Niels Bohr introduced the atomic Hydrogen model in 1913. He described it as a positively charged nucleus, comprised of protons and neutrons, surrounded by a negatively charged electron cloud. In the model, electrons orbit the nucleus in atomic shells. The atom is held together by electrostatic forces between the positive nucleus and negative surroundings.

Bohr correctly proposed that the energy and radii of the orbits of electrons in atoms are quantized, with energy for transitions between orbits given by
\(\Delta E=h v=E_{i}-E_{f}\) Where \(\Delta E\) is the change in energy between the initial and final orbits and hv is the energy of an absorbed or emitted photon.
(i) In the Bohr model of the hydrogen atom, discrete radii and energy states result when an electron circles the atom in an integer number of

(ii) The angular speed of the electron in the n^th orbit of Bohr's hydrogen atom is

(iii) When electron jumps from n = 4 level to n = 1 level, the angular momentum of electron changes by

(iv) The lowest Bohr orbit in hydrogen atom has

(v) Which of the following postulates of the Bohr modelled to the quantization of energy of the hydrogen atom?

In 1911, Rutherford, along with his assistants, H. Geiger and E. Marsden, performed the alpha particle scattering experiment. H. Geiger and E. Marsden took radioactive source \(\left(\begin{array}{c} 214 \\ 83 \end{array} \mathrm{Bi}\right)\) for a-particles. A collimated beam of a-particles of energy 5.5 MeV was allowed to fall on 2.1 x 10^-7 m thick gold foil. The a-particles were observed through a rotatable detector consisting of a Zinc sulphide screen and microscope. It was found that a-particles got scattered. These scattered a-particles produced scintillations on the zinc sulphide screen. Observations of this experiment are as follows.
(I) Most of the a-particles passed through the foil without deflection.
(II) Only about 0.14% of the incident a-particles scattered by more than 1°.
(III) Only about one a-particle in every 8000 a-particles deflected by more than 90°.
These observations led to many arguments and conclusions which laid down the structure of the nuclear model of an atom.

(i) Rutherford's atomic model can be visualised as

(ii) Gold foil used in Geiger-Marsden experiment is about 10^-8 m thick. This ensures

(iii) In Geiger-Marsden scattering experiment, the trajectory traced by an a-particle depends on

(iv) In the Geiger-Marsden scatteririg experiment, in case of head-on collision, the impact parameter should be

(v) The fact only a small fraction of the number of incident particles rebound back in Rutherford scattering indicates that

At room temperature, most of the H-atoms are in ground state. When an atom receives some energy (i.e., by electron collisions), the atom may acquire sufficient energy to raise electron to higher energy state. In this condition, the atom is said to be in excited state. From the excited state, the electron can fall back to a state of lower energy emitting a photon equal to the energy difference of the orbit.

In a mixture of H-He⁺ gas (He+ is single ionized He atom), H-atoms and He⁺ ions are excited to their respective first excited states. Subsequently, H-atoms transfer their total excitation energy to He⁺ ions (by collisions).
(i) The quantum number n of the state finally populated in He⁺ ions is

(ii) The wavelength of light emitted in the visible region by He⁺ ions after collisions with H-atoms is

(iii) The ratio of kinetic energy of the electrons for the H-atoms to that of He⁺ ion for n = 2 is

(iv) The radius ofthe ground state orbit of H-atoms is

(v) Angular momentum of an electron in H-atom in first excited state is

Bohr's model explains the spectral lines of hydrogen atomic emission spectrum. While the electron of the atom remains in the ground state, its energy is unchanged. When the atom absorbs one or more quanta of energy, the electrons moves from the ground state orbit to an excited state orbit that is further away.

The given figure shows an energy level diagram of the hydrogen atom. Several transitions are marked as I, II, III and so on. The diagram is only indicative and not to scale.
(i) In which transition is a Balmer series photon absorbed?

(ii) The wavelength of the radiation involved in transition II is

(iii) Which transition will occur when a hydrogen atom is irradiated with radiation of wavelength 103 nm?

