Hydrogen Spectrum: Lyman, Balmer, Paschen, Brackett

The hydrogen spectrum refers to the spectrum of light emitted or absorbed by hydrogen atoms. It is an example of the line emission spectrum or atomic emission spectrum.

When an electric discharge is passed through hydrogen gas contained in a discharge tube under the low pressure, the hydrogen atoms absorb energy. As a result, the electrons in these atoms move to higher energy levels, depending on the amount of energy absorbed.

For example, the electrons from the first energy level (n = 1) may get excited to the second (n = 2), third (n = 3), fourth (n = 4), or at a higher energy level. These higher energy states are unstable and the excited electrons in the higher energy levels quickly return to the lower energy levels. During this process, the energy so released is given out in the form of bluish light (i.e. electromagnetic radiation) of certain definite frequencies (or wavelengths).

When this light radiation is passed through a prism or diffraction grating, a discontinuous line spectrum is obtained which consists of several isolated sharp lines. The light radiation emitted is then analyzed with the help of a spectroscope.

The bright lines recorded on the photographic plate forms the atomic emission spectrum of hydrogen as shown in the below figure. These lines correspond to specific wavelengths, which are found in different regions of the electromagnetic spectrum, such as visible, ultraviolet (UV), and infrared (IR).

J.J. Balmer in 1884 observed the four prominent colored lines in the visible spectrum of the hydrogen atom, which belong to the Balmer series. These lines are produced when electrons transition from higher energy levels (n₂ = 3, 4, 5, 6) to the second energy level (n₁ = 2) in the hydrogen atom. The details of the four lines are as follows:

• a red line with a wavelength of 6563 Å.
• a blue-green line with a wavelength 4861 Å.
• a blue line with a wavelength 4340 Å.
• a violet line with a wavelength 4102 Å.

Origin of Spectral Lines in Hydrogen Spectrum: Ṣpectral Series

The hydrogen spectrum consists of several series of lines named by the name of scientists who discovered them. All these lines fall into different regions of the electromagnetic spectrum. The lines observed in the emission spectrum of hydrogen can be classified into five series that are as:

The complete spectrum of hydrogen is shown in the below figure.

Rydberg Formula for Wavelength Calculation

The Rydberg formula presented a mathematical formula to calculate the wavelengths of spectral lines in the hydrogen spectrum. It is given by:

Where,

• R is a universal constant known as Rydberg’s constant, whose value is 109678 cm^-1 or 1.097 × 10⁷ m⁻¹ for hydrogen.
• n₁ is the lower energy level (smaller orbit).
• n₂ is the higher energy level (larger orbit).
• n₁ and n₂ are integers such as n₂ > n₁.

For a given spectral series, n₁ remains constant, while n₂ varies from line to line in the same series.
The value of n₁ = 1, 2, 3, 4 and 5 for the Lyman, Balmer, Paschen, Brackett and Pfund series, respectively. n₂ is greater than n₁ by at least 1.

From the above mathematical equation, it is clear that the wave number and wavelength of the radiation absorbed or emitted depends only on the values of n1 and n2.

Values of n1 and n2 for various series:

The locations of the spectral lines in the hydrogen spectrum depend on the energy levels involved in the electronic transitions. The wavelength (or frequency) of the emitted or absorbed radiation is determined by the difference in energy between the initial and final energy levels of the electron.

When an electron transitions from the nth excited state to the first energy level (n = 1), the maximum number of spectral lines can be calculated using the formula:

This formula occurs because the electron can transition to any lower energy level, and each transition corresponds to a spectral line. For example, if an electron falls from the 5th energy level (n = 5) to the first energy level (n = 1):

Thus, 10 spectral lines of H-spectrum are possible. Each line corresponds to a transition, such as n = 5 → n = 4, n = 5 → n = 3, n = 5 → n = 2, and so on.

Spectral Lines of Hydrogen Spectrum

The electronic transitions which produce various series of spectral lines in the hydrogen spectrum are shown in the below diagram. The below diagram shows the Lyman, Balmer, Paschen, Brackett, and Pfund series.

1. Lyman Series:

• Lyman series is the first series of spectral series of hydrogen spectrum.
• It was found out in the ultraviolet (UV) region in 1898 by the American physicist Theodore Lyman.
• In the Lyman series, the value of n₁ is always 1, which represents the ground state of the hydrogen atom. The value of n₂ can be any integer greater than 1 (such as 2, 3, 4, etc.), which represents the excited states of the electron present in the hydrogen atom.
• The spectral lines of this series are produced when the electrons transitions 2nd, 3rd, 4th and higher energy level to the first energy level (n₁ = 1).
• The energy difference is released in the form of electromagnetic radiation, which corresponds to the spectral lines observed in the ultraviolet (UV) region of the hydrogen spectrum.
• If the electron goes from n₂ = 2 to n₁ = 1, it is the first spectral line in the Lyman series.
• If the electron goes from n₂ = 3 to n₁ = 1, this is the second line in the Lyman series.
• If the electron moves from n₂ = 4 to n₁ = 1, this is the third spectral line in the Lyman series, and so on.
• 1/λ = R_H (1/1² – 1/n₂²) where n₂ > 1 always.
• The wavelength of marginal line = n₁²/R_H for all series. So, for Lyman series λ = 1/R_H

