Solve the given problems as indicated. If an electric...

Solve the given problems as indicated. If an electric discharge is passed through hydrogen gas, a spectrum of isolated parallel lines, called the Balmer series, is formed. See Fig. $30.3 .$ The wavelengths $\lambda$ (in $\mathrm{nm}$ ) of the light for these lines is given by the formula $\frac{1}{\lambda}=1.097 \times 10^{-2}\left(\frac{1}{2^{2}}-\frac{1}{n^{2}}\right) \quad(n=3,4,5, \ldots)$ Find the wavelengths of the first three lines and the shortest wavelength of all the lines of the series.

Solve the given problems as indicated.
If an electric discharge is passed through hydrogen gas, a spectrum of isolated parallel lines, called the Balmer series, is formed. See Fig. $30.3 .$ The wavelengths $\lambda$ (in $\mathrm{nm}$ ) of the light for these lines is given by the formula $\frac{1}{\lambda}=1.097 \times 10^{-2}\left(\frac{1}{2^{2}}-\frac{1}{n^{2}}\right) \quad(n=3,4,5, \ldots)$ Find the wavelengths of the first three lines and the shortest wavelength of all the lines of the series.

Key Concepts

Emission Spectrum

An emission spectrum is generated when electrons in an atom transition from a higher to a lower energy state, emitting photons with characteristic wavelengths. Each element has its own distinct set of energy levels, which results in a unique pattern of spectral lines. This property is fundamental to identifying elements based on their light emissions and is widely used in fields such as astrophysics and chemical analysis.

Balmer Series

The Balmer series is a subset of the hydrogen emission spectrum corresponding to electron transitions from higher energy levels (n > 2) down to the n = 2 energy level. These transitions result in spectral lines that are primarily in the visible region of the electromagnetic spectrum. The Balmer series was historically significant as it helped lay the groundwork for the development of quantum theory by illustrating the quantization of atomic energy levels.

Quantum Energy Levels

In atomic physics, electrons within an atom occupy discrete energy levels. Transitions between these levels involve the absorption or emission of photons with energies corresponding to the difference between the initial and final states. This quantization of energy explains the appearance of specific spectral lines, rather than a continuous spectrum, as electrons drop from higher to lower levels.

Rydberg Formula

The Rydberg formula provides a mathematical relationship for determining the wavelengths of the spectral lines of hydrogen and hydrogen-like elements. It relates the inverse wavelength of the emitted or absorbed light to the difference between the inverse squares of two quantum numbers, using the Rydberg constant. This formula plays a crucial role in predicting and understanding the patterns observed in hydrogen's emission spectrum.

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