Question

Suppose an electron is attracted towards the origin by a force k/r, where k is a constant and r is the distance of the electron from the origin. By applying Bohr model to this system, the radius of n^th orbit of the electron is found to be $r_{n}$ and the kinetic energy of the electron is found to be $T_{n}$ Then which of the following is true?

• A. $T_{n} \propto \frac{1}{n^{2}}$
• B. $T_{n}$is independent of n; $r_{n} \propto n$
• C. $T_{n} \propto \frac{1}{n}\text{ and }r_{n}$
• D. $T_{n} \propto \frac{1}{n}\text{ and }r_{n} \propto n^{2}$

Answer 1

Answer: (B)

$T_{n}$is independent of n; $r_{n} \propto n$

Answer 2

Applying Bohr model to the given system,

$\frac{mv^{2}}{r_{n}} = \frac{k}{r_{n}}$ ……. (i)

And $mvr_{n} = \frac{nh}{2\pi}$ or $v = \frac{nh}{2\pi mr_{n}}$

Put in (i),

$\frac{m}{r_{n}} \times \frac{n^{2}h^{2}}{4\pi^{2}m^{2}r_{n}^{2}} = \frac{k}{r_{n}}$

$r_{n}^{2} = \frac{n^{2}h^{2}}{4\pi^{2}mk}$ ……… (ii)

$\therefore r_{n}^{2} \propto n^{2}$ or $r_{n} \propto n$

K.E. of the electron,

$T_{n} = \frac{1}{2}mv^{2}\frac{1}{2}m\frac{n^{2}h^{2}}{4\pi^{2}m^{2}r_{n}^{2}} = \frac{n^{2}h^{2}}{8\pi^{2}mr_{n}^{2}}$

Using (ii), we get

$T_{n} = \frac{n^{2}h^{2}4\pi^{2}mk}{8\pi^{2}mn^{2}h^{2}} = \frac{k}{2}$

$\therefore T_{n}$ is independent of n.