Notes-Part-1-Class-12-Physics-Chapter-15-Structure of Atoms and Nuclei-MSBSHSE

Topics to be Learn : Part-1 Introduction Thomson's atomic model Geiger — Marsden experiment Rutherford's atomic model Atomic spectra Bohr’s atomic model Topics to be Learn : Part-2 Atomic nucleus Nuclear binding energy Radioactive decay Law of radioactive decay Nuclear energy

Introduction :

• Greek philosophers Leucippus (-370 BC) and Democritus (460 – 370 BC) propose, that matter is made of indivisible parts called atoms.
• According to Dalton (1766-1844) to his atomic theory (i) matter is made up of indestructible particles, (ii) atoms of a given element are identical and (iii) atoms can combine with other atoms to form new substances.
• In 1887, J. J. Thomson discovered electrons and developed his atomic model.
• In 1911, Ernest Rutherford proposed the nuclear atom model.
• In 1913, Niels Bohr proposed the quantum theory of the electronic structure of the atom.
• In 1924, Wolfgang Pauli discovered the exclusion principle.
• In 1932, Sir James Chadwick discovered the neutron.Bohr proposed the liquid drop model of nucleus in 1936.
• In 1942, Enrico Fermi constructed the first atomic reactor (nuclear reactor) and achieved the first controlled self-sustaining nuclear reaction.
• The first atomic bombs, based on nuclear fission, were detonated in 1945.
• In 1952, Edward Teller and others studied the idea of a fusion bomb (the hydrogen bomb), and the H-bomb was detonated.
• In 1951, the first nuclear reactor [Experimental Breeder Reactor 1 (EBR 1)] was developed. It produced electrical energy. Walter Zinn (1906-2000) was the lead scientist on this project.
• Various types of nuclear reactors are currently in use for generating electrical energy, producing radioisotopes, and conducting research.

Thomson’s Atomic Model:

• Sir J. J. Thomson (1856-1940), a British physicist, proposed the first model of an atom with a substructure in 1898.
• An atom, according to this model, is a sphere with a uniform distribution of positive charge and electrons embedded in it, resulting in an electrically neutral and stable atom.

Drawbacks : This model, known as the plum-pudding model, failed to account for the observed scattering of α-particles and spectra of various elements. It can be shown that the Thomson atom cannot be stable.

Geiger-Marsden Experiment:

α -particle scattering experiment :

• Following his pioneering work on radioactivity and the characteristics of α-particles, Sir Ernest Rutherford (1871 — 1937), a New Zealand British scientist, observed  that a narrow stream of α-particles gets somewhat broadened or scattered on passing through a thin metal foil or mica sheet.
• Since most of the α -particles remained undeviated, this suggested that atoms could not be solid spheres as proposed in Thomson's model and kinetic theory of gases.
• To probe into the effects of the distribution of an atom’s mass and charge on the α-particles, he suggested his collaborator Geiger and the latter’s student Marsden to see if any α -particles are scattered through a large angle.

Atomic Spectra:

Emission spectrum of a substance :

• The distribution of electromagnetic radiations emitted by a substance when heated or attacked by electrons, ions, or photons is referred to as its emission spectrum.
• Unless the temperature of the substance is very high and the distribution is continuous, the distribution is sorted in order of rising (or decreasing) frequency (or wavelength).
• The spectrum could be either a line spectrum or a band spectrum. The intensities associated with certain frequencies differ.
• The absorption spectrum is formed by absorption of electromagnetic radiation when the substance is exposed to radiation of all frequencies.

Wavelengths of lines in the hydrogen spectrum :

• Hydrogen atom also emits radiation at some other values of wavelengths in the ultraviolet (UV), the infrared (IR) and at longer wavelengths.

• The spectral lines can be divided into groups known as series with names of the scientists who studied them.
• The series, starting from shorter wavelengths and going to larger wavelengths are called Lyman series, Balmer series, Paschen series, Brackett series, Pfund series, etc.
• In each series, the separation between successive lines decreases as we go towards shorter wavelength and they reach a limiting value.
• Fig. shows the first three series. The limiting value of the wavelength for each series is shown by dotted lines.

