Atomic Spectrum: Line Spectrum
When an elemental gas or vapour is placed in a discharge tube at a pressure of a few millimetres of mercury and passed through it, the tube becomes a bright light source. It is commonly called a discharge lamp. Neon, sodium, mercury, and various halogen gas discharge lamps are used in everyday work.
When the light from the discharge lamp is dispersed with the help of a prism or some other device, the light is separated into different primary colours according to different wavelengths. This is called the atomic spectrum. This atomic spectrum is observed by placing it on a suitable screen. In addition to visible light, this spectrum usually includes ultraviolet and infrared rays. The wavelength of each elemental ray can be determined with the help of a special test system. Atomic spectra are different for different elements.
The characteristics of the atomic spectrum are that this spectrum is generally composed of several narrow and bright lines, which are separated from each other; that is, between any two bright lines there is a slightly dark area. This spectrum is called the line spectrum.
On the other hand, the spectrum from a heated solid-state light source (e.g., an incandescent tungsten lamp) is a continuous spectrum. In this, different colours of light create a continuous brightness on the screen, with no dark areas in between. Again, molecular spectra are usually band spectra. Instead of a narrow, bright line, a slightly wider, bright band is formed on the screen, with a dark area between any two bands.
Balmer Series of Hydrogen Spectrum
Four bright lines are found in the visible region of the atomic spectrum of hydrogen. The experimental values of their wavelengths are 6563 Å, 4861 Å, 4341 Å, and 4102 Å. These four lines are collectively known as the Balmer series of the hydrogen spectrum. Long before Bohr’s model was proposed, scientist Balmer tried to express these wavelengths through a relationship. The relationship he refers to is,
Here, λ = wavelength of the spectrum; R = constant, which is later referred to as the Rydberg constant of hydrogen; and n = 3, 4, 5, …, ∞. A unit length will have a full wave, indicated by the ratio 1/λ. Hence, 1/λ is called the wave number.
By setting R = 109706 cm-1, the test wavelength of the spectral lines is obtained from relation (1). For example,
If n = 3, λ = 6563 Å; if n = 4, λ = 4861 Å.
If n = 5, λ = 4341 Å; if n = 6, λ = 4102 Å
Also, setting n = 7, …, ∞ gives different values of λ; these also belong to the Balmer series. But these wavelengths are in the ultraviolet region, not the visible region. Balmer was able to arrange the line spectrum of hydrogen with precise rules but could not determine theoretically the relation (1).
Other Series of Hydrogen Spectrum
Lyman Series
This series index relationship is,
The values of λ obtained from this relation, using the same value of R, correspond to the wavelengths of the lines found in the ultraviolet region of the hydrogen spectrum. For example,
If n = 2, λ = 1216 Å; If n = 3, λ = 1026 Å
Paschen Series
This series index relationship is,
From this relationship the wavelengths of some spectral lines in the infrared region of the hydrogen spectrum can be obtained. For example, if n = 4, λ = 1875 Å.
Brackett Series & Pfund Series
Besides the Paschen series, there are other series in the infrared region of the hydrogen spectrum, such as the Brackett series and the Pfund series. But in this case the auxiliary Brackett series index relation is,
and the Pfund series index relationship is,
Rydberg Formula
Some times after the discovery of the Balmer series, Rydberg expressed the following general equation for the series of the spectrum. This is known as the Rydberg formula.
Where R (Rydberg constant) is a constant for a particular element, a and b are characteristic constants for a particular series, m is an integer which is constant for a given series and n is a variable integer whose different values correspond to different lines of the series. With the help of equation no. (2), the spectral classes of most elements can be expressed almost exactly.
Explanation of the Hydrogen Spectrum from Bohr's Theory
Rydberg Constant
The isolated energy levels of hydrogen atoms are,
c = speed of light in vacuum = 3 × 10⁸ m/s and n = 1, 2, 3, …….
ground level (n = 1 level) energy,
E1 = -Rch
By substituting the exact values of the various constants in equation no. (2), we get,
R ≈ 1.09625 × 10⁷ m-1
Note that the value of R is slightly lower than the value of Rydberg constant (1.09625 × 10⁷ m-1) obtained from the Bammer class analysis of the hydrogen spectrum. Correcting for the mass of the hydrogen nucleus no longer makes this difference. So, the R constant is actually the Rydberg constant and equation no. (2) is its index.
The value of Rydberg constant in CGS method is R = 109625 cm-1.
Wavelength of the Emitted Radiation
Suppose, the electron in hydrogen atom is transferred from a higher energy level Ei to a lower energy level Ef. According to the board, this would result in the emission of a photon from the hydrogen atom. If the frequency of this photon is ν (wavelength λ = c/ν),
Balmer Series
If the electron of hydrogen atom falls from any energy level E3, E4, E5, … etc. to E2 energy level, then in equation (4). i = 3, 4, 5, …….. and f = 2, the equation becomes,
This relationship indicates the Balmer series of the atomic spectrum of hydrogen.
Other Series
Similarly, putting i = 2, 3, 4, …… and f = 1 in equation no. (4), we get Lyman Series,
Again, putting i = 4, 5, 6, …. and f = 3 gives the Paschen Series,
Putting i = 5, 6, 7, ….. and f = 4 gives the Brackett Series,
Putting i = 6, 7, 8, ….. and f = 5 gives the Pfund Series,
Thus, it was found that the relations that Balmer and other scientists had shown for the various wavelengths of the atomic spectrum of hydrogen could be established accurately from Bohr’s theory. In fact, despite significant deviations from classical physics, the ability to accurately explain the hydrogen spectrum was the basis for the success of Bohr’s theory. Different classes of the atomic spectrum of hydrogen are shown in the figure.
Absorption Spectrum of Hydrogen
Let us say that when an electron in a hydrogen atom falls from energy level Ei to energy level Ef, a photon of wavelength λ is emitted. Then, according to equation no. (3),
Ei – Ef = hc/λ
So, in the reverse process, if the atom absorbs a photon of wavelength λ, the electron will rise again from the Ef to the Ei solid state. Since the spacing of the energy levels is predetermined, the wavelength of the absorbed photon and the emitted photon will be exactly the same. As a result, if the continuous spectrum from a source (e.g., an incandescent tungsten lamp) passes through hydrogen gas, some spectral lines remain dark for absorption. These dark, i.e., black lines are called absorption spectra. The same bright lines in the emission spectrum from a hydrogen emission lamp; the same lines are darkened in the continuous spectrum of other sources when absorbed by hydrogen.