Question

The Brackett series lines in the atomic spectrum of hydrogen result from transitions from \(\mathbf{n}>4\) to \(\mathbf{n}=4\) (a) Calculate the wavelength, in nanometers, of a line in this series resulting from the \(\mathbf{n}=5\) to \(\mathbf{n}=4\) transition. (b) In what region of the spectrum are these lines formed?

Short answer

Question: Calculate the wavelength corresponding to the transition from n=5 to n=4 in the Brackett series of the hydrogen atomic spectrum, and determine which region of the spectrum these lines are formed in. Answer: The wavelength corresponding to the transition from n=5 to n=4 in the Brackett series is 1940 nm. The Brackett series lines are formed in the infrared region of the spectrum.

Step-by-step solution

Using the Rydberg formula for hydrogen's atomic spectrum

The Rydberg formula for hydrogen's atomic spectrum is given by: $$\frac{1}{\lambda}=R_H\left(\frac{1}{n^2_1} - \frac{1}{n^2_2}\right)$$ Where \(\lambda\) is the wavelength, \(n_1\) and \(n_2\) are the initial and final energy levels, and \(R_H\) is the Rydberg constant for hydrogen which is equal to \(1.097\times10^7 m^{-1}\). For this problem, we are given the transition from \(n=5\) to \(n=4\). So, \(n_1=4\) and \(n_2=5\).

Calculate the reciprocal of the wavelength

Plugging the values into the Rydberg formula, we get: $$\frac{1}{\lambda}=R_H\left(\frac{1}{4^2} - \frac{1}{5^2}\right)$$ Replacing the Rydberg constant: $$\frac{1}{\lambda} = (1.097\times10^7) \left(\frac{1}{16}-\frac{1}{25}\right)$$

Calculate the wavelength

To find the wavelength in meters, we must take the reciprocal of the value calculated in step 2: $$\lambda = \frac{1}{(1.097\times10^7) (\frac{1}{16}-\frac{1}{25})}$$ Calculating this result, we obtain the wavelength in meters: $$\lambda = 1.94\times10^{-6}\,m$$

Convert wavelength to nanometers

Now we must convert the wavelength from meters to nanometers. There are \(10^9\) nanometers in a meter, so we can multiply the result by this factor: $$\lambda_{nm} = \lambda \times 10^9 = (1.94\times10^{-6})(10^9)$$ This gives us the wavelength in nanometers: $$\lambda = 1940\, nm$$

Solution for part (a)

The wavelength corresponding to the transition from \(n=5\) to \(n=4\) in the Brackett series is \(1940\,nm\).

Identify the region of the spectrum

The electromagnetic spectrum has different regions based on wavelength. To determine in which region the Brackett series lines are formed, we must compare the wavelength calculated in part (a) to those regions: - Ultraviolet: \(10-400\,nm\) - Visible light: \(400-700\,nm\) - Infrared: \(700\,nm - 1\,mm\) With a wavelength of \(1940\,nm,\) the lines corresponding to the Brackett series fall in the infrared region of the spectrum.

Solution for part (b)

The Brackett series lines are formed in the infrared region of the spectrum.

Key concepts

Brackett Series

In the hydrogen atomic spectrum, the Brackett series involves electron transitions where the electrons move to the fourth energy level, denoted as \( n=4 \), from any higher energy level \( n > 4 \). This means transitions could be from levels \( n=5, 6, 7, \ldots \) down to \( n=4 \). It is part of several spectral series, each named after scientists, including Balmer, Lyman, Paschen, and more. The Brackett series specifically is most notable in spectroscopy for its distinct position and characteristics in the electromagnetic spectrum.

This series was named after physicist Frederick Brackett, who discovered it in the early 20th century. An important aspect of the Brackett series is that its spectral lines extend across the infrared region, making them valuable for observing celestial phenomena that cannot be seen in visible light. The study of these series helps scientists understand fundamental processes in atomic physics and astrophysics.

Rydberg Formula

The Rydberg formula is a cornerstone in understanding the hydrogen atomic spectrum. This formula, named after Swedish physicist Johannes Rydberg, is used to calculate the wavelengths of electromagnetic radiation emitted or absorbed by electrons in hydrogen atoms during transitions between energy levels.

The formula is expressed as:

• \( \frac{1}{\lambda}=R_H\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \)
• Here, \( \lambda \) is the wavelength of the spectral line.
• \( R_H \) is the Rydberg constant for hydrogen, approximately \( 1.097\times10^7 \, m^{-1} \).
• \( n_1 \) and \( n_2 \) are the principal quantum numbers of the electron's initial and final energy levels.

The formula showcases how energy differences between levels lead to different wavelengths. For instance, in the Brackett series, transitions from higher levels like \( n=5 \) to \( n=4 \) result in spectral lines of specific wavelengths visible as discrete lines within the hydrogen emission spectrum.

By inserting the appropriate values for \( n_1 \) and \( n_2 \), scientists can calculate exact wavelengths which, when measured, confirm the presence of specific transitions and their series designation.

Infrared Region

The infrared region of the electromagnetic spectrum is a range that extends from the edge of the visible light section to microwave radiation. It includes wavelengths longer than visible light, with typical boundaries around 700 nm to 1 mm. This region is further divided into near-infrared, mid-infrared, and far-infrared areas.

The Brackett series of hydrogen falls within this infrared region, specifically at the longer wavelengths beyond the visible spectrum. With calculated wavelengths, such as the \( 1940 \, nm \) line from the \( n=5 \) to \( n=4 \) transition, these lines clearly lie within the infrared range. As a result, they are not visible to the naked eye but can be detected using specialized instruments.

Infrared radiation is critical in numerous applications, including:

• Remote sensing and climate monitoring.
• Astronomical observations to see through dust clouds in space.
• Medical imaging techniques like thermal imaging.

Understanding the infrared properties of spectral series like the Brackett series enables scientists to delve deeper into areas such as the thermal characteristics and behavior of distant celestial bodies.