5. (Hydrogen Atom) In class, we will see that classical mechanics predicts atoms described by the Rutherford model, wherein electrons are pictured to circularly orbit nuclei, are not stable. a. Use your own words to describe why classical mechanics cannot explain the stability of the hydrogen atom (a single electron orbiting a proton). b. Neils Bohr was able to explain the stability of atomic nuclei by hypothesizing the angular momentum of electrons orbiting nuclei can only take on discrete values. =mevr=nwheren=1,2,3, In the above equation, is the orbiting electron's angular momentum, me, is the electron's mass is electron speed, r the radius of its orbit, and =h/2. Can you use this information to calculate the allowed radii and energies, and velocities of the hydrogen atom. c. According to Bohr's model for the hydrogen atom, what is the speed of an electron placed in the atom's lowest allowed energy state (n=1) ? What about the speed of an electron placed into the n=2 or n=3 state? Why do electrons placed into higher energy states move more slowly? d. One of the biggest arguments supporting Bohr's "quantum hypothesis" is its ability to explain the experimentally measured emission spectrum of hydrogen atom, which contains a series of discrete emission lines. Rydberg showed the frequencies at which these lines occur can be described by: v=R(n121n221) where R is a constant, n1=1,2,3, and n2=n1+1,n1+2,n1+3, Using your result from part (b), write an expression for the Rydberg constant, R, in terms of fundamental constants, such as the electron mass, Planck's constant, and the speed of light. Evaluate your expression and compare to the experimentally measured value of R of 109,678 cm1. 5. (Hydrogen Atom) In class, we will see that classical mechanics predicts atoms described by the Rutherford model, wherein electrons are pictured to circularly orbit nuclei, are not stable. a. Use your own words to describe why classical mechanics cannot explain the stability of the hydrogen atom (a single electron orbiting a proton). b. Neils Bohr was able to explain the stability of atomic nuclei by hypothesizing the angular momentum of electrons orbiting nuclei can only take on discrete values. =mevr=nwheren=1,2,3, In the above equation, is the orbiting electron's angular momentum, me, is the electron's mass is electron speed, r the radius of its orbit, and =h/2. Can you use this information to calculate the allowed radii and energies, and velocities of the hydrogen atom. c. According to Bohr's model for the hydrogen atom, what is the speed of an electron placed in the atom's lowest allowed energy state (n=1) ? What about the speed of an electron placed into the n=2 or n=3 state? Why do electrons placed into higher energy states move more slowly? d. One of the biggest arguments supporting Bohr's "quantum hypothesis" is its ability to explain the experimentally measured emission spectrum of hydrogen atom, which contains a series of discrete emission lines. Rydberg showed the frequencies at which these lines occur can be described by: v=R(n121n221) where R is a constant, n1=1,2,3, and n2=n1+1,n1+2,n1+3, Using your result from part (b), write an expression for the Rydberg constant, R, in terms of fundamental constants, such as the electron mass, Planck's constant, and the speed of light. Evaluate your expression and compare to the experimentally measured value of R of 109,678 cm1
