Briefly explain the following terms: wave-particle duality and Born interpretation of the...

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Briefly explain the following terms: wave-particle duality and Born interpretation of the wavefunction. 2. Use the formulae to explain why the wavefunctions of a particle in a one-dimensional box are not eigenfunctions of the momentum operator. 3. If a diatomic molecule vibrates harmonically, and the value of its zero-point energy is J, then what is the value of the gap between consecutive vibrational energy levels? 4. If the collisional cross-section for oxygen is 0.41 nm2 , calculate the mean free path for oxygen stored in a cylinder at 20 bar and 212 K. 5. Draw a fully-labelled diagram to show how a nematic liquid crystal display works. 6. Explain, using a sketch of the phase diagram for CO2, what is meant by the critical point. Turn Over CHEM121001 May/June XXXXXXXXXXThe base of a glass tube of internal radius 0.05 cm is placed in a bowl of hexane. Calculate how far up the tube the hexane will rise as a result of capillary action. The surface tension of hexane is 18.4 mN m-1 and its density is 0.65 g cm-3 . You may find the formula below useful [g = 9.8 m s-2]. 8. On the same graph, draw the Maxwell distributions for the molecular speeds of He, N2 and CO2 at the same temperature; label each curve with the gas to which it corresponds. Turn Over CHEM121001 May/June 12 3 Section B Answer all questions (Questions carry 20 marks each) 1. (a) A particle is set to move across a potential energy barrier. The figure below shows how the transmission coefficient changes with the ratio between E (the translational energy of the particle) and V (the height of the barrier). (i) Under the conditions illustrated by this plot, is the particle energy larger or smaller than the barrier height? (ii) At which of the points labelled a, b, … , l are the transmission coefficient values not as one would expect in classical mechanics? Justify your answer. (iii) Consider an experiment where the particle approaches the barrier from the left and a particle detector is placed to the right of the barrier. If the experiment were repeated ten million times with , how many times would a particle be detected? (iv) Calculate the reflection coefficient at points c and f. (12 marks) Question continues Turn Over CHEM121001 May/June 12 4 (b) “System A” consists of a particle of mass m trapped in a one-dimensional box of length m; the de Broglie wavelength is m. “System B” is similar, except that the box length is m, and the de Broglie wavelength is m. (i) For which of the two systems are quantum effects most relevant? Justify your answer. (ii) Suggest a measurement, or set of measurements, whose result(s) would show quantum effects to be important for one of the two systems, but not for the other. (8 marks) 2. (a) Draw a labelled diagram to illustrate the variation of electronic potential energy V with inter-atomic distance r for the case of two Kr atoms forming the dimer (Kr2). Assume that the potential energy follows the Lennard-Jones XXXXXXXXXXpotential. Briefly account for the long- and short-range terms of this potential. (8 marks) (b) A piece of Na metal is placed into a chamber which is evacuated. The chamber is then heated to 900 K. The solid Na melts and partially evaporates to produce a vapour consisting of Na atoms and dimers. You are asked to calculate the pressure of both species above the molten metal: (i) If the boiling point of Na is 1156 K, and the heat of vaporisation is 96.1 kJ mol-1 , use the van’t Hoff relation to calculate the vapour pressure (in bar) of Na atoms at 900 K. (ii) For the equilibrium Na + Na ?Na2 ?H ? = -81.7 kJ mol-1 and ?S ? = -83.1 J K-1 mol-1 . Use this information to calculate the equilibrium vapour pressure (in bar) of Na2 at 900 K. (iii) If N2 is now added into the chamber so that the total pressure is 2 bar, will the pressures of Na or Na2 change? (12 marks) Turn Over CHEM121001 May/June XXXXXXXXXXa) Draw a diagram of a typical galvanic cell. Label the electrodes, locations of chemical oxidation and reduction, and direction of electron flow. What additional component would be found in an electrolytic cell? (8 marks) (b) In an alkaline fuel cell, the chemistry can be described by the following halfreactions: O2 + 2H2O + 4e-? 4OH- E ? = 0.401 V 2 H2O + 2e- ? H2 + 2OH- E ? = XXXXXXXXXXV (i) Which of these half cells will form the cathode, and which the anode? (ii) What will be the overall cell reaction, and the standard cell potential (E ? )? (iii) If the pressures of H2 and O2 supplied to the fuel cell are 15.2 and 10.4 bar, respectively, what will be the zero-current cell potential at 300 K?

Robert · Accepted Answer

Section A 
Problem 1 
a)Wave particle duality states that at one moment matter can behave like a wave and in another moment matter 
can behave like a particle . The wavelength of a moving particle is given by DeBroglie Hypothesis 
h
where h plank 's constant and p= momentum of the particle
p
    
b) Born interpretation of the wave function establishes relationship between the wave function and the 
probability that an observable as a specific value 
if  (x, y, z)  is a wave function in x,y,z position , then the probability of finding the particle at (x,y,z) is given by 
2
P(x, y, z) (x, y, z)* (x, y, z) (x, y, z)      
Problem 2 
the given wave function is  n
2 n x
sin
L L
 
   
 
 
And momentum operator is given  by x
d
P̂ i
dx
   
nd 2 n x ni i cos
dx L L L
    
     
  
  
This is not in the form of 
x n nP̂ p where p eigenvalues     
n  is not an Eigen function of momentum operator 
if n  is also an Eigen function of the momentum operator then it will violate the uncertainty principle. 
Two quantities can be measured simultaneously if they have a common set of Eigen functions , that means if n  
is also the Eigen function momentum operator then we can measure the position and momentum of the particle 
simultaneously (then xx P 0   ) , which is not true according to the uncertainty principle which states 
xx P
2
    
For any two conjugate operator A and B we have A B
2
   and here xx,P  are conjugate operators 
Problem 3 
The energy of the nth energy level for a harmonic oscillator is given by  n
1
E n
2
 
   
 
 
Now, energy gap between consecutive vibrational energy levels is 
n 1 n
1 1
E E E n 1 n
2 2

   
             
   
 
given that zero point energy is 
201.5x10 J  and zero point energy corresponds to n=0 
0
20 20
0
1
E for n 0
2
2xE 2x1.5x10 J 3x10 J 
 
    
 
   
 
Hence  
20E 3x10 J     
Problem 4 
The collision cross-section for oxygen is given by  
2
o od where d diameter of oxygen     
o
9 9
o
0.41
d x10 m 0.3610 m 3.6A 

    
 
 
Given data is  
5Pr essure(P) 20bar 20x10 Pa
Temperature(T) 212K
 

 
The mean free path   is given by  
2
A o
RT
2 N Pd
 

 
Substituting the values we get  
92.54x10 m   
Problem 5 
Liquid Crystals:   A liquid crystal is a liquid substance that has solid-like properties. liquid crystals typically having 
rod-shaped molecules which are free to move about is of importance for Liquid crystal display .Liquid crystal 
substances exist as crystals at low temperatures, and upon heating, they will slowly become a liquid while 
retaining some of their crystalline properties. At higher temperatures, they become completely liquid.
A pair of polarising filters work with the liquid crystal to control the emitted light .As it passes through the first 
filter only vertical component of the light is passed, if the liquid crystal are in their natural state without any 
application of voltage they are twisted , which causes light to turn horizontal. If the electric field is applied, the 
liquid crystal straighten and the cell doesn’t bend the light.

Briefly explain the following terms: wave-particle duality and Born interpretation of the wavefunction. 2. Use the formulae to explain why the wavefunctions of a particle in a one-dimensional box are...

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