A free iron atom emits K_ X-rays having an energy of 6.4 keV . Determine the recoil kinetic energy of the atom. The mass of an iron atom is 9.3 10⁻²⁶ kg .

Step-by-step solution

By conservation of momentum, the recoil momentum of the atom is equal to the momentum of the emitted X-ray photon. p = E c The recoil kinetic energy of the atom is given by: K = p^2 2m = E^2 2mc^2 Given: E = 6.4 keV = 6.4 10^3 eV m = 9.3 10⁻²⁶ kg c = 3 10^8 m/s First, we find the rest mass energy of the iron atom in eV: mc^2 = 9.3 10⁻²⁶ (3 10^8)^2 1.6 10⁻¹⁹ eV = 52.3125 10^9 eV Substituting the values into the kinetic energy formula: K = (6.4 10^3)^2 2 52.3125 10^9 eV K = 40.96 10^6 104.625 10^9 eV K 3.9 10⁻⁴ eV