Calculate the atomic number of the element which gives minimum X-ray wavelength of 0 . 1 nm of K-series. R = 1 . 09737 × 10 7 m - 1

Given that, λ m i n = 0 . 1 nm , for K series = 1 A ∘ , and R = 1 . 09737 × 10 7 m - 1 . For K-series 1 λ = Z - b 2 R 1 1 - 1 ∞ 1 1 × 10 - 10 = Z - b 2 × R Z - b 2 = 1 1 × 10 - 10 × 1 . 09737 × 10 7 Z - b 2 = 1000 1 . 09737 Z - b = 1000 1 . 09737 Z - b = 30 . 1827 ≈ 30 b = 1 ( k - series ) Z = 31 .

Wilson and Sommerfield have defined a fine structure constant as α= e 2 4 π ε 0 h ' c in their theory, here h ' = h 2 π . Find the speed of electrons in the orbit n = 1 in terms of α .

The velocity of an electron in the orbit around the nucleus is given by , v = 2 πKe 2 nh . For the first orbit. n = 1 . Substituting the various values we get the velocity in the first orbit as, v = 2 . 2 × 10 6 m / s .....................................................(1) This value is about 1 137 of the velocity of light.. The value of the Sommerfield fine structure constant is also α = 1 137 . Therefore we can say that the speed of electrons in the first orbit of the hydrogen atom is v = α × c .

Wilson and Sommerfield have defined a fine structure constant as α= e 2 4 π ε 0 h ' c in their theory, here h ' = h 2 π . Express energy of hydrogen atom in terms of α .

The energy of hydrogen atom is given by, E = - 2 π 2 mK 2 e 4 h 2 Z 2 n 2 Substituting the various value for the ground state this becomes, E = - 13 . 6 n 2 eV or - 13 . 6 eV ,..................................................(1) The value of Sommerfield fine structure constant is given by, α= e 2 4 π ε 0 h ' c . Putting different value this comes out to be, α = 1 137 . Therefore we have , E α = 13 . 6 eV 1 137 or E = 1863 α eV This is the required relation.

Wilson and Sommerfield have defined a fine structure constant as α= e 2 4 π ε 0 h ' c in their theory, here h ' = h 2 π . Find the dimensions of α .

According to Sommerfeld, (i) The path of an electron around the nucleus, in general, is an ellipse with the nucleus at one of its foci. (ii) The velocity of the electron moving in an elliptical orbit varies at different parts of the orbit. This causes relativistic variation in the mass of the moving electron. The fine-structure constant, also known as Sommerfeld's constant, is a fundamental physical constant that quantifies the strength of the electromagnetic interaction between elementary charged particles. The constant α quantifies the gap in the fine structure of the spectral lines of the hydrogen atom, α= e 2 4 π ε 0 h ' C The dimensions of α = e 2 ε h c = AT 2 M - 1 L - 3 T 4 A 2 M 1 L 2 T - 1 LT - 1 = 0 Thus the Sommerfield constant is dimensionless.

Discuss an experimental setup off Geiger -Marsden experiment of α - scattering.

The experiments were performed between 1908 and 1913 by Hans Geiger and Ernest Marsden under the direction of Ernest Rutherford at the Physical Laboratories of the University of Manchester. In the experiment, Rutherford sent a beam of alpha particles (helium nuclei) emitted from a radioactive source against a thin gold foil (the thickness of about 0.0004 mm, corresponding to about 1000 atoms). Surrounding the gold foil it was placed a zinc sulfide screen that would show a small flash of light when hit by a scattered alpha particle. It was observed that the majority of the particles passes undisturbed, or with small angles of deflection, through the “empty” atom, some particles, however, passing close to the nucleus are diverted with a high angle or even very few bounced backwards.

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EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR PHYSICS>Chapter 12 - Atoms>Gujarat Board-2020>Q 14

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What is the shortest wavelength present in the Paschen series of spectral lines? (Rydberg constant $= 1.097 \times 10^{7} m^{- 1}$ )

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Important Questions on Atoms

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EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR PHYSICS>Chapter 12 - Atoms>Gujarat Board-2020>Q 16

In accordance with the Bohr's model, find the quantum number that characterises the earth's revolution around the Sun in an orbit of radius $1.5 \times 10^{11} m$ with orbital speed $3 \times 10^{4} {ms}^{- 1}$ .(Mass of the earth $= 6 \times 10^{24} kg$ )