Atomic Spectra

All condensed matter (solids, liquids, and dense gases) emit

electromagnetic radiation at all temperatures. Also, this radiation has a

continuous distribution of several wavelengths with different

intensities. This is caused by oscillating atoms and molecules and their

interaction with the neighbours. In the early nineteenth century, it was

established that each element is associated with a characteristic

spectrum of radiation, known as Atomic Spectra. Hence, this suggests

an intimate relationship between the internal structure of an atom and

the spectrum emitted by it.

Atomic Spectra

When an atomic gas or vapour is excited under low pressure by

passing an electric current through it, the spectrum of the emitted

radiation has specific wavelengths. It is important to note that, such a

spectrum consists of bright lines on a dark background. This is an

emission line spectrum. Here is an emission line spectrum of hydrogen

gas:

The emission line spectra work as a ‘fingerprint’ for identification of

the gas. Also, on passing a white light through the gas, the transmitted

light shows some dark lines in the spectrum. These lines correspond to

those wavelengths that are found in the emission line spectra of the

gas. This is the absorption spectrum of the material of the gas.

Spectral Series of Atomic Spectra

Normally, one would expect to find a regular pattern in the

frequencies of light emitted by a particular element. Let’s look at

hydrogen as an example. Interestingly, at the first glance, it is difficult

to find any regularity or order in the atomic spectra. However, on

close observation, it can be seen that the spacing between lines within

certain sets decreases in a regular manner. Each of these sets is a

spectral series.

Five spectral series identified in hydrogen are

1. Balmer Series

2. Lyman Series

3. Paschen Series

4. Brackett Series

5. Pfund Series

Further, let’s look at the Balmer series in detail.

Balmer Series

In 1885, when Johann Balmer observed a spectral series in the visible

spectrum of hydrogen, he made the following observations:

● The longest wavelength is 656.3 nm

● The second longest wavelength is 486.1 nm

● And the third is 434.1 nm

● Also, as the wavelength decreases the lines appear closer

together and weak in intensity

● He found a simple formula for the observed wavelengths:

Further, for n=∞, you can get the limit of the series at a wavelength of

364.6 nm. Also, you can’t see any lines beyond this; only a faint

continuous spectrum.Furthermore, like the Balmer’s formula, here are

the formulae for the other series:

Lyman Series

Paschen Series

Brackett Series

Pfund Series

Solved Examples for You

Question: The emission line spectra works as a ______________ for

identification of the gas.

A. clue

B. fingerprint

C. deterrent

D. confuser

Solution: Fingerprint

Question: Name the five spectral series identified in hydrogen gas?

Solution: Balmer series, Lyman series, Paschen series, Brackett series

and Pfund series.

Alpha-Particle Scattering and Rutherford’s Nuclear Model of Atom

In 1911, Rutherford, along with his assistants, H. Geiger and E.

Marsden, performed the Alpha Particle scattering experiment, which

led to the birth of the ‘nuclear model of an atom’ – a major step

towards how we see the atom today.

J.J Thomson’s Plum-pudding Model

In 1897-98, the first model of an atom was proposed by J.J. Thomson.

Famously known as the Plum-pudding model or the watermelon

model, he proposed that an atom is made up of a positively charged

ball with electrons embedded in it. Further, the negative and positive

charges were equal in number, making the atom electrically neutral.

Figure 1 shows what Thomson’s plum-pudding model of an atom

looked like. Ernest Rutherford, a former research student working

with J.J. Thomson, proposed an experiment of scattering of alpha

particles by atoms to understand the structure of an atom.

Rutherford, along with his assistants – H. Geiger and E. Marsden –

started performing experiments to study the structure of an atom. In

1911, they performed the Alpha particle scattering experiment, which

led to the birth of the ‘nuclear model of an atom’ – a major step

towards how we see the atom today.

Figure 1. Source: Wikipedia

The Alpha Particle Scattering Experiment

They took a thin gold foil having a thickness of 2.1×10-7 m and placed

it in the centre of a rotatable detector made of zinc sulfide and a

microscope. Then, they directed a beam of 5.5MeV alpha particles

emitted from a radioactive source at the foil. Lead bricks collimated

these alpha particles as they passed through them.

After hitting the foil, the scattering of these alpha particles could be

studied by the brief flashes on the screen. Rutherford and his team

expected to learn more about the structure of the atom from the results

of this experiment.

