The energy (in $J$) released when an excited electron of the $5^{th}$ orbit of a hydrogen atom returns to its ground state is

The radius of the electron in the first excited state of a hydrogen atom is:

According to the Bohr theory,which of the following transitions in the hydrogen atom will give rise to the least energetic photon?

Which transition in a $H$ atom has the same wavelength as the transition from $n = 4$ to $n = 2$ in a $He^{+}$ ion?

Answer the following by appropriately matching the lists based on the information given in the paragraph. Consider Bohr's model of a one-electron atom where the electron moves around the nucleus. In the following,$List-I$ contains some quantities for the $n^{\text{th}}$ orbit of the atom and $List-II$ contains options showing how they depend on $n$.

$List-I$	$List-II$
$(I)$ Radius of the $n^{\text{th}}$ orbit	$(P) \propto n^{-2}$
$(II)$ Angular momentum of the electron in the $n^{\text{th}}$ orbit	$(Q) \propto n^{-1}$
$(III)$ Kinetic energy of the electron in the $n^{\text{th}}$ orbit	$(R) \propto n^0$
$(IV)$ Potential energy of the electron in the $n^{\text{th}}$ orbit	$(S) \propto n^1$
-	$(T) \propto n^2$
-	$(U) \propto n^{1/2}$

$(1)$ Which of the following options has the correct combination considering $List-I$ and $List-II$?
$(1) (II), (R)$ $(2) (I), (P)$ $(3) (I), (T)$ $(4) (II), (Q)$
$(2)$ Which of the following options has the correct combination considering $List-I$ and $List-II$?
$(1) (III), (S)$ $(2) (IV), (Q)$ $(3) (IV), (U)$ $(4) (III), (P)$
Give the answer for questions $(1)$ and $(2)$.

The threshold frequency of a metal is $10^{15} \ s^{-1}$. The ratio of maximum kinetic energies of the photoelectrons, when the metal is irradiated with radiations of frequencies $1.5 \times 10^{15} \ s^{-1}$ and $2.0 \times 10^{15} \ s^{-1}$ respectively, is