Mathematics ATOMIC SPECTRA AND BOHR MODEL OF THE HYDROGEN ATOM

Topic Covered

`color{blue}{star}` ATOMIC SPECTRA
`color{blue}{star}` BOHR MODEL OF THE HYDROGEN ATOM

ATOMIC SPECTRA

`color{blue} ✍️`As we know, each element has a characteristic spectrum of radiation, which it emits. When an atomic gass or vapor is excited at low pressure, usually by passing an electric current through it, the emitted radiation has a spectrum which contains certain specific wavelengths only.

`color{blue} ✍️`A spectrum of this kind is termed as emission line spectrum and it consists of bright lines on a dark background. The spectrum emitted by atomic hydrogen is shown in Fig. 12.5.



`color{blue} ✍️`Study of emission line spectra of a material can therefore serve as a type of “fingerprint” for identification of the gas. When white light passes through a gas and we analyse the transmitted light using a spectrometer we find some dark lines in the spectrum.

`color{blue} ✍️`These dark lines correspond precisely to those wavelengths which were found in the emission line spectrum of the gas. This is called the absorption spectrum of the material of the gas.

`color{brown}bbul("Spectral series")`
`color{blue} ✍️`We might expect that the frequencies of the light emitted by a particular element would exhibit some regular pattern. Hydrogen is the simplest atom and therefore, has the simplest spectrum.

`color{blue} ✍️`In the observed spectrum, however, at first sight, there does not seem to be any resemblance of order or regularity in spectral lines. But the spacing between lines within certain sets of the hydrogen spectrum decreases in a regular way (Fig. 12.5).

`color{blue} ✍️`Each of these sets is called a spectral series. In 1885, the first such series was observed by a Swedish school teacher Johann Jakob Balmer in the visible region of the hydrogen spectrum. This series is called Balmer series (Fig. 12.6).



`color{blue} ✍️`The line with the longest wavelength, 656.3 nm in the red is called Hα; the next line with wavelength 486.1 nm in the bluegreen is called Hβ, the third line 434.1 nm in the violet is called Hγ; and so on.

`color{blue} ✍️`As the wavelength decreases, the lines appear closer together and are weaker in intensity. Balmer found a simple empirical formula for the observed wavelengths

`color{navy}(1/lamda = R (1/(2^2) - 1/(n^2)))`

......(12.5)

where `λ` is the wavelength, `R` is a constant called the Rydberg constant, and `n` may have integral values `3, 4, 5`, etc.
The value of `R` is `color{purple}(1.097 × 10^7 m^(–1)).` This equation is also called Balmer formula.

`"Paschen series:"`

`color{blue} ✍️`Taking n = 3 in Eq. (12.5), one obtains the wavelength of the `Hα` line:

`color{purple}(1/lamda = 1.097xx 10^7( 1/(3^2) - 1/(n^2))`

`color{purple}(=1.522 xx10^6m^(-1))`

`color{purple}(i.e., lamda= 656.3nm)`

`color{blue} ✍️`For `n = 4,` one obtains the wavelength of `H_β` line, etc. For `n = ∞,` one obtains the limit of the series, at `λ = 364.6 nm.` This is the shortest wavelength in the Balmer series. Beyond this limit, no further distinct lines appear, instead only a faint continuous spectrum is seen. Other series of spectra for hydrogen were subsequently discovered. These are known, after their discoverers, as Lyman, Paschen, Brackett, and Pfund series. These are represented by the formulae:

`color{brown}bbul("Lyman series:")`

`color{navy}(1/lamda = R (1/(1^2) - 1/(n^2)) \ \ \ \ n = 2,3,4)`

......... (12.6)

`color{brown}bbul("Paschen series:")`

`color{blue}(1/lamda = R (1/(3^2) - 1/(n^2)) \ \ \ \ n = 4,5,6)`

......... (12.7)

`color{brown}bbul("Brackett series:")`

`color{blue}(1/lamda = R(1/(4^2)-1/(n^2)) \ \ \ n = 5,6,7)`

........... (12.8)

`color{brown}bbul("Pfund series:")`

`color{navy}(1/lamda = R(1/(5^2)-1/(n^2)))`

............ (12.9)

`color{blue} ✍️`The Lyman series is in the ultraviolet, and the Paschen and Brackett series are in the infrared region. The Balmer formula Eq. (12.5) may be written in terms of frequency of the light, recalling that

`color{purple}(c = v lamda)`

or `color{purple}(1/lamda= v/c)`

`color{blue} ✍️`Thus, Eq. (12.5) becomes

`color{blue}(v = Rc (1/2^2-1/n^2))`

.... (12.10)

`color{blue} ✍️`There are only a few elements (hydrogen, singly ionised helium, and doubly ionised lithium) whose spectra can be represented by simple formula like Eqs. (12.5) – (12.9).

`color{blue} ✍️`Equations (12.5) – (12.9) are useful as they give the wavelengths that hydrogen atoms radiate or absorb. However, these results are empirical and do not give any reasoning why only certain frequencies are observed in the hydrogen spectrum.