Unit 20: Atomic Spectra — Long Questions

SPECTROSCOPY

The branch of physics which deals with the investigation of wavelengths and intensities of electromagnetic radiations emitted or absorbed by atoms. There are three types:

(i) Continuous spectra

The electromagnetic radiation emitted by the black body is an example of continuous spectra.

(ii) Band Spectra

The molecular spectra are the example of band spectra.

(iii) Line Spectra

The atomic spectra are the example of line spectra.

ATOMIC SPECTRUM / SPECTRA

When an atomic gas or vapour at much less than atmospheric pressure is suitably excited, usually by passing an electric current through it, the emitted radiation has a spectrum, which contains certain specific wavelengths only. An idealized arrangement for observing such atomic spectra is shown in figure. Actual spectrometer uses diffraction grating for better results. The impression on the screen is in the form of lines if the slit in front of the source S is narrow rectangle. It is for this reason that the spectrum is referred to as line spectrum. The fact that the spectrum of any element contains wavelengths that exhibit definite regularities was utilized in the second half of the 19th century in identifying different elements.

These regularities were classified into certain groups called the spectral series. The first such series was identified by J.J Balmer in 1885 in the spectrum of hydrogen. This series, called the Balmer series, is shown in figure, and is in the visible region of the electromagnetic spectrum.

The results obtained by Balmer were expressed in 1896 by J.R Rydberg in the following mathematical form

1/lambda = R_H(1/n₂² - 1/n²) ...... (i)

where R_H is the Rydberg's constant. Its value is 1.0974 x 10⁷ m^-1. Since then many more series have been discovered and proved helpful in predicting the arrangement of the electrons in different atoms.

Atomic Spectrum of Hydrogen

The Balmer series contain wavelengths in the visible portion of the hydrogen spectrum. The spectral lines of hydrogen in the ultraviolet and infrared regions fall into several other series. In the ultraviolet region, the Lyman series contains the wavelengths given by the formula

1/lambda = R_H(1/1² - 1/n²) ...... (ii)

where n = 2, 3, 4, ......

In the infrared region, three spectral series have been found whose lines have the wavelengths specified by the formulae:

Paschen series

1/lambda = R_H(1/3² - 1/n²) ......

where n = 4, 5, 6, ......

Brackett series

1/lambda = R_H(1/4² - 1/n²) ......

where n = 5, 6, 7, ......

Pfund series

1/lambda = R_H(1/5² - 1/n²) ...... (v)

where n = 6, 7, 8, ......

The existence of these regularities in the hydrogen spectrum together with similar regularities in the spectra of more complex elements, proposes a definite test for any theory of atomic structure.

BOHR'S MODEL OF THE HYDROGEN ATOM

In 1913 Bohr formulated a model of hydrogen atom based on classical physics and Planck's Quantum Theory. His atomic model based on the following three postulates:

An electron bound to the nucleus in an atom can moves around the nucleus in certain stationary orbits without radiating energy, these orbits are called discrete stationary states of the atom.

Only these stationary orbits are allowed for which the angular momentum is equal to an integral multiple of h/2π i.e.,

mvr = nh/2π

where n = 1, 2, 3, ...... and n is called principle quantum number, m and v are the mass and velocity of the orbiting electron respectively and h is the Planck, constant.

Whenever, an electron jumps from higher energy state E_n to the lower energy state E_p, a photon of energy hf is emitted, so that

hf = E_n - E_p

where f = c/lambda is the frequency of the radiation emitted.

de-Brogle's Interpretation of Bohr's Orbits

At the time of formulation of Bohr theory there was no justification for the first two postulates. But postulates (III) have some roots / justification in Planck's theory. Later on, postulate (II) proved by de-Brogle. Consider a string of length l as shown in figure. If this string is put into stationary vibrations i.e., l = nlambda. Suppose that, the string is bent into circle of radius r then,

l = 2πr = nlambda

2πr = nlambda

lambda = 2πr/n

By using de-Brogle equation

p = h/lambda

p = h/2πr/n

p = nh/2πr

mv = nh/2πr

mvr = nh/2π

Which is postulate I.

