The radius of a hydrogen atom in the ground state is $0.53 \ Å$. After collision with an electron,it is found to have a radius of $2.12 \ Å$. The principal quantum number $n$ of the final state of the atom is:

$A$ free hydrogen atom after absorbing a photon of wavelength $\lambda_{a}$ gets excited from the state $n=1$ to the state $n=4$. Immediately after that,the electron jumps to $n=m$ state by emitting a photon of wavelength $\lambda_{e}$. Let the change in momentum of the atom due to the absorption and the emission be $\Delta p_{a}$ and $\Delta p_{e}$,respectively. If $\lambda_{a} / \lambda_{e} = 1/5$,which of the following options is/are correct?
[Use $hc = 1242 \text{ eV nm}$; $1 \text{ nm} = 10^{-9} \text{ m}$,$h$ and $c$ are Planck's constant and speed of light,respectively]
$(1)$ $\lambda_{e} = 418 \text{ nm}$
$(2)$ The ratio of kinetic energy of the electron in the state $n=m$ to the state $n=1$ is $1/4$
$(3)$ $m=2$
$(4)$ $\Delta p_{a} / \Delta p_{e} = 1/2$

Write the theoretical value and experimental value of the Rydberg constant,including the unit.

The ratio of the area of the first excited state to the ground state orbit of a hydrogen atom is:

Let $r, v, E$ be the radius of orbit, speed of electron, and total energy of electron respectively in a $H$-atom. Which of the following quantities, according to Bohr theory, is proportional to the quantum number $n$?

The following parameter is the same for all hydrogen-like atoms and ions in their ground state.