CBSE Class 12 · 2022 · Modern PhysicsShort Answer

(a) Use Bohr's postulate to prove that the radius of the

n^{th}

orbit in a hydrogen atom is proportional to

n^{2}

.
(b) How will the energy of a hydrogen atom change if

n

increases from 1 to

\infty

Answer: $r_{n} \propto n^{2}$ ; energy rises from $- 13.6$ eV to $0$ .

Solution

(a) Quantisation:

m v r = \frac{nh}{2 π}

(1)

. Coulomb = centripetal:

\frac{m v ^{2}}{r} = \frac{1}{4 π ε _{0}} \frac{e ^{2}}{r ^{2}}

(2)

.
Eliminating

v

between (1) and (2):

r_{n} = \frac{n ^{2} h ^{2} ε _{0}}{π m e ^{2}} \Rightarrow r_{n} \propto n^{2} .

(b) The energy is

E_{n} = - \frac{13.6}{n ^{2}} eV

. As

n

goes from 1 to

\infty

E

rises from

- 13.6 eV

0

, i.e. the energy increases (becomes less negative); at

n = \infty

the atom is just ionized.

r_{n} \propto n^{2}; E : - 13.6 eV \to 0

CBSE Class 12 Board · 2022 · Subjective