QUANTIZED RADII

Consider, a hydrogen atom in which electron is moving with velocity V_n in stationary circular orbit of radius r_n then, by using Bohr's second postulate.

mv_n r_n = nh/2π

v_n = nh/2πmr_n ...... (i)

In this condition the centripetal force is provided by the Coulomb's force. Therefore,

F_c = F_e

mv_n²/r_n = ke²/r_n²

mv_n² = ke²/r_n

r_n = ke²/mv_n² ...... (ii)

Putting the values of v_n in this equation

r_n = ke²/(m(nh/2πmr_n)²)

r_n = ke²/(m) × (4π² m² r_n²)/(n² h²)

r_n = (4π² k e² m r_n)/(n² h²)

r_n = (n² h²)/(4π² k e² m)

= (4π² k e² m r_n)/(n² h²)

r_n = (n² h²)/(4π² k e² m)

where,

r₁ = h²/(4π² k e² m) = 0.053 nm

So,

r_n = n² r₁

This agrees with experimental measured value and is called first Bohr's orbit radius of the H-atom.

This according to Bohr theory the radii of different stationary orbit of the electron in H-atom are given by

r_n = r₁, 4r₁, 9r₁, 16r₁, ......

Putting value of r_n from eq. (ii) in eq. (i)

∴ V_n = nh/(2πm) × ke²/(2πm × ke²/mv_n²)

V_n = (nhV_n²)/(2πke²)

V_n = (2πke²)/(nh)

∴ V_n = (2πke²)/(nh)

This gives speed of electron in nth orbit.

QUANTIZED ENERGY

Let us calculate the total energy E_n of the electron which is the sum of K.E and P.E that is

E_n = K.E + P.E ...... (i)

Consider a hydrogen atom in which electron moving with velocity V_n is in stationary circular orbit of radius r_n. For this

F_c = F_e

mV_n²/r_n = ke²/r_n²

mV_n² = ke²/r_n

Multiply both sides by 1/2

∴ (1/2) m V_n² = ke²/(2r_n)

∴ K.E. = ke²/(2r_n)

and its P.E is

P.E = -ke²/r_n

w = -ke²/r_n

This work is stored as P.E.

P.E = -ke²/r_n

This according to Bohr theory the radii of different stationary orbit of the electron in H-atom are given by

r_n = r₁, 4r₁, 9r₁, 16r₁, ......

Then,

E_n = -(1/2) ke²/r_n

But,

r_n = (n² h²)/(4π² k e² m)

Then,

E_n = -ke²/(2 × (n² h²)/(4π² k e² m))

E_n = (ke²/(2)) × (4π² k e² m)/(n² h²)

E_n = (1/n²)[2π² (ke)² m/h²]

Where,

2π² (ke)² m/h² = Constant = E_o

So,

E_n = E_o/n² = -13.6 eV/n²

Here

E_o = 13.6 eV

As electric potential due to a charge q at a distance r is V = kq/r.

Which is the energy required to completely remove an electron from the first Bohr orbit. This is commonly described by collision with an external electron. The minimum potential through which this external electron should be accelerated so that it supply the requisite ionization energy is known as ionization potential.

Thus, for n = 1, 2, 3, ......

Then

E_n = -E_o, -E_o/4, -E_o/9, ......

The experimentally measured value of the binding energy of the electron in H-atom is in perfect agreement with the value predicted by Bohr theory.

HYDROGEN EMISSION SPECTRUM

The results derived above for the energy levels along with postulate III can be used to arrive at the expression for the wavelength of the hydrogen spectrum. Suppose that the electron in the hydrogen atom is in the excited state n with energy E_n and makes a transition to a lower state p with energy E_p, where E_p < E_n, then

hf = E_n - E_p

where E_n = -E_o/n² and E_p = -E_o/p²

Hence

hf = -E_o(1/p² - 1/n²)

Substituting for f = c/lambda

hc/lambda = -E_o(1/p² - 1/n²)

1/lambda = E_o/hc(1/p² - 1/n²)

or

1/lambda = R_H(1/p² - 1/n²)

Where R_H = E_o/hc and is constant called Rydberg constant. Its value is 1.0974 x 10⁷ m^-1 which agrees well with the latest measured value of H-atom.

