Atomic absorption and emission spectrum

Atomic spectrum is an impotent tool for the determination of the structure of an atom. Bohr’s theory explained the existence of various lines in the hydrogen spectrum. But the emission energy of hydrogen always produces a series of line spectra.

Thus the measurements of the atomic emission spectrum are feasible. Because each hydrogen atom has a definite energy level in which the electron can stay. This is the ground state of an atom.

Atomic emission spectrum of hydrogen

So on the addition of thermal or electrical energy, the electron moved to the higher energy level of an atom.

Thus when this excited electron returns to the ground state it emits a definite frequency of radiation.

Frequency of atomic spectra

In 1901 plank proposed a hypothesis in which he connected photon energy and frequency of the emitted light.

ΔE = hν
or, ν = ΔE/h
where ν = frequency of emitted light
h = plank constant

The emission spectrum of hydrogen

The energy corresponding to a particular line in the emission and absorption spectra or spectrum of hydrogen is the energy difference between the ground level and the exited level.

Bohr’s theory provides the energy of an electron at a particular energy level. Thus the energy of an electron in the hydrogen

    \[ E _{n}= \frac{-2\pi ^{2}me^{4}}{n^{2}h^{2}} \]

But ΔE = E2 – E1

    \[ \Delta E = \frac{-2\pi ^{2}me^{4}}{n_{2}^{2}h^{2}}-\left ( \frac{-2\pi ^{2}me^{4}}{n_{1}^{2}h^{2}} \right ) \]

    \[ \boxed{=\frac{2\pi ^{2}me^{4}}{h^{2}}\left ( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right )} \]

But the frequency of emitted light from the electromagnetic spectrum related to energy by plank equation

ν = ΔE/h

    \[ \Delta E = \frac{2\pi ^{2}me^{4}}{h^{2}}\left ( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right ) \]

    \[ \nu = \frac{2\pi ^{2}me^{4}}{h^{3}}\left ( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right ) \]

    \[ \boxed{\nu = R\left ( \frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}} \right )} \]

where R = Rydberg constant

Rydberg constant value for hydrogen

Rydberg equation can measure the values of rydberg constant for the hydrogen.

    \[ \therefore R = \frac{2\pi ^{2}me^{4}}{h^{3}} \]

= 3.2898 × 1015 cycles sec-1

    \[ \lambda \nu = c\, and\, \nu= \frac{c}{\lambda }= c\bar{\nu } \]

where wavenumber = ν/c
and c = velocity of light

Thus the frequency (ν) indicates the number of waves passing a given point per second and is expresses as cycles per second.

But wavenumber stands for the number of waves connected in unit length that is per centimeter (cm-1) or per meter (m-1).

∴ R = 109737 cm-1
= 10973700 m-1

The experimental values showing a remarkable agreement between the experiment and values from the atomic theory.

Emission and absorption spectra of hydrogen

Putting n = 1, n = 2, n = 3, etc in the Rydberg equation we get the energies of the different stationary states for the hydrogen electron.

Thus the transitions energies calculated from the Rydberg equation exhibited several series of lines.

How many spectral lines are in hydrogen?

Hydrogen is given several spectral lines because any given sample of hydrogen contains an almost infinite number of atoms.

Under normal conditions, the electron of each hydrogen atom remains in the ground state near the nucleus of an atom that is n = 1 (K – Shell).

When heat or electrical energy is supplied to hydrogen, it absorbed different amounts of energy to give absorption spectra or spectrum.

Some of the atoms absorbed such energy to shift their electron to third energy level, while some others may absorb a large amount of energy to shift their electron to the fourth, fifth, sixth and seventh energy levels.

But the electrons of hydrogen in the excited state are relatively unstable and hence drop back to the ground state by the emission of energy in the form of the spectrum. Thus for different drop back given different lines of the hydrogen spectrum.

Line spectrum of the hydrogen atom

Transition to the ground state to excited states constitute the Lyman, Balmer, Pashen, Brackett, Pfund and Hampe series of spectral lines. These emission spectraum lie from the ultraviolet region to the far IR region.

    \[ \boxed{\bar{\nu } = \frac{1}{\lambda }= R\left ( \frac{1}{n_{1}^{2}}- \frac{1}{n_{2}^{2}} \right )} \]

When we putting the values of n1 and n2 on the above equation we obtained the frequency of different spectral lines.

Series of lines n1 n2 Spectral region Wavelength
Lyman Series 1 > 1 UV < 4000
Balmer Series 2 > 2 Visible 4000 to 7000
Paschen Series 3 > 3 Near IR > 7000
Brackett Series 4 > 4 Far IR > 7000
Pfund Series 5 > 5 Far IR > 7000

This is the theoretical basis for the formation of emission line of the hydrogen spectrum. The above discussion also tells us that as we go to the higher to still higher energy level the energy gap between two-level is continued decreases.