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Correct option is (C) \(\cfrac h{2\pi}\)

In the rarefied gases ,the separation between atoms or molecules are farther apart. Hence the atoms give discrete wavelengths without any interaction with the neighbouring atoms. 

Hydrogen atom has a single electron. Its spectrum consists of series of spectral lines. Lyman series : Lyman Series consists of spectral lines corresponding to the transition of an electron from higher energy orbits n = 1 and n₂ = 2,3,4… These lines belong to Ultraviolet region. 

Balmer series : Balmer series consists of spectral lines emitted during transitions of electrons from higher energy orbits to the second orbit. n₁ = 2 and n₂ = 3,4,5… .These lines lie is the visible region. 

Paschen series : Paschen series consists of spectral lines emitted when electron jumps higher energy orbits to the third orbit n₁ = 3 and n₂ = 4,5,6 … .These lines lie in the infrared region. 

Brackett series : Brackett series consists of spectral lines emitted during transitions of electrons from higher energy orbits to fourth orbit. n₁ = 4 and n₂ = 5,6,7… 

Pfund series : Pfund series consists spectral lines emitted during transition of electrons from higher energy orbits to the fifth orbit. . n₁ = 5 and n₂ = 6,7,8… . 

These lines lie in infrared region . The transition from (n₁ + 1) to n₁ corresponding to I^st member or longest wavelength of the series. The transition from (infinity) state to ‘n₁’ state corresponds to the last number or series limit or shortest wavelength of the series. 

The three basic characteristics of nuclear forces are as follows: 

1. Nuclear forces are very strong attractive forces. 

2. Nuclear forces are short range forces. 

3. Heavy nuclei for their stability must also have a repulsive component such that the nuclear forces are on the whole attractive.

Radioactivity is the phenomenon in which unstable nuclei of an element spontaneously distintegrate into nuclei of another element by emitting a or β particles accompanied by γrays. Such transformation is known as radioactive transformation or radioactive decay.

Nature of α-particles : 

(1) An alpha particle is a helium nucleus, i.e., a doubly ionized helium atom. It consists of two protons and two neutrons. 

(2) Mass of the α-particle ≅ 4u Charge on the α-particle = 2 × charge on the proton

Properties of α-particles :

(1) Of the three types of radioactive radiations, α- particles have the maximum ionizing power. It is about 100 times that of β-particles and 104 times that of γ-rays.

(2) They have the least penetrating power, about 100 times less than that of β-particles and 10⁴ times less than that of γ-rays. They can pass through very thin sheets of paper but are scattered by metal foils and mica. Since αparticles produce intense ionization in a medium, they lose their kinetic energy quickly. As a result, they do not penetrate more than a few centimetres (about 2.7 cm to 8.6 cm) in air under normal conditions.

(3) They are deflected by electric and magnetic fields since they are charged particles. Their deflection is less than that of β-particles in the same field.

(4) They affect photographic plates. 

(5) They cause fluorescence in fluorescent materials such as zinc sulphide.

(6) They emerge from the nuclei with tremendous speeds in the range of 1/100th to 1/10th of the speed of light in free space.

(7) They destroy living cells.

[Note : α-rays and β-rays were discovered by Henri BecquereL]

Rydberg’s formula is given as 

\(\frac{hc}{\lambda} = 21.76 \times 10^{-19}(\frac{1}{n_1^2}-\frac{1}{n_2^2})\)

h = 6.6 x 10^-34 is the plank's constant.

c = 3 x 10⁸ is speed of life.

The shortest wavelength present in the Paschen series of the spectral lines is given for values n₁ = 3 and n₂ = \(\infty\)

From the above values,

\(\lambda = 819nm\)

The element which exhibits the property of radioactivity, i.e., spontaneous disintegration of unstable nuclei of the element by emission of α or β particles accompanied by γ-rays is called a radioactive element. Examples : Uranium, thorium, polonium, radium, actinium.

1. The number of protons in the nucleus of an atom of an element is called the atomic number of the element. It is also known as the proton number. It is denoted by Z. 

2. The number of nucleons (protons and neutrons) in the nucleus of an atom is called the mass number or the atomic mass number. It is denoted by A.

[Notes : (i) The number of neutrons in the nucleus of an atom is known as the neutron number, denoted by N. It is usually greater than Z, with the exceptions of helium (2 protons, 2 neutrons) and hydrogen (1 proton, no neutron) (ii) A = Z + N.]

An atom is represented as \(^A_ZX\), where X is the chemical symbol for the element, Z is the atomic number and A is the mass number. Examples : Fluorine, \(^{19}_9F\); Phosphorus, \(^{31}_{15}p\); Gold, \(^{197}_{73}Au.\)

The atomic number of an element, which is the number of protons in the nucleus of an atom of the element, is characteristic of the element. It equals the number of electrons in the atom and hence determines the chemical properties of the element and its place in the modern periodic table.

The condition for stationary wave formation is that the total distance travelled between the nodes(two) up and down or given path is integral multiple of ‘λ’ 

i.e ., 2πr_n = nλ where, n = 1,2,3,……. 

But λ = h/mv (from de-Broglie’s hypothesis) 

2πr_n = nh/mv_n 

mv_nr_n = n (h/2π) 

i.e. integral multiple of (h/2π) should be equal to the angular momentum of electron in the orbit. 

(i) mvr= nh/2π → quantisation of angular momentum 

(ii) E₂ – E₁ = hν → quantisation of energy states and resulting transition. 

(i) The theory is applicable only for hydrogen atom. 

(ii) The relativistic variation of mass is not taken into account in the theory. 

(iii) The fine structure of spectral lines cannot be accounted for. 

(iv) The theory fails to account for relative intensities of spectral lines. 

Protons and Neutrons. 

Mass number.

Infinite. The last (sixth) of the named series in the hydrogen spectrum is the Humphreys series which results from transitions to n_f = 6 from n_i = 7, 8, 9, … etc. in the far-infrared region. Further series are unnamed but follow the same pattern. Their lines are increasingly faint showing that they correspond to increasingly rare atomic events.

 [Note : There are infinite number of lines in each series. The spacing between adjacent lines decreases with decreasing wavelength, converging to the so-called series limit or short-wavelength limit.]

(C) a higher orbit to the second orbit

Correct option : (3) charged particles emitted by the nucleus

Explanation : 

From the basic studies of the phenomenon of radioactivity, it is known that β particles are fast moving electrons emitted by the nucleus. One can confirm this by determining the e/m values of these particles.