the development of quantum theory, as it demonstrated that electron energies in atoms are not continuous but occur in specific quantized states.

Rydberg Formula

The Rydberg formula is an empirical equation that relates the wavelengths of the spectral lines in hydrogen to the inverse of the difference between the squares of integers representing the electron’s initial and final energy levels. It is a cornerstone of atomic spectroscopy, providing precise predictions of the spectral line positions in the hydrogen atom.

Quantum Numbers and Energy Transitions

Quantum numbers (such as n and n1) characterize the energy levels in an atom, where electrons reside in orbits at certain distances from the nucleus. Transitions between these quantum states involve the absorption or emission of photons with energies (or wavelengths) determined by the differences in energy levels, as described by the Rydberg formula.

Spectral Series (Lyman, Balmer, Paschen)

Spectral series group together lines in the hydrogen spectrum based on the final stationary energy level of the electron transition. For example, the Lyman series involves transitions with the electron ending at the ground state (n1 = 1), the Balmer series ends at the first excited state (n1 = 2), and the Paschen series ends at a higher energy level (n1 = 3). Each series covers a different region of the electromagnetic spectrum.

Series Limit

The series limit in a spectral series corresponds to the wavelength (or frequency) reached when the electron transitions from an infinitely high energy level down to a fixed lower energy level. This limit represents the boundary beyond which no spectral lines of that series can be observed, and it is derived by taking the appropriate limit in the Rydberg formula as the initial quantum number approaches infinity.

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