Source: Wikipedia

Observations

Here is what they found:

● Most of the alpha particles passed through the foil without

suffering any collisions

● Around 0.14% of the incident alpha particles scattered by more

than 1o

● Around 1 in 8000 alpha particles deflected by more than 90o

These observations led to many arguments and conclusions which laid

down the structure of the nuclear model on an atom.

Conclusions and arguments

The results of this experiment were not in sync with the plum-pudding

model of the atom as suggested by Thomson. Rutherford concluded

that since alpha particles are positively charged, for them to be

deflected back, they needed a large repelling force. He further argued

that for this to happen, the positive charge of the atom needs to be

concentrated in the centre, unlike scattered in the earlier accepted

model.

Hence, when the incident alpha particle came very close to the

positive mass in the centre of the atom, it would repel leading to a

deflection. On the other hand, if it passes through at a fair distance

from this mass, then there would be no deflection and it would simply

pass through.

He then suggested the ‘nuclear model of an atom’ wherein the entire

positive charge and most of the mass of the atom is concentrated in the

nucleus. Also, the electrons are moving in orbits around the nucleus

akin to the planets and the sun. Further, Rutherford also concluded

from his experiments that the size of the nucleus is between 10-15 and

10-14 m.

According to Kinetic theory, the size of an atom is around 10-10 m or

around 10,000 to 100,000 times the size of the nucleus proposed by

Rutherford. Hence, the distance of the electrons from the nucleus

should be around 10,000 to 100,000 times the size of the nucleus.

This eventually implies that most of the atom is empty space and

explains why most alpha particles went right through the foil. And,

these particles are deflected or scattered through a large angle on

coming close to the nucleus. Also, the electrons having negligible

mass, do not affect the trajectory of these incident alpha particles.

Alpha Particle Trajectory

The trajectory traced by an alpha particle depends on the impact

parameter of the collision. The impact parameter is simply the

perpendicular distance of each alpha particle from the centre of the

nucleus. Since in a beam all alpha particles have the same kinetic

energy, the scattering of these particles depends solely on the impact

parameter.

Hence, the particles with a small impact parameter or the particles

closer to the nucleus, experience large angle of scattering. On the

other hand, those with a large impact parameter suffer no deflection or

scattering at all. Finally, those particles having ~zero impact

parameter or a head-on collision with the nucleus rebound back.

Coming to the experiment, Rutherford and his team observed that a

really small fraction of the incident alpha particles was rebounding

back. Hence, only a small number of particles were colliding head-on

with the nucleus. This, subsequently, led them to believe that the mass

of the atom is concentrated in a very small volume.

Electron Orbits

In a nutshell, Rutherford’s nuclear model of the atom describes it as:

● An electrically neutral sphere with

○ A small and positively charged nucleus at the centre

○ Surrounded by revolving electrons in their

dynamically stable orbits

The centripetal force that keeps the electrons in their orbits is an

outcome of:

● The electrostatic force of attraction between-

○ The positively charged nucleus and

○ The negatively charged revolving electrons.

Solved Example for You

Question: Rutherford, Geiger and Marsden, directed a beam of alpha

particles on a foil of which metal

A. Platinum

B. Tungsten

C. Gold

D. Silver

Solution: Gold

Bohr Model of the Hydrogen Atom

Bohr Model of the hydrogen atom attempts to plug in certain gaps as

suggested by Rutherford’s model by including ideas from the newly

developing Quantum hypothesis. Bohr postulated that in an atom,

electrons could revolve in stable orbits without emitting radiant

energy.

Bohr Model

Bohr model of the hydrogen atom attempts to plug in certain gaps as

suggested by Rutherford’s model by including ideas from the newly

developing Quantum hypothesis. According to Rutherford’s model, an

atom has a central nucleus and electron/s revolve around it like the

sun-planet system.

However, the fundamental difference between the two is that, while

the planetary system is held in place by the gravitational force, the

nucleus-electron system interacts by Coulomb’s Law of Force. This is

because the nucleus and electrons are charged particles. Also, an

object moving in a circle undergoes constant acceleration due to the

centripetal force.

Further, electromagnetic theory teaches us that an accelerating

charged particle emits radiation in the form of electromagnetic waves.

