First slide

Bohr's atom model

Question

Suppose an electron is attracted towards the origin by a force $\frac{k}{r}$ , where, k is a constant and r is the distance of the electron from the origin. By applying Bohr’s model to this system, the radius of the $n^{t h}$ orbital of the electron is found to be $r_{n}$ and the kinetic energy of the electron to be $T_{n}$ . Then, which of the following is true?

Moderate

A

$T_{n} \propto \frac{1}{n^{2}}, r_{n} \propto n^{2}$
B

$T_{n} independent of n, r_{n} \propto n$
C

$T_{n} \propto n, r_{n} \propto n$
D

$T_{n} \propto \frac{1}{n}, r_{n} \propto n^{2}$

Solution

1. Given Force $f = \frac{k}{r}$
We know Centripetal force $f_{c} = \frac{m v^{2}}{r}$
$\Rightarrow \frac{m v^{2}}{r} = \frac{k}{r} .... (1)$
2. (From Bohr’s theory)
$m v r = \frac{n h}{2 π} ..... (2)$
From (1)
$v = \sqrt{\frac{k}{m}}$
From (2)
$m \sqrt{\frac{k}{m}} r = \frac{n h}{2 π}$
$∴ r = \frac{n h}{2 π \sqrt{m k}}$
Kinetic energy
$T_{n} = K E = \frac{1}{2} m v^{2} = \frac{1}{2} m \frac{k}{m} = \frac{k}{2}$
$r_{n} \propto n and T_{n} is independent of n$
Therefore, the correct answer is (B).

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