UNIVERSITY OF LONDON

BSc/MSci DEGREE – June 2003, for Internal Students of Imperial College London

This paper is also taken for the relevant examination for the Associateship

Physical Chemistry 2

Thursday 19th June, 2003, 2.00 – 5.00pm

ANSWER ALL QUESTIONS

USE A SEPARATE ANSWER BOOK FOR EACH QUESTION. WRITE YOUR EXAMINATION NUMBER ON EACH ANSWER

BOOK.

Y2PHYJune2003 Turn over

Q1 Molecular Thermodynamics

Answer part (a) and EITHER part (b) OR part (c).

a) Answer ALL parts of this question

The rotational partition function of a diatomic molecule is given by:

2

28h

TIkq Br σ

π=

(i) What does the rotational partition function represent?

( 2 marks) (ii) Write an expression for the total rotational partition function for N molecules of oxygen.

(3 marks) (ii) Calculate the rotational partition function for a molecule of O2 at 500 K given that the mass of oxygen is 15.9994 g mol-1 and the bond length is 0.121 nm.

(5 marks)

b) Answer ALL parts of this question

(i) Given that the translational partition function for one atom is:

,2 2/3

2 Vh

Tmkq Bt

=π

Derive an expression for the entropy of a monoatomic ideal gas. State clearly any assumptions you make (You may wish to use the Stirling’s approximation: ln N! ~ N ln N – N).

(7 marks)

(ii) Calculate the translational contribution to the molar entropy of 1 mole of CH4 at 300 K and 1x105 Pa, assuming ideal gas behaviour, compare with the experimental value of 186.2 J mol-1 K-1. Discuss the origin of the difference between the calculated and the experimental value.

(8 marks) c) Use the definition of the Gibbs free energy derive a relationship between the equilibrium constant and the molecular partition functions for the reactant and product of the following reaction:

2 Na (g) ↔Na2 (g) (10 marks)

List the data that would be required in order to predict the above equilibrium constant.

(5 marks)

Q2 Electrochemistry and Electrochemical Kinetics. Answer ANY TWO of the three parts (a), (b) and (c) of this question.

(a) Answer ALL parts of this question.

i) Sketch the variation of molar conductivity (Λm) with concentration for both acetic acid (CH3COOH, a weak electrolyte) AND potassium chloride ( KCl , a strong electrolyte) using the same graph. Make sure that your graph is labelled clearly and that the relative sizes of Λm for each electrolyte are clearly visible across the whole range of concentration.

(5.5 marks)

ii) With reference to your graph where necessary, explain why the conductivities differ for the high concentration limit.

(3 marks)

iii) State and explain ONE reason why Λm varies with concentration for a strong electrolyte.

(4 marks) (b) Answer ALL parts of this question.

i) For the following half-cell reaction, calculate the pH value at which the equilibrium potential equals 0.50V? Assume a temperature of 298.15K and that the pressure of oxygen is 1 atmosphere.

O2 + 2H2O + 4e ↔4OH- Eo = 0.401V

(8 marks) ii) Explain how the Nernst equation for a metal electrode (M) in contact with a solution of metal ions (Mz+) can be used to find the concentration of the Mz+ ions.

(4.5 marks)

(c) Answer ALL parts of this question.

i) The table below shows some values of current density (j) as a function of overpotential (η) for a particular oxidation process at an electrode. Use the data to find the value of the exchange current density for this process. (8 marks)

η/V 1010 j/Acm-2 0.11 4.25 0.15 9.26 0.20 24.5 0.25 64.7 0.30 171

ii) Explain why data for overpotentials that are less than 0.1V cannot be used to find the

exchange current density for an oxidation process. (4.5 marks)

Q3 Liquids & Liquid Interfaces Answer part (a) and EITHER part (b) OR part (c). a) Answer ALL parts

i) Describe why oil spreads across water, whereas water forms a droplets on a polyethylene surface

(3 marks) ii) The degree to which one liquid spreads across another is characterised by a parameter called the spreading coefficient. State (without derivation) the relationship between the spreading coefficient and the surface tension of each of the phases present. Make sure you define each of the parameters involved. Under what conditions does one phase spread over another?

(3 marks) iii) The van der Waals equation may be regarded as a modification to the ideal gas law. State the van der Waals equation, defining all parameters used

(3 marks) iv) Describe how the van der Waals equation overcomes many of the limitations of the ideal gas equation

(3 marks) b) Answer ALL parts

For a certain sample of a synthetic poly(amino acid) in water at 303.15K (density 0.996 g cm-3), osmotic pressure determinations gave the following values for the difference in height ∆h between the liquids in the capillary tubes in the diagram below

∆h / cm 2.18 3.58 6.13 9.22 m B / g dm-3 3.71 5.56 8.34 11.12

(i) State the law relating osmotic pressure to molarity, defining all of the terms in the equation

(3 marks) (ii) Convert the height readings in the table above to pressures and using the above equation, find the number average molecular weight of the polymer

(10 marks)

Water

Water + polymer

∆h

c) Answer ALL parts Below is the phase diagram of the NH3-N2H4 system

(i) What are the melting points of NH3, and N2H4?

(2 marks) (ii) At what composition does the system have its lowest freezing-point. What is this

composition called? (2 marks)

(iii) Describe what happens when liquid with an NH3 mole fraction of 0.1 is cooled from

10oC to –81oC (4 marks)

(iv) Over suitable temperature and pressure ranges, many gases and liquids can

coexist. Draw an example of a pressure-molar volume graph with several isotherms and describe how discontinuities (tie-lines) in these isotherms provide information about such two-phase regions. As the temperature changes, these tie lines decrease in length, until eventually they disappear – what is special about the point at which they disappear?

(5 marks)

Q4 Quantum Chemistry II

Answer part (a) and EITHER part (b) OR part (c).

a) Answer ALL parts of this