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Chem 107B

Summer 2016

Ames

Problem Set 2 (due Thu Sep 1, 2016).

1. Consider an electron with a mass of 9.11 x 10-31 kg and a 50-g tennis ball that are both moving with

a velocity of 20 m s-1.

(a) Calculate the momentum of the electron (p = mv).

(b) Calculate the momentum of the tennis ball.

(c) What is the uncertainty in the position of the electron (x) if the uncertainty in its momentum

(p) is equal to 1% of p for the electron?

(d) What is the uncertainty in the position of the tennis ball (x) if the uncertainty in momentum

(p) is equal to 1% of p for the tennis ball?

(e) Comment on how the uncertainty in position (x) compares to the overall size in each case.

2. Consider an electron in a one-dimensional particle-in-a-box with a box length of L=1 x 10-9 m and

mass equal to m 9.111031 kg.

(a) For the two lowest energy wavefunctions (1 and 2), evaluate the following integral:

L

0 2 1 dx

2 L

2x

x

sin dx ?

0 sin

L

L

L

(b) What is the probability of locating the electron in the region between L/4 and 3L/4 when the

electron is in its lowest energy state (1)?

(c) What is the probability of locating the electron in the region between L/4 and 3L/4 when the

electron is in the second-lowest energy state (2)?

(d) Calculate the energy difference between the n=2 and n=1 states (E = E2 ? E1).

(e) What is the frequency of a quantum transition from the n=1 to n=2 state?

(f) Calculate the wavelength of electromagnetic radiation (in nanometers) that will excite a

quantum transition from the n=1 to n=2 state.

1

Chem 107B

Summer 2016

Ames

3. When the spacing between translational energy levels is small compared to the thermal energy

(kBT), classical mechanics is a good approximation for quantum mechanics. Consider whether the

translational motion of each particle below can be treated classically. For each system below,

calculate the energy of the particle in a box for the n=1 and n=2 states and determine whether E (ie

E2 ? E1) is smaller or larger than the available thermal energy, kBT.

(a) A helium atom in a 1000 Å box at 298 K (L = 10-7 m).

(b) A protein with a molecular weight of 50 kDa in a 100 Å box at 298 K (L = 10-8 m).

(c) A helium atom in a 1 Å box at 1 K (L = 10-10 m).

ax 2

) , where N is the

2

normalization constant and ?a? is a constant that depends on the mass (m) and force constant (k).

4. The ground state wavefunction of a harmonic oscillator is ( x) N exp(

ax2

(a) Normalize this wavefunction by setting 2 ( x)dx Ne 2

2

dx 1 and solve for N.

(b) A harmonic oscillator describes the one-dimensional periodic displacement (x) of two objects

(with mass = m) attached at each end of a spring. The net displacement is zero (x = 0) when the

spring is at equilibrium. The net displacement is positive (x &gt; 0) when the spring is stretched

and the two objects move apart. The net displacement is negative (x &lt; 0) when the spring is

compressed and the two objects move close together.

d ( x)

What is the most probable displacement (x)? (Hint: set

0 and solve for x)

dx

(c) Calculate the fundamental vibrational frequency (in units of s-1) of carbon monoxide that

m m

consists of a carbon and oxygen atom ( C O 1.14 1026 kg ) connected by a triple

mC mO

-1

bond with a force constant (k) of 1860 N m . (Hint: assume the C?O stretching vibration

behaves like a harmonic oscillator that has a fundamental vibrational frequency of

1

0

2

1/ 2

k

)

2

Chem 107B

Summer 2016

Ames

(d) In infrared spectroscopy it is common to convert vibrational frequency (units of s-1) into

vibrational wavenumber ( ~0 / c) that has units of cm-1. What is the vibrational wavenumber

of the C?O stretching vibration?

5. The electrons of metal porphyrins, such as the iron-heme of hemoglobin or the magnesiumporphyrin of chlorophyll, can be described energetically using a simple model of free electrons in a

2

h2 nx2 n y

two-dimensional particle in a box: Enx , n y

where nx 1, 2, 3 &amp; n y 1, 2, 3

8m a 2 a 2

(a) The porphyrin square structure measures about 1 nm on each side (a = 1 nm). Calculate the

energy ( Enx , n y ) for each of the 15 lowest energy states in a two-dimensional box with a = 1 nm

(E11, E12, E21, E22, E23, E32, E33, E34, E43, E44, E54, E45, E55, E65, E56). Sketch an energy-level

diagram showing the proper energy spacing between each of the states.

(b) A heme porphyrin contains 26 electrons. Place the 26 electrons (2 at a time) in the 13 lowest

energy level states in your diagram from part a. Each state (called an orbital) holds two

electrons at a time, including the states that are degenerate in energy (E12 = E21 or E23 = E32).

(c) Calculate the wavelength of electromagnetic radiation (in units of nm) that will excite the lowest

energy absorption band (E55 E56) called a * transition. (Experimentally these bands

3

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This question was answered on: Feb 21, 2020

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