Formula: 1/λ = \(R[\frac{1}{n^2}-\frac{1}{m^2}]\)  where λ is the wavelength of a line in the hydrogen spectrum, R is a constant, now called the Rydberg constant, and n and m are integers with n = 1,2,3,...and m = n+ 1, n+2, n+3,

For a fixed value of n, λ decreases as m increases and has minimum value as m → ∞. λ = n²/R as m → ∞.

Bohr’s Atomic Model:

He made three postulates which defined his atomic model. These are given below.

• (i) The electron revolves with a constant speed in a circular orbit around the nucleus. The centripetal forcenecessary for the circular motion is providedby the electrostatic force of attraction betweenthe electron and the nucleus.
• (ii) The electron can revolve without radiating energy only in certain orbits, called allowed or stable orbits, in which the angular momentum of the electron is equal to an integral multiple of h/2π where h is Planck's constant.
• (iii) An electron can make a transition from one of its orbit to another orbit having lower energy. In doing so, it emits a photon of energy equal to the difference in its energies in the two orbits.

Radii of the Orbits:

Consider an electron revolving in the n^th orbit around the nucleus of an atom with the atomic number Z. Let m and —e be the mass and charge of the electron, r the radius of the orbit and v the linear speed of the electron.

According to Bohr’s first postulate, centripetal force on the electron = electrostatic force of attraction exerted on the electron by the nucleus.

∴ \(\frac{mv^2}{r}=\frac{Ze^2}{4πε_0r^2}\)          ...(1)

where ε₀ is the permittivity of free space.

∴ v² = \(\frac{Ze^2}{4πε_0mr}\)     ……(2)

According to Bohr’s second postulate, the orbital angular momentum of the electron,

mvr = \(\frac{nh}{2π}\)

where h is Planck's constant and n is the principal quantum number which takes integral values 1, 2, 3, ..., etc.

∴ v = \(\frac{nh}{2πmr}\)

∴ v² = \(\frac{n^2h^2}{4π^2m^2r^2}\)       ….(4)

Equating the right hand sides of Eqs. (2) and (4),

\(\frac{Ze^2}{4πε_0mr}\) = \(\frac{n^2h^2}{4π^2m^2r^2}\)

∴ r = \((\frac{ε_0h^2}{πmZe^2})n^2\)  ….(5)

Since, ε₀, h, Z, m and e are constants, it follows that r ∝ n², i.e., the radius of a Bohr orbit of the electron in an atom is directly proportional to the square of the principal quantum number.

Centripetal acceleration of an electron in a Bohr orbit of a hydrogen atom :

Centripetal acceleration a = v²/r =

Substituting v² From eq. (2)

∴ a = \(\frac{Ze^2}{4πε_0mr}.\frac{1}{r}\) = \(\frac{Ze^2}{4πε_0mr^2}\)

Hence substituting for r from eq. (5)

a = \(\frac{Ze^2π^2m^2Z^2e^4}{4πε_0mε_0^2n^4h^4}\) = \((\frac{πmZ^3e^6}{4ε_0^3h^4}).\frac{1}{n^4}\)

Since all the quantities within the bracket are constants,

a ∝ 1/n⁴

Hence the centripetal acceleration of an electron in a Bohr orbit of a hydrogen atom is inversely proportional to the fourth power of the principal quantum number.

 Q. Derive an expression for the linear speed of an electron in a Bohr orbit. Hence, show that it is inversely proportional to the principal quantum number.

Stationary orbit : In the Bohr model of the hydrogen atom, a stationary orbit refers to any of the discrete allowed orbits such that the electron does not radiate energy while it is in such orbits.

Energy of the Electrons:

Binding energy : Binding energy of an electron in an atom is defined as the minimum energy that should be provided to an orbital electron to remove it from the atom such that its total energy is zero.

If we provide the energy \(\frac{mZ^2e^4}{8ε_0^2h^2n^2}\) [ref. eq no (6) from above] to the electron, its total energy will become zero, i.e., it will become just free from the influence of the host atom.

Therefore, the binding energy of the electron,

BE = −E_n = \(\frac{mZ^2e^4}{8ε_0^2h^2n^2}\)  ….(7)

Excitation energy of an electron in an atom : The energy required to transfer an electron from the ground state to an excited state (a state of higher energy) is called the excitation energy of the electron in that state.

