Figure 37.7 identified the wavelengths of four lines in the Balmer series of hydrogen. Predict the wavelength of the fifth line in the spectrum.

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Understand the Balmer series: The Balmer series describes the wavelengths of light emitted when an electron in a hydrogen atom transitions from a higher energy level (n > 2) to the second energy level (n = 2). The formula for the wavelength of the emitted light is given by the Rydberg formula:

\frac{1}{λ} = R (\frac{1}{n^{2}} - \frac{1}{m^{2}})

, where R is the Rydberg constant (approximately 1.097 × 10⁷ m⁻¹), n = 2 for the Balmer series, and m > 2.

Identify the fifth line in the Balmer series: The fifth line corresponds to the transition where the electron moves from the energy level m = 7 to n = 2. This is because the first four lines correspond to m = 3, 4, 5, and 6, respectively.

Substitute the values into the Rydberg formula: Use n = 2 and m = 7 in the formula. The equation becomes:

\frac{1}{λ} = R (\frac{1}{2^{2}} - \frac{1}{7^{2}})

Simplify the terms inside the parentheses: Calculate the values of

\frac{1}{2^{2}}

and

\frac{1}{7^{2}}

. Subtract the second term from the first to find the difference.

Find the wavelength: After calculating the difference, take the reciprocal to find

λ

. This will give the wavelength of the fifth line in the Balmer series. Ensure the units are consistent, and convert the result to nanometers (1 nm = 10⁻⁹ m) if needed.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Balmer Series

The Balmer series is a set of spectral lines corresponding to transitions of electrons in a hydrogen atom from higher energy levels to the second energy level. These transitions emit light at specific wavelengths, which can be calculated using the Rydberg formula. The series is significant in understanding atomic structure and the quantization of energy levels.

Rydberg Formula

The Rydberg formula provides a mathematical relationship to predict the wavelengths of spectral lines in hydrogen and other hydrogen-like atoms. It is expressed as 1/λ = R_H (1/n1² - 1/n2²), where R_H is the Rydberg constant, n1 and n2 are integers representing the energy levels. This formula is essential for calculating the wavelengths of lines in the Balmer series.

Energy Level Transitions

Energy level transitions refer to the movement of electrons between different energy states within an atom. When an electron transitions from a higher energy level to a lower one, it releases energy in the form of light, resulting in the emission spectrum. Understanding these transitions is crucial for predicting the wavelengths of emitted light in the Balmer series.