Question

How is the necessary force provided to an electron to keep it moving in a circular orbit according to Bohr model of hydrogen atom? Derive an expression for the total energy of an electron moving in an orbit of radius \( r \) in hydrogen atom. Give the significance of the negative sign in this expression.

Hint:

The negative sign in the total energy formula represents the binding energy of the electron in the atom. It is a measure of the energy required to remove the electron from the atom (ionization).

Answer 1

In Bohr’s model of the hydrogen atom, the electron moves in a circular orbit around the nucleus under the influence of the electrostatic (Coulomb) force. The necessary centripetal force to keep the electron in its orbit is provided by the electrostatic force between the electron and the proton in the nucleus. (i) Necessary force to keep the electron moving in a circular orbit: According to Coulomb’s law, the electrostatic force between the electron and the proton is given by: \[ F_{\text{electrostatic}} = \frac{1}{4 \pi \epsilon_0} \cdot \frac{e^2}{r^2} \] where: - \( e \) is the charge of the electron (\( e = 1.6 \times 10^{-19} \, \text{C} \)), - \( r \) is the radius of the orbit, - \( \epsilon_0 \) is the permittivity of free space. For an electron to move in a circular orbit, the centripetal force required is: \[ F_{\text{centripetal}} = \frac{m v^2}{r} \] where: - \( m \) is the mass of the electron, - \( v \) is the velocity of the electron. According to Bohr’s postulate, the centripetal force is provided by the electrostatic force, so: \[ \frac{m v^2}{r} = \frac{1}{4 \pi \epsilon_0} \cdot \frac{e^2}{r^2} \] (ii) Derivation of the total energy of the electron: To find the total energy of the electron, we need to calculate both the kinetic energy \( K.E. \) and the potential energy \( P.E. \) of the electron. 1. Kinetic Energy: The kinetic energy of the electron is given by: \[ K.E. = \frac{1}{2} m v^2 \] From the centripetal force equation: \[ m v^2 = \frac{1}{4 \pi \epsilon_0} \cdot \frac{e^2}{r} \] Thus: \[ K.E. = \frac{1}{2} \cdot \frac{1}{4 \pi \epsilon_0} \cdot \frac{e^2}{r} \] \[ K.E. = \frac{1}{8 \pi \epsilon_0} \cdot \frac{e^2}{r} \] 2. Potential Energy: The potential energy between two charges (electron and proton) is given by: \[ P.E. = - \frac{1}{4 \pi \epsilon_0} \cdot \frac{e^2}{r} \] The negative sign indicates that the electron is bound to the nucleus (attractive force). 3. Total Energy: The total energy \( E \) of the electron is the sum of its kinetic and potential energies: \[ E = K.E. + P.E. \] Substituting the expressions for \( K.E. \) and \( P.E. \): \[ E = \frac{1}{8 \pi \epsilon_0} \cdot \frac{e^2}{r} - \frac{1}{4 \pi \epsilon_0} \cdot \frac{e^2}{r} \] \[ E = - \frac{1}{8 \pi \epsilon_0} \cdot \frac{e^2}{r} \] Thus, the total energy of the electron is: \[ E = - \frac{1}{8 \pi \epsilon_0} \cdot \frac{e^2}{r} \] Significance of the negative sign: The negative sign in the expression for total energy indicates that the electron is bound to the nucleus by the electrostatic force. If the total energy were positive, it would mean that the electron is not bound to the nucleus and would escape (ionization). Therefore, the negative sign reflects the fact that the electron is in a bound state, and work is required to remove the electron from the atom.