Ionization energy : Ionization energy of an electron in an atom is defined as the minimum energy required to remove the least strongly bound electron from a neutral atom such that its total energy is zero.

Remeber : Since a hydrogen atom has only one electron, the ionization energy is also the binding energy of the electron in the first orbit.

Ground state of an atom  : Ground state of an atom is defined as the lowest stable energy state of the atom.

Bohr's formula for a hydrogen spectral line (expression for the wave number) :

According to Bohr’s third postulate for the model of the hydrogen atom, an atom radiates energy only when an electron jumps from a higher energy state to a lower energy state and the energy of the quantum of electromagnetic radiation emitted in this process is equal to the energy difference between the two states of the electron. This emission of radiation gives rise to a spectral line.

The energy of the electron in a hydrogen atom, when it is in an orbit with the principal quantum number n, is

E_n = \(-\frac{me^4}{8ε_0^2h^2n^2}\) 

where m =mass of electron, e = electronic charge, h = Planck's constant and ε₀ = permittivity of free space.

Let E_m be the energy of the electron in a hydrogen atom when it is in an orbit with the principal quantum number m and E_n, its energy in an orbit with the principal quantum number n, n < m. Then

E_m = \(-\frac{me^4}{8ε_0^2h^2m^2}\)   and E_n = \(-\frac{me^4}{8ε_0^2h^2n^2}\)

Therefore, the energy radiated when the electron jumps from the higher energy state to the lower energy state is

E_m − E_n = \(-\frac{me^4}{8ε_0^2h^2m^2}-(-\frac{me^4}{8ε_0^2h^2n^2})\)

              = \(\frac{me^4}{8ε_0^2h^2}(\frac{1}{n^2}-\frac{1}{m^2})\)

This energy is emitted in the form of a quantum of radiation (photon) with energy hv, where v is the frequency of the radiation.

∴ E_m − E_n = hv

∴ v = \(\frac{E_m-E_n}{h}\) = \(\frac{me^4}{8ε_0^2h^3}(\frac{1}{n^2}-\frac{1}{m^2})\)

The wavelength of the radiation is λ = c/v, where c is the speed of radiation in free space.

The wave number, \(\bar{v}\) = 1/λ = v/c

∴ \(\bar{v}\) = 1/λ = \(\frac{me^4}{8ε_0^2h^3c}(\frac{1}{n^2}-\frac{1}{m^2})\)= \(R(\frac{1}{n^2}-\frac{1}{m^2})\)

where R = \(\frac{me^4}{8ε_0^2h^3c}\) is a constant called the Rydberg constant.

This expression gives the wave number of the radiation emitted and hence that of a line in hydrogen spectrum.

[Note : Z =1 for hydrogen atom]

For the Laymen series : n= 1, m =2,3,4,..., ∞

1/λ_L  = \(R(\frac{1}{1^2}-\frac{1}{m^2})\)

and for the shortest wavelength line in this series,

1/λ_Ls  = \(R(\frac{1}{1^2})\)  as m = ∞

For the Balmer series : n= 2, m = 3,4,5, ..., ∞

1/λ_B  = \(R(\frac{1}{2^2}-\frac{1}{m^2})\)

and for the shortest wavelength line in this series,

1/λ_Bs  = \(R(\frac{1}{4})\) as m = ∞

Q. Show that the energy of the electron in the n^th stationary orbit in the hydrogen atom is E_n = —Rch/n².

Limitations of Bohr’s Model:

• It could not explain the line spectra of atoms other than hydrogen. Even for hydrogen, more accurate study of the observed spectra showed multiple components in some lines which could not be explained on the basis of this model.
• The intensities of the emission lines seemed to differ from line to line and Bohr’s model had no explanation for that.
• On the theoretical side, the model was similarly unsatisfactory because it arbitrarily assumed orbits obeying a specific criteria to be stable. That assumption has no theoretical support. The model cannot account for the Zeeman effect and Stark effect

De Broglie provided some explanation for the third shortcoming (i.e., the theoretical basis for the second postulate in Bohr's atomic model).