INNER-SHELL TRANSITIONS AND CHARACTERISTIC X-RAYS

The transitions of electrons in the hydrogen or other light elements results in the emission of spectral lines in the infrared, visible or ultraviolet region of electromagnetic spectrum due to small energy difference in the transition levels. In heavy atoms, the electrons are supposed to be arranged in concentric shells named as K, L, M, N, O. The K-shell being closed to the nucleus, next is L shell and so on. The inner shell electron are tightly bounded and large amount of energy is required to excite them. After excitation, when an atom returns to the ground state photons of larger energy are emitted. Thus, transition of inner shell electron in heavy atoms gives rise to the emission of light energy or photons or X-rays. These X-rays consist of series of wavelengths or frequencies and hence are called characteristics X-rays. The study of characteristic X-rays spectra has played a very important role in the study of atomic structure and the periodic table of elements.

Production of X-rays

The figure shows an arrangement for production of X-rays which consists of X-rays are electromagnetic radiation of short wavelength (10^-9 m - 10^-11 m). X-rays were discovered by Rontgen in 1895 when he was investigating cathode rays.

An evacuated glass tube or high vacuum tube called X-rays tube.

A source of electron, filament is used as source of electron.

A target, usually tungsten is used as target.

A high voltage battery of several thousands volts.

WORKING

When the cathode is heated by filament F it emits electrons which are accelerated towards the anode T. If V is the potential difference between cathode and anode, the K.E with which the electron strikes the target is given by

K.E = Ve

Suppose that these fast moving electrons strike a target of heavy atom. It is possible that the electrons from K shell will be knocked out. Suppose that, one of the electron in the K-shell is removed creating a hole in that shell. The electron from L-shell jumps to occupy the space / hole in K-shell, emitting a photon of energy hf_Ka is called K_a, X-rays given by

hf_Ka = E_L - E_K

It is also possible that the electron from the M shell might also jump to occupy the hole in the K shell. The photons emitted are K_β X-ray with energies

hf_Kβ = E_M - E_K

these photons give rise to K_β X-ray and so on.

The photons emitted in such transitions i.e., inner shell transitions are called characteristic X-rays, because their energies depend upon the type of target material.

The holes created in the L and M shells are occupied by transitions of electrons from higher states creating more X-rays. The characteristic X-rays appear as discrete lines on a continuous spectrum as shown.

Continuous X-rays

The continuous spectrum is due to an effect known at bremsstrahlung or braking radiation. When the fast moving electrons bombarded the target, they are suddenly slowed down on impact with the target. We know that an accelerating charge emits electromagnetic radiation. Hence, these impacting electrons emit radiation as they are strongly decelerated by the target. Since the rate of deceleration is so large, the emitted radiation correspond to short wavelength and so the bremsstrahlung is in the X-ray region. In the case when the electrons lose all their kinetic energy in the first collision, the entire kinetic energy appears as a X-ray photon of energy hf_max, i.e.,

K.E. = hf_max

The wavelength λ_min corresponds to frequency f_max is as shown. Other electrons do not lose all their energy in the first collision. They may suffer a number of collisions before coming to rest. This will give rise to photons of smaller energy or X-rays of longer wavelength. Thus the continuous spectrum is obtained due to deceleration of impacting electrons.

Properties and Uses of X-rays

Properties

They are not deflected by electric and magnetic field, so, this shows that they are chargeless.

They are diffracted by crystals.

They cause ionization in gases.

X-rays can cause photoelectric effect on striking some metal surface.

X-rays are highly penetrating radiations. So, they can pass many substances.

Since X-rays are electromagnetic waves, they exhibit wave properties such as diffraction, interference.

Uses

High energy X-rays are used to destroy cancer cells with in the body.

X-rays are used at custom and security posts to detect arms.

X-rays are widely use in industry e.g., X-rays photographs show hidden flaws such as cracks.

