CHEM*2070
1.
Problem Set #1
For each of the following particle-in-a-box systems calculate the
energy difference, in units of kJ/mol, eV and cm−1, between the
n = 2 and n = 1 energy levels. Also calculate the frequency (υ)
and wavelength (λ) of light that is needed to “promote” the
“particle” from the n = 1 to n = 2 energy level. What is the
region of the electromagnetic spectrum of the light?
a) electron trapped inside a nucleus (L = 10 fm)
b) electron in a carbon-carbon π-bond of length 1.34 Å.
c) an electron delocalized through a series of alternating single
and double bonds (i.e. a π-electron in a dye molecule) that is
21.8 Å long.
d) electron in a “quantum wire” that is 1 μm long.
e) Xenon atom trapped in a cavity of an inclusion compound
(L= 1 nm).
f) a billiard ball of mass 200g moving back-and-forth on a
billiard table that is 1.5 m long.
2.
The dye problem in question 1 c) is not realistic in that there are
many π-electrons in the system and these occupy the lowest
energy levels. Consider the molecule β-carotene
The energy levels are occupied by 20 π-electrons. Remember
that you must pair the electrons; two per energy level (↑↓), to fill
up the lowest 10 levels. Therefore the transition from the
highest occupied level to the lowest unoccupied level must be
considered.
3.
a)
Draw the energy level diagram, filling the lowest 10 levels
with pairs of electrons (note: be careful of the scale, recall
that 12 = 1 and 112 = 121).
b)
Draw the 10 → 11 transition on the diagram.
c)
Calculate the wavelength of the transition, in nm.
d)
In what region of the spectrum is the transition.
e)
There are two additional modifications that we should
make to the particle-in-a-box model in order to make it
more realistic in this case. What are they?
The energy levels of a helium atom trapped inside a nanotube
(i.e. a buckytube) were calculated by Prof. Saul Goldman
(Chem), Prof. Chris Gray (Phys.) and Dr. Chris Joslin (Chem. &
Phys.) using a rather sophisticated model [Chem. Phys. 227
405 (1994)]. As an approximation we can model this system as
a particle in a 1-D box.
Calculate ΔE, υ and λ for the n = 1 to n = 2 transition for the
system if the tube is 0.7 nm long and when the tube is 2.0 nm
long. In what region(s) of the spectrum would we observe these
transitions?
4.
In the area of material science (solid state chemistry) recent
advances have been made in “nanotechnology”. For example it
is possible to make nanocrystals of CdSe such that the colour
of these crystals depends on crystal size. We can model these
crystals as 3-D cubes with electrons trapped inside. If we
consider only the E1,2,1 → E1,1,1 transition, what would be the
size of the cube have to be (i.e. what is L) in order for the
crystals to absorb (or emit) light that is a) red, b) green or c)
violet?
5.
6.
After sodium atoms absorb light and the 3s electrons jump up
to the 4p atomic orbitals, the atoms remains in this state for
only 1.6 x 10−8 s, and then emits light of wavelength 589 nm
while returning to the ground state.
a)
what is the colour of the emitted light?
b)
what is the wavelength spread of the spectral line?
A spectral emission line of 48Ti+8 in a distant star was found to
be shifted from a wavelength of 654.2 to 706 nm and to be
broadened to 61.8 pm. What is the speed of recession and the
surface temperature of the star? Hint: convert the emission
wave lengths to frequencies. Also the following relationships
 '

' 
are useful:
7.
and
 
.



Estimate the lifetime of a state that gives rise to a line of width
a) 0.1 cm−1 b) 1 cm−1 and c) 100 MHz.