CBSE Class 12 · 2015 · Delhi · Modern PhysicsShort Answer

Show that the radius of the orbit in a hydrogen atom varies as

n^{2}

, where

n

is the principal quantum number of the atom.

Answer: $r_{n} = \frac{n ^{2} h ^{2}}{4 π ^{2} k m e ^{2}} \propto n^{2}$ .

Solution

Coulomb force as centripetal force:

\frac{k e ^{2}}{r ^{2}} = \frac{m v ^{2}}{r}

, so

m v^{2} = \frac{k e ^{2}}{r}

.
Bohr quantisation:

m v r = \frac{nh}{2 π}

, i.e.

v = \frac{nh}{2 π m r}

.
Substituting:

\frac{m}{r} (\frac{nh}{2 π m r})^{2} = \frac{k e ^{2}}{r ^{2}} \Rightarrow r = \frac{n ^{2} h ^{2}}{4 π ^{2} k m e ^{2}} \Rightarrow r \propto n^{2} .

r_{n} \propto n^{2}

CBSE Class 12 Board · 2015 · Delhi · Subjective