X-rays are most useful in medical treatment.

X-rays have many practical application in medicine and industry.

CAT-SCANNER

In the recent past, several vastly improved X-ray techniques have been developed. One widely used system is computerized axial tomography; the corresponding instrument is called CAT-Scanner. The X-ray source produces a thin fan-shaped beam that is detected on the opposite side of the subject by an array of several hundred detectors in a line. Each detector measures absorption of X-ray along a thin line through the subject. The entire apparatus is rotated around the subject in the plane of the beam during a few seconds. The changing reactions of the detector are recorded digitally; a computer processes this information and reconstructs a picture of different densities over an entire cross section of the subject. Density differences of the order of one percent can be detected with CAT-Scans. Tumors, and other anomalies much too small to be seen with older techniques can be detected.

BIOLOGICAL EFFECTS OF X-RAYS

X-rays cause damage to living tissue. As X-ray photons are absorbed in tissues, they break molecular bonds and create highly reactive free radicals (such as H and OH), which in turn can disturb the molecular structure of the proteins and especially the genetic material. Young and rapidly growing cells are particularly susceptible; hence X-rays are useful for selective destruction of cancer cells. On the other hand a cell may be damaged by radiation but survive, continue dividing and produce generation of defective cells. Thus X-rays can cause cancer. Even when the organism itself shows no apparent damage, excessive radiation exposure can cause changes in their productive system that will affect the organism's offspring.

UNCERTAINTY WITH IN THE ATOM

One of the characteristics of dual nature of atom is a fundamental limitations in the accuracy of the measurement of the position and momentum of a particle. Heisenberg showed that this is given by the equation

ΔP . Δx ≈ h

or

ΔP . Δx ≥ h

However, these limitations are significant within the atom. Whether the electron are present in a nucleus.

As the typical nuclei are less than 10^-14 m in diameter for an electron to be confined within such a nucleus, the uncertainty in the position is of the order of 10^-14 m. Thus the uncertainty in the electron's momentum is

ΔP ≥ h/Δx

ΔP ≥ 6.63 x 10^-34/10^-14

Therefore,

mΔV ≥ 6.63 x 10^-34/10^-14

ΔV ≥ 6.63 x 10^-34/(9.1 x 10^-31 x 10^-14)

ΔV ≥ 7.3 x 10¹0 m/sec.

Hence, for the electron to be confined to a nucleus, its speed is greater than 10¹0 m/sec. That is greater than the speed of light. Because, this is impossible. So, we must conclude that an electron can never be inside the nucleus.

Can an electron reside inside the atom?

To find this we have to calculate the speed of electron by considering the radius of the H-atom, that is 5 × 10^-11 m. Thus by using uncertainty principle

ΔP ≥ h/Δx

mΔV ≥ h/Δx

ΔV ≥ h/(Δx)

ΔV ≥ 6.63 x 10^-34/(9.1 x 10^-31 × 5 × 10^-11)

ΔV ≥ 1.46 x 10⁷ m/sec.

This speed of the electron is less than the speed of light. Therefore, the electron can exist in the atom but outside the nucleus.

LASER

The word laser stands for light amplification by the stimulated emission of radiation. A laser is the device which produces very narrow beam of light having the following properties.

It is a monochromatic.

It is a phase coherent.

It is a uni-directional.

There are some requirements necessary for laser action which are:

(i) Laser active medium.

(ii) Population inversion.

(iii) Existence of metastable state.

(iv) Stimulated emission.

(v) Resonators or reflecting mirrors.

Uses of Laser

(1) Laser beams are used as surgical tool, for welding detached retina.

(2) Narrow intense beam of laser can be used to destroy tissue in a localized area.

(3) Lasers are used to break the stone in human kidney.

(4) Laser can be used for telecommunication along optical fibres.

(5) A laser beam can be used to drill tiny holes in the hardest materials.

(6) Laser also have military applications and laser guided missile.

(7) A laser beam can be used to develop hidden finger print.

(8) A laser can be used for the photographic recording of output data of a computer.

(9) It can be used to generate three-dimensional images of objects in a process called holography.