2. Balmer Series:

• Balmer series is the second series of spectral series of hydrogen spectrum.
• It was found out in the visible region in 1885 by the Swiss mathematician and physicist Johann Jakob Balmer.
• In the Balmer series, the value of n₁ is always 2, which represents the ground state of the hydrogen atom. The value of n₂ can be any integer greater than 1 (such as 3, 4, 5, etc.), which represents the excited states of the electron present in the hydrogen atom.
• The spectral lines of this series are produced when the electrons transitions 3rd, 4th, or higher energy level to the second energy level (n₁ = 2).
• The energy difference is released in the form of electromagnetic radiation, which corresponds to the spectral lines observed in the visible region of the hydrogen spectrum.
• If the electron goes from n₂ = 3 to n₁ = 2, it is the first spectral line in the Balmer series.
• If the electron goes from n₂ = 4 to n₁ = 2, this is the second line in the Balmer series.
• If the electron moves from n₂ = 5 to n₁ = 3, this is the third spectral line in the Balmer series, and so on.
• 1/λ = R_H (1/2² – 1/n₂²) where n₂ > 2 always.
• The wavelength of marginal line = n₁²/R_H = 2²/R_H = 4/R_H for all series.

3. Paschen Series:

• Paschen series is the third series of spectral series of hydrogen spectrum.
• It was discovered in the infrared (IR) region in 1892 by the German physicist Friedrich Paschen.
• In the Paschen series, the value of n₁ is always 3, which represents the ground state of the hydrogen atom. The value of n₂ can be any integer greater than 1 (such as 4, 5, 6, etc.), which represents the excited states of the electron present in the hydrogen atom.
• The spectral lines of this series are formed when the electrons transitions 4th, 5th, 6th or higher energy level to the third energy level (n₁ = 3).
• If the electron goes from n₂ = 4 to n₁ = 3, it is the first spectral line in the Paschen series.
• If the electron goes from n₂ = 5 to n₁ = 3, this is the second line in the Paschen series.
• If the electron moves from n₂ = 6 to n₁ = 3, this is the third spectral line in the Paschen series, and so on.
• 1/λ = R_H (1/3² – 1/n₂²) where n₂ > 3 always.
• The wavelength of marginal line of Paschen series = n₁²/R_H = 3²/R_H = 9/R_H for all series.

4. Brackett Series:

• Brackett series is the fourth series of spectral series of hydrogen spectrum.
• It was discovered in the infrared (IR) region in 1922 by the American physicist Frederick Sumner Brackett.
• In the Brackett series, the value of n₁ is always 4, which represents the ground state of the hydrogen atom. The value of n₂ can be any integer greater than 1 (such as 5, 6, 7, etc.), which represents the excited states of the electron present in the hydrogen atom.
• The spectral lines of this series are formed when the electrons jumps from 5th, 6th or higher energy level to the fourth energy level (n₁ = 4).
• If the electron jumps from n₂ = 5 to n₁ = 4, it is the first spectral line in the Brackett series.
• If the electron jumps from n₂ = 6 to n₁ = 4, this is the second line in the Brackett series.
• If the electron jumps from n₂ = 7 to n₁ = 5, this is the third spectral line in the Brackett series, and so on.
• 1/λ = R_H (1/4² – 1/n₂²) where n₂ > 4 always.
• The wavelength of marginal line of Brackett series = n₁²/R_H = 4²/R_H = 16/R_H .

5. Pfund Series:

• Pfund series is the fifth series of spectral series of hydrogen spectrum.
• It was discovered in the infrared (IR) region in 1924 by the American physicist August Herman Pfund.
• In the Pfund series, the value of n₁ is always 5, which represents the ground state of the hydrogen atom. The value of n₂ can be any integer greater than 1 (such as 6, 7, 8, etc.), which represents the excited states of the electron present in the hydrogen atom.
• The spectral lines of this series are formed when the electrons goes from 6th, 7th, or higher energy level to the fifth energy level (n₁ = 5).
• If the electron jumps from n₂ = 6 to n₁ = 5, it is the first spectral line in the Pfund series.
• If the electron jumps from n₂ = 7 to n₁ = 5, this is the second line in the Pfund series.
• If the electron jumps from n₂ = 8 to n₁ = 5, this is the third spectral line in the Pfund series, and so on.
• 1/λ = R_H (1/5² – 1/n₂²) where n₂ > 5 always.
• The wavelength of marginal line of Brackett series = n₁²/R_H = 5²/R_H = 25/R_H .

Solved Examples based on Hydrogen Spectrum

Example 1: Calculate the frequency and wavelength of light emitted when the electron in the hydrogen atom jumps from the energy level with n = 3 to the energy level with n = 1? Which region of the electromagnetic spectrum does correspond to this wavelength?

Solution:

According to Rydberg formula,

Here, R = 109677 cm^-1 and n₂ = 3, n₁ = 1.