Therefore, the energy of such an electron should constantly decrease

and the electron should collapse into the nucleus. This would make the

atom unstable.

The classical electromagnetic theory also states that the frequency of

the electromagnetic waves emitted by an accelerating electron is equal

to the frequency of revolution. This would mean that, as the electron

spirals inwards, it would emit electromagnetic waves of changing

frequencies. In other words, it would emit a continuous spectrum.

However, actual observation tells us that the electron emits a line

spectrum.

Watch Modern Atomic Theory –

Bohr Model Postulates

Bohr, in an attempt to understand the structure of an atom better,

combined classical theory with the early quantum concepts and gave

his theory in three postulates:

Postulate I

In a radical departure from the established principles of classical

mechanics and electromagnetism, Bohr postulated that in an atom,

electron/s could revolve in stable orbits without emitting radiant

energy. Further, he stated that each atom can exist in certain stable

states. Also, each state has a definite total energy. These are stationary

states of the atom.

Postulate II

Bohr defined these stable orbits in his second postulate. According to

this postulate:

● An electron revolves around the nucleus in orbits

● The angular momentum of revolution is an integral multiple of

h/2p – where hàPlanck’s constant [h = 6.6 x 10-34 J-s].

● Hence, the angular momentum (L) of the orbiting electron is: L

= nh/2p

Postulate III

In this postulate, Bohr incorporated early quantum concepts into the

atomic theory. According to this postulate, an electron can transition

from a non-radiating orbit to another of a lower energy level. In doing

so, a photon is emitted whose energy is equal to the energy difference

between the two states. Hence, the frequency of the emitted photon is:

hv = Ei – Ef

(Ei is the energy of the initial state and Ef is the energy of the final

state. Also, Ei > Ef).

Some important equations

Radii of Bohr’s stationary orbits

● n – integer

● rn – radius of the nth orbit

● h – Planck’s constant

● ε0 – Electric constant

● m – Mass of the electron

● Z – the Atomic number of the atom

● e – Elementary charge

Since ε0, h, m, e, and p are constants and for a hydrogen atom, Z = 1,

rn α n2

The velocity of Electron in Bohr’s Stationary Orbits

Since ε0, h, and e are constants and for a hydrogen atom, Z = 1, rn α

(1/n)

Total Energy of Electron in Bohr’s Stationary Orbits

The negative sign means that the electron is bound to the nucleus.

Although these equations were derived under the assumption that

electron orbits are circular, subsequent experiments conducted by

Arnold Sommerfeld reaffirm the fact that the equations hold true even

for elliptical orbits.

Energy Levels

When the electron is revolving in an orbit closest to the nucleus, the

energy of the atom is the least or has the largest negative value. In

other words, n = 1. For higher values of n, the energy is progressively

larger.

The state of the atom wherein the electron is revolving in the orbit of

smallest Bohr radius (a0) is the ‘Ground State’. In this state, the atom

has the lowest energy. The energy in this state is:

E1 = -13.6 eV

Hence, the minimum energy required to free an electron from the

ground state of an atom is 13.6 eV. This energy is the ‘Ionization

Energy’ of the hydrogen atom. This value agrees with the

experimental value of ionization energy too.

Now, a hydrogen atom is usually in ‘Ground State’ at room

temperature. The atom might receive energy from processes like

electron collision and acquire enough energy to raise the electron to

higher energy states or orbits. This is an ‘excited’ state of the atom.

Therefore, the energy required by the atom to excite an electron to the

first excited state is:

E2 – E1 = -3.40 eV – (-13.6) eV = 10.2 eV

Similarly, to excite the electron to the second excited state, the energy

needed is:

E3 – E1 = -1.51 eV – (-13.6) eV = 12.09 eV

Remember, that the electron can jump to a lower energy state by

emitting a photon. Also, note that, as the excitation of the hydrogen

atom increases, the minimum energy required to free the electron

decreases.

Solved Examples for You

Question: How many postulates are present in the Bohr model of a

hydrogen atom?

Solution: Bohr model of a hydrogen atom has three postulates. The

postulate of the circular orbit, postulate of the selected orbit and

postulate of the origin of spectral lines.

Question: According to the Bohr model, what is the energy of the

atom in the ground state?

Solution: According to the Bohr model, the energy of the atom in the

ground state is -13.6 eV.