Question

In the Bohr's model of hydrogen atom, the ratio of the kinetic energy to the total energy of the $\rho$ in the $nth$ quantum state is:

• A. 2
• B. -2
• C. +1
• D. -1

Answer 1

Answer: (D)

The energy $E$ of an electron in an orbit is the sum of kinetic and potential energies. The kinetic energy of the electron is
$K=\frac{1}{2} m v^{2}=\frac{Z e^{2}}{8 \pi \varepsilon_{ o }r}$
where $Z$ is atomic number, $e$ is charge, $r$ is radius.
The potential energy of an electron in an orbit of radius $r$ due to electrostatic attraction by the nucleus is given by
$U =\frac{1}{4 \pi \varepsilon_{0}} \frac{(Z e)(-e)}{r} $
$=-\frac{1}{4 \pi \varepsilon_{o}} \frac{Z e^{2}}{r}$
Total energy of the electron is therefore,
$E=K +U=\frac{Z e^{2}}{8 \pi \varepsilon_{o} r}-\frac{Z e^{2}}{4 \pi \varepsilon_{0} r}=-\frac{Z e^{2}}{8 \pi \varepsilon_{0} r} $
$\therefore \frac{K}{E}=-1$