Quantum Physics

934 Chapter 28 Atomic Physics6. In a Rutherford scattering experiment, an -particle(charge 2e) heads directly toward a goldnucleus (charge 79e). The -particle had a kineticenergy of 5.0 MeV when very far ( r : )from the nucleus. Assuming the gold nucleus tobe fixed in space, determine the distance of closestapproach. [Hint: Use conservation of energywith PE k e q 1 q 2 /r.]Section 28.3 The Bohr Theory of Hydrogen7. A hydrogen atom is in its first excited state (n 2).Using the Bohr theory of the atom, calculate (a) theradius of the orbit, (b) the linear momentum of theelectron, (c) the angular momentum of the electron,(d) the kinetic energy, (e) the potential energy,and (f ) the total energy.8. For a hydrogen atom in its ground state, use theBohr model to compute (a) the orbital speed of theelectron, (b) the kinetic energy of the electron, and(c) the electrical potential energy of the atom.9. Show that the speed of the electron in the nth Bohrorbit in hydrogen is given byv n k ee 2n10. A photon is emitted as a hydrogen atom undergoesa transition from the n 6 state to the n 2 state.Calculate (a) the energy, (b) the wavelength, and(c) the frequency of the emitted photon.11. A hydrogen atom emits a photon of wavelength656 nm. From what energy orbit to what lower energyorbit did the electron jump?12. Following are four possible transitions for a hydrogenatomI. n i 2; n f 5 II. n i 5; n f 3III. n i 7; n f 4 IV. n i 4; n f 7(a) Which transition will emit the shortest-wavelengthphoton? (b) For which transition will theatom gain the most energy? (c) For which transition(s)does the atom lose energy?13. What is the energy of a photon that, when absorbedby a hydrogen atom, could cause (a) an electronictransition from the n 3 state to the n 5 stateand (b) an electronic transition from the n 5state to the n 7 state?14. A hydrogen atom initially in its ground state (n 1) absorbs a photon and ends up in the state forwhich n 3. (a) What is the energy of the absorbedphoton? (b) If the atom eventually returnsto the ground state, what photon energies couldthe atom emit?15. Determine both the longest and the shortest wavelengthsin (a) the Lyman series (n f 1) and (b) thePaschen series (n f 3) of hydrogen.16. Show that the speed of the electron in the first(ground-state) Bohr orbit of the hydrogen atom maybe expressed asv (1/137)c.17. A monochromatic beam of light is absorbed by acollection of ground-state hydrogen atoms in such away that six different wavelengths are observedwhen the hydrogen relaxes back to the groundstate. What is the wavelength of the incident beam?18. A particle of charge q and mass m, moving with aconstant speed v, perpendicular to a constant magneticfield, B, follows a circular path. If in this casethe angular momentum about the center of this circleis quantized so that mvr 2n,show that the allowedradii for the particle arer n √2 nqBwhere n 1, 2, 3, . . .19.(a) If an electron makes a transitionfrom the n 4 Bohr orbit to the n 2 orbit,determine the wavelength of the photon created inthe process. (b) Assuming that the atom was initiallyat rest, determine the recoil speed of the hydrogenatom when this photon is emitted.20. Consider a large number of hydrogen atoms, withelectrons all initially in the n 4 state. (a) Howmany different wavelengths would be observed inthe emission spectrum of these atoms? (b) What isthe longest wavelength that could be observed? Towhich series does it belong?21. Analyze the Earth–Sun system by following the Bohrmodel, where the gravitational force between Earth(mass m) and Sun (mass M) replaces the Coulombforce between the electron and proton (so that F GMm/r 2 and PE GMm/r). Show that (a) the totalenergy of the Earth in an orbit of radius r is givenby (a) E GMm/2r, (b) the radius of the nth orbitis given by r n r 0 n 2 , where r 0 2 /GMm 2 2.32 10 138 m, and (c) the energy of the nth orbit is givenby E n E 0 /n 2 , where E 0 G 2 M 2 m 3 /2 2 1.71 10 182 J. (d) Using the Earth–Sun orbit radius ofr 1.49 10 11 m, determine the value of the quan-