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Could you please help me with question 2. Thank you!
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Could you please help me with question 2. 

Thank you!


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 > 0) when the spring is stretched

 

and the two objects move apart. The net displacement is negative (x < 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 & 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

 

occur at about 600 nm).

 


 

3

 


 

 







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