Atoms

Consider a hydrogen atom with $v_k,r_k$, and $K_k$ denoting the velocity, orbital radius and kinetic energy of the electron in the $k^{\text{th}}$ orbit, respectively. The electron undergoes a transition from the $n^{\text{th}}$ orbit, emitting radiation corresponding to the Lyman series. Considering $h$ to be the Planck's constant and $\epsilon_0$ the permittivity of the free space, the correct statement(s) is/are:

Angular momentum of an electron in a hydrogen atom is $\dfrac{3h}\pi$, then the energy of the electron is ________ eV.

Using Bohr's model, calculate the ratio of the magnetic fields generated due to the motion of the electrons in the 2nd and 4th orbits of hydrogen atom ________.

In Rutherford's alpha-particle scattering experiment, only a few alpha particles rebound back because

A. The size of gold nucleus is very small as compared to the size of gold atom.

B. Alpha particle and gold nucleus have equal charge.

C. The impact parameter is minimum for a few alpha particles.

D. A few alpha particles have very high kinetic energy.

E. Only a few alpha particles undergo head-on collision with the nuclei.

Choose the correct answer from the options given below:

In the hydrogen atom, the electron makes a transition from the higher orbit ($i$) to a lower orbit ($f$). The ratio of the radius of the orbits in given by $r_i:r_f=16:4$. The wavelength of photon emitted due to this transition is ________ nm.
(Given Rydberg constant $=1.0973\times10^7$ /m)

The ratio of momentum of the photons of the 1st and 2nd line of Balmer series of Hydrogen atoms is $\alpha/\beta$. The possible values of $\alpha$ and $\beta$ are:-

Consider an electron in the $n = 3$ orbit of a hydrogen-like atom with atomic number $Z$. At absolute temperature $T$, a neutron having thermal energy $k_BT$ has the same de Broglie wavelength as that of this electron. If this temperature is given by $T = \dfrac{Z^2h^2}{\alpha\pi^2a_0^2m_Nk_B}$, (where $h$ is the Planck's constant, $k_B$ is the Boltzmann constant, $m_N$ is the mass of the neutron and $a_0$ is the first Bohr radius of hydrogen atom) then the value of $\alpha$ is _____.

List-I shows various functional dependencies of energy ($E$) on the atomic number ($Z$). Energies associated with certain phenomena are given in List-II.

Choose the option that describes the correct match between the entries in List-I to those in List-II.

A hydrogen atom, initially at rest in its ground state, absorbs a photon of frequency $\nu_1$ and ejects the electron with a kinetic energy of 10 eV. The electron then combines with a positron at rest to form a positronium atom in its ground state and simultaneously emits a photon of frequency $\nu_2$. The center of mass of the resulting positronium atom moves with a kinetic energy of 5 eV. It is given that positron has the same mass as that of electron and the positronium atom can be considered as a Bohr atom, in which the electron and the positron orbit around their center of mass. Considering no other energy loss during the whole process, the difference between the two photon energies (in eV) is _____.

A particle of mass $m$ is moving in a circular orbit under the influence of the central force $F(r) = -kr$, corresponding to the potential energy $V(r) = kr^2/2$, where $k$ is a positive force constant and $r$ is the radial distance from the origin. According to the Bohr's quantization rule, the angular momentum of the particle is given by $L = n\hbar$, where $\hbar = h/(2\pi)$, $h$ is the Planck's constant, and $n$ a positive integer. If $v$ and $E$ are the speed and total energy of the particle, respectively, then which of the following expression(s) is(are) correct?

A metal target with atomic number $Z = 46$ is bombarded with a high energy electron beam. The emission of X-rays from the target is analyzed. The ratio $r$ of the wavelengths of the $K_\alpha$-line and the cut-off is found to be $r = 2$. If the same electron beam bombards another metal target with $Z = 41$, the value of $r$ will be

A Hydrogen-like atom has atomic number $Z$. Photons emitted in the electronic transitions from level $n = 4$ to level $n = 3$ in these atoms are used to perform photoelectric effect experiment on a target metal. The maximum kinetic energy of the photoelectrons generated is 1.95 eV. If the photoelectric threshold wavelength for the target metal is 310 nm, the value of $Z$ is _____.

[Given: $hc = 1240$ eV-nm and $Rhc = 13.6$ eV, where $R$ is the Rydberg constant, $h$ is the Planck's constant and $c$ is the speed of light in vacuum]

Choose the correct option from the following options given below :

In Bohr's atomic model of hydrogen, let K, P and E are the kinetic energy, potential energy and total energy of the electron respectively. Choose the correct option when the electron undergoes transitions to a higher level :

The ratio for the speed of the electron in the 3$^{\rm rd}$ orbit of $\mathrm{He^+}$ to the speed of the electron in the 3$^{\rm rd}$ orbit of hydrogen atom will be :

A hydrogen atom in its ground state absorbs 10.2 eV of energy. The angular momentum of electron of the hydrogen atom will increase by the value of :

(Given, Planck's constant $=6.6\times10^{-34}$ Js).

A beam of monochromatic light is used to excite the electron in $\mathrm{Li^{++}}$ from the first orbit to the third orbit. The wavelength of monochromatic light is found to be $x\times10^{-10}$ m. The value of $x$ is ______.

[Given $hc=1242$ eV nm]

Given below are two statements:

Statement I: In hydrogen atom, the frequency of radiation emitted when an electron jumps from lower energy orbit ($E_1$) to higher energy orbit ($E_2$), is given as $hf=E_1-E_2$

Statement II: The jumping of electron from higher energy orbit ($E_2$) to lower energy orbit ($E_1$) is associated with frequency of radiation given as $f=(E_2-E_1)/h$

This condition is Bohr's frequency condition.

In the light of the above statements, choose the correct answer from the options given below:

$\sqrt{d_1}$ and $\sqrt{d_2}$ are the impact parameters corresponding to scattering angles 60° and 90° respectively, when an $\alpha$ particle is approaching a gold nucleus. For $d_1=x\,d_2$, the value of $x$ will be ______.

A hydrogen atom in ground state absorbs 12.09 eV of energy. The orbital angular momentum of the electron is increased by :