(iv) The electron in a hydrogen atom makes a transition from n = n₁ to n = n₂ state. The time period of the electron in the initial state is eight times that in the final state. The possible values of n₁ and n₂ are

(v) The Balmer series for the H-atom can be observed

The spectral series of hydrogen atom were accounted for by Bohr using the relation \(\bar{v}=R\left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right)\), 
where R = Rydberg constant = 1.097 x 10⁷ m.
Lyman series is obtained when an electron jumps to first orbit from any subsequent orbit. Similarly, Balmer series is obtained when an electron jumps to 2^nd orbit from any subsequent orbit, Paschen series is obtained when an electron jumps to 3^rd orbit from any subsequent orbit. Whereas Lyman series lies in U.V. region, Balmer series is in visible region and Paschen series lies in infrared region. Series limit is obtained when n₂ = \(\infty\)
(i) The wavelength of first spectral line of Lyman series is

(ii) The wavelength limit of Lyman series is

(iii) The frequency of first spectral line of Balmer series is

(iv) Which of the following transitions in hydrogen atoms emit photons of highest frequency?

(v) The ratio of minimum to maximum wavelength in Balmer series is

Hydrogen is the simplest atom of nature. There is one proton in its nucleus and an electron moves around the nucleus in a circular orbit. According to Niels Bohr, this electron moves in a stationary orbit. When this electron & I is in the stationary orbit, it emits no electromagnetic radiation. The angular momentum of the electron is quantized, i.e., mvr = \((n h / 2 \pi)\) where m = mass of the electron, v = velocity of the electron in the orbit, r = radius of the orbit and n = 1,2, 3, .... When transition takes place from K^th orbit to J^th orbit, energy photon is emitted. If the wavelength of the emitted photon is \(\lambda\)_y we find that \(\frac{1}{\lambda}=R\left[\frac{1}{J^{2}}-\frac{1}{K^{2}}\right]\) where R is Rydberg's constant.
On a different planet, the hydrogen atom's structure was somewhat different from ours. The angular momentum of electron was P = 2n(h/2\(\pi\) ), i.e., an even multiple of (h/2\(\pi\)).

(i) The minimum permissible radius of the orbit will be

(ii) In our world, the velocity of electron is V₀ when the hydrogen atom is in the ground state. The velocity of electron in this state on the other .planet should be

(iii) In our world, the ionization potential energy of a hydrogen atom is 13.6 ev On the other planet, this ionization potential energy will be

(iv) Check the correctness of the following statements about the Bohr model of hydrogen atom
(i) The acceleration of the electron in n = 2 orbit is more than that in n = 1 orbit.
(ii) The angular momentum of the electron in n = 2 orbit is more than that in n = 1 orbit.
(iii) The kinetic energy of the electron in n = 2 orbit is less than that in n = 1 orbit.

(v) In Bohr's model of hydrogen atom, let PE represent potential energy and TE the total energy. In going to a higher orbit

Hydrogen spectrum consists of discrete bright lines in a dark background and it is specifically known as hydrogen emission spectrum. There is one more type of hydrogen spectrum that exists where we get dark lines on the bright background, it is known as absorption spectrum. Balmer found an empirical formula by the observation of a small part of this spectrum and it is represented by \(\frac{1}{\lambda}=R\left(\frac{1}{2^{2}}-\frac{1}{n^{2}}\right), \text { where } n=3,4,5, \ldots .\)
For Lyman series, the emission is from first state to n^th state, for Paschen series, it is from third state to n^th state, for Brackett series, it is from fourth state to n^th state and for Pfund series, it is from fifth state to n^th state,
(i) Number of spectral lines in hydrogen atom is

(ii) Which series of hydrogen spectrum corresponds to ultraviolet region?

(iii) Which of the following lines of the H-atom spectrum belongs to the Balmer series?

(iv) Rydberg constant is

(v) Hydrogen atom is excited from ground state to another state with principal quantum number equal to 4. Then the number of spectral lines in the emission spectra will be