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 orbits the nucleus circularly due to the electrostatic (Coulomb) force. This electrostatic force provides the necessary centripetal force to maintain the electron's orbit.(i) Force for Circular Electron Orbit:Coulomb’s law states the electrostatic force between the electron and proton as:\[F_{\text{electrostatic}} = \frac{1}{4 \pi \epsilon_0} \cdot \frac{e^2}{r^2}\]where:- \( e \) is the electron's charge (\( e = 1.6 \times 10^{-19} \, \text{C} \)),- \( r \) is the orbital radius,- \( \epsilon_0 \) is the permittivity of free space.The centripetal force required for circular motion is:\[F_{\text{centripetal}} = \frac{m v^2}{r}\]where:- \( m \) is the electron's mass,- \( v \) is the electron's velocity.Bohr's postulate dictates that the electrostatic force equals the centripetal force:\[\frac{m v^2}{r} = \frac{1}{4 \pi \epsilon_0} \cdot \frac{e^2}{r^2}\](ii) Derivation of Electron's Total Energy:The total energy is the sum of the electron's kinetic energy \( K.E. \) and potential energy \( P.E. \).1. Kinetic Energy: Given by:\[K.E. = \frac{1}{2} m v^2\]From the centripetal force equation, \( m v^2 = \frac{1}{4 \pi \epsilon_0} \cdot \frac{e^2}{r} \). Therefore:\[K.E. = \frac{1}{2} \cdot \frac{1}{4 \pi \epsilon_0} \cdot \frac{e^2}{r}\]\[K.E. = \frac{1}{8 \pi \epsilon_0} \cdot \frac{e^2}{r}\]2. Potential Energy: The potential energy between the electron and proton is:\[P.E. = - \frac{1}{4 \pi \epsilon_0} \cdot \frac{e^2}{r}\]The negative sign signifies the electron's bound state due to the attractive force.3. Total Energy: Summing kinetic and potential energies:\[E = K.E. + P.E.\]Substituting the derived expressions:\[E = \frac{1}{8 \pi \epsilon_0} \cdot \frac{e^2}{r} - \frac{1}{4 \pi \epsilon_0} \cdot \frac{e^2}{r}\]\[E = - \frac{1}{8 \pi \epsilon_0} \cdot \frac{e^2}{r}\]Consequently, the electron's total energy is:\[E = - \frac{1}{8 \pi \epsilon_0} \cdot \frac{e^2}{r}\]Meaning of the Negative Sign:The negative total energy indicates the electron is bound to the nucleus by electrostatic attraction. A positive total energy would imply the electron is not bound and would ionize. Thus, the negative sign signifies a bound state, requiring energy input for electron removal.