Wavenumber (ṽ) = (109677 * 8) / 9 = 97490.7 cm^-1

Wavelength (λ) = 1/ṽ = 1/97490.7 = 103 * 10^-7 cm^-1 = 103 * 10^-9 m = 103 nm.

Frequency (f) = c/λ = 3 * 10⁸ ms^-1 / 103 * 10^-9 m = 2.91 * 10¹⁵ s^-1

If the wavelength of light emitted by an electron in a hydrogen atom is 103 nm, it falls in the ultraviolet (UV) region of the electromagnetic spectrum.

Example 2: Find out the electronic state from which an electron in the hydrogen atom undergoes transition to emit radiations with wavelength 1212 Å and give a spectral line in the Lyman series of hydrogen spectrum (R = 109678 cm^-1).

Solution:

Wavelength (λ) = 1212 Å = 1212 * 10^-8 cm (given)

We know that wavenumber (ṽ) = 109678 cm^-1 (1/n₁² – 1/n₂²)

Or, 1/λ = 109678 cm^-1 (1/n₁² – 1/n₂²)

But, λ = 1212 * 10^-8 cm and for Lyman series, n₁ = 1.

So, 1/1212 *10^-8 = 109678 cm^-1 (1/1² – 1/n₂²)

1/n₂² = 1 – 3/4 = 1/4

n₂² = 4 or n₂ = 2.

Hence, 2 is the second electronic state from which an electron jumps into the first orbit to emit radiation.

Example 3: Calculate the wavelength of the light emitted when an electron in a hydrogen atom transitions from the energy level n = 4 to n = 2. Also, determine the color corresponding to this wavelength.

Solution:

According to Rydberg formula,

Here, R = 109677 cm^-1 and n₂ = 4, n₁ = 2.

Wavenumber (ṽ) = (109677 * 3) / 16 = 20564.4375 cm^-1

Wavelength (λ) = 1/ṽ = 1/20564.4375 = 486 * 10^-7 cm^-1 = 486 nm.

A wavelength of 486 nm falls in the visible light spectrum, specifically in the blue-green region.

Example 4: Find out the wavelength of the second line in the Balmer series if the wavelength of the first spectral line in this series is 656.1 nm.

Solution:

For the first spectral line in the Balmer series, n₁ = 2 and n₂ = 3.

Then, for the second spectral line in the Balmer series, n₁ = 2 and n₂ = 4.

According to Rydberg formula, 1/λ = R_H (1/n₁² – 1/n₂²)

1/656.1 = R_H (1/2² – 1/3²) = 5R_H/36 . . . . (1)

1/λ = R_H (1/2² – 1/4²) = 3R_H/16 . . . (2)

Dividing eq. (1) by (2)

λ/656.1 nm = 5R_H/36 * 16/3R_H = 80/108

λ = 80 * 656.1 nm / 108 = 486.0 nm.

Example 5: What is the maximum number of spectral lines when the excited electrons of a hydrogen atom in n = 6 falls to the ground state?

Solution:

Maximum number of spectral lines = n(n – 1)/2 = 6(6 – 1)/2 = 6 * 5 / 2 = 15.

MCQ on Hydrogen Spectrum

1. Which series in the hydrogen spectrum lies in the visible region?

• a) Lyman
• b) Balmer
• c) Paschen
• d) Brackett

Answer: b) Balmer

2. What is the Rydberg constant in SI units?

• a) 6.626 × 10^-34 J
• b) 1.097 × 10⁷ m^-1
• c) 9.109 × 10^-31 kg
• d) 1.602 × 10^-19 C

Answer: b) 1.097 × 10⁷ m^-1

3. In the Lyman series, which energy level does the electron transition to?

• a) n = 1
• b) n = 2
• c) n = 3
• d) n = 4

Answer: a) n = 1

4. What causes the hydrogen spectrum to form?

• a) Continuous radiation from hydrogen
• b) Energy transitions of hydrogen electrons
• c) Reflection of light by hydrogen
• d) Hydrogen gas breaking into ions

Answer: b) Energy transitions of hydrogen electrons

5. In which region does the Balmer series fall?

• a) Ultraviolet
• b) Visible
• c) Infrared
• d) X-ray

Answer: b) Visible

6. Which series is observed in the ultraviolet region?

• a) Lyman
• b) Balmer
• c) Paschen
• d) Pfund

Answer: a) Lyman

7. Which of the following series are part of the emission spectrum of hydrogen?

• a) Balmer Series
• b) Paschen Series
• c) Brackett Series
• d) Pfund Series
• e) All of the above

Answer: e) All of the above.

Note that the Balmer, Paschen, Brackett, and Pfund series are all observed in the emission spectrum of hydrogen.

8. Which of the following is true about the hydrogen spectrum?

• a) The hydrogen spectrum is an example of a continuous spectrum.
• b) The hydrogen spectrum is an example of a line spectrum.
• c) The hydrogen spectrum is an example of a broad-spectrum emission.
• d) The hydrogen spectrum does not produce any spectral lines.

Answer: b) The hydrogen spectrum is an example of a line spectrum.

The hydrogen spectrum consists of discrete spectral lines. These spectral lines are observed as a line spectrum and not as a continuous spectrum.