Sheet 4
1- (a) What of ni is associated with the 94.96 nm spectral line in the Lyman series of hydrogen?(b)
What If? Could this wavelength be associated with the Paschen or Balmer series?
2- (a) Compute the shortest wavelength in each of these hydrogen spectral series: Lyman, Balmer,
Paschen, and Brackett. (b) Compute the energy (in electron volts) of the highest-energy photon
produced in each series.
3- According to classical physics, a charge e moving with an acceleration a radiates at a rate
dE/dt = - (1/6πƐo) (e2a2/c3)
(a) Show that an electron in a classical hydrogen atom spirals into the nucleus at a rate
dr/dt = - 𝑒 4 /12π2 Ɛ20 𝑟 2 𝑚𝑒2 𝑐 3
(b) Find the time interval over which the electron will reach r=0 , starting from
𝑟0 = 2.00 𝑋10−10 𝑚
4- In the Rutherford scattering experiment , 4.00-MeV alpha particles ( 4𝐻𝑒 nuclei
containing 2 protons and 2 neutrons) scatter off gold nuclei (containing 79 protons and
118 neutrons). Assume that a particular alpha particle makes a direct head-on collision
with the gold nucleus and scatters back-ward at 180𝑜 . Determine (a) the distance of
closest approach of the alpha particle to the gold nucleus , and (b) the maximum force
exerted on the alpha particle . Assume that the gold nucleus remains fixed throughout
the entire process.
5- For a hydrogen atom in its ground state , use the Bohr model to compute (a) the orbital
speed of the electron, (b) the kinetic energy of the electron, and (c) the electric potential
energy of the atom.
6- Four possible transition for a hydrogen atom are as follows :
(i)
𝑛𝑖 = 2; 𝑛𝑓 = 5
(ii)
𝑛𝑖 = 5; 𝑛𝑓 = 3
(iii)
𝑛𝑖 = 7; 𝑛𝑓 = 4
(iv)
𝑛𝑖 = 4; 𝑛𝑓 = 7
(a) In which transition is light of the shortest wavelength emitted?
(b) In which transition dose the atom the atom gain the most energy?
(c) In which transition (s) does the atom lose energy?
7- A hydrogen atom is in its first excited state (n=2). Using the Bohr theory of the atom ,
calculate (a) the radius of the orbit, (b)the linear momentum of the electron, (c) the
angular momentum of the electron , (d) the kinetic energy of the electron , (e) the
potential energy of the system, and (f) the total energy of the system.
8- How much energy is required to ionize hydrogen (a) when it is in the ground state? (b)
When it is in the state for n=3?
9- A photon is emitted as a hydrogen atom undergoes a 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.
10- Show that the speed of the electron in the nth Bohr orbit in hydrogen is given by
𝑘𝑒 𝑒 2
𝑣𝑛 =
𝑛ℎ
11- Two hydrogen atoms collide head-on and up with zero kinetic energy. Each atom then
emits light with a wave-length of 121.6 nm (n=2 to n=1 transition). At what speed were
the atoms moving before the collision?
12- A monochromatic beam of light is absorbed by a collection of grand state hydrogen
atoms in such a way that six different wavelengths are absorbed when the hydrogen
relaxes back to the ground state. What is the wavelength of the incident beam?
13- An electron of momentum p is at a distance r from a stationary proton. The electron has
kinetic energy K=𝑝2 /2𝑚𝑒 . the atom has potential energy U = - 𝑘𝑒 𝑒 2/r, and total energy
E= K+U. If the electron is bound to the proton to form a hydrogen atom, its average
position is at the proton, but the uncertainty in its position is approximately equal to the
radius r of its orbit. The electron’s average vector momentum is zero, but its average
squared momentum is approximately equal to squared uncertainty in its momentum, as
given by the uncertainty principle. Treating the atom as a one-dimensional system, (a)
estimate the uncertainty in the electron’s momentum in terms of r. (b) Estimate the
electron’s kinetic, potential, and total energies in terms of r. (c) The actual value of r is
the one that minimizes the total energy, resulting in a stable atom. Find that value of r
and the resulting total energy. Compare your answer with the predictions of the Bohr
Theory.
14- The ground-state wave function for the electron in a hydrogen atom is ψ(r) =
1
√𝜋𝑎03
𝑒 −𝑟/𝑎0
where r is the radial coordinate of the electron and 𝑎0 𝑖𝑠 𝑡ℎ𝑒 𝐵𝑜ℎ𝑟 𝑟𝑎𝑑𝑖𝑢𝑠 . (a) Show that
the wave function as given in normalized. (b) Find the probability of locating the electron
𝑎
𝑎
between 𝑟1 = 20 𝑎𝑛𝑑 𝑟2 = 3 20 .
15- The wave function for an electron in the 2p state of hydrogen is
𝜓2𝑝 =
1
(2𝑎0 )3 /2 √3
𝑒 −𝑟/2𝑎0
What is the most likely distance from the nucleus to find an electron in the 2p state?
16- List the possible sets of quantum numbers for electrons in (a) the 3d subshell and (b) the
3p subshell.
17- Calculate the angular momentum for an electron in (a) the 4d state and (b) the 6f state .