Question

Consider a hydrogen atom modeled as a stationary proton with an electron orbiting in uniform circular motion. The radius of the orbit is a value given the name Bohr radius (you will have to look up this value). Calculate the total energy required to ionize the hydrogen atom.

Answer 1

Only force on electron is electrostatic force which is acting towards the center of the circle.

$\frac{e^2}{4\pi \epsilon_0 r^2}=\frac{mv^2}{r}$

$\frac{e^2}{4\pi \epsilon_0 r}={mv^2}$ ---------------(1)

Energy of the hydrogen atom is $E=PE+KE$

$E=\frac{-e^2}{4\pi \epsilon_0r}+\frac{mv^2}{2}$ ---------------------(2)

using (1) and (2)

$E=\frac{-e^2}{4\pi \epsilon_0r}+\frac{e^2}{8\pi \epsilon_0r}=\frac{-e^2}{8\pi \epsilon_0r}$

Energy required to ionize the hydrogen atom is $E=\frac{e^2}{8\pi \epsilon_0r}=\frac{8.99\times10^9(1.6\times10^{-19})^2}{2\times5.29177\times10^{-11}}=2.174\times10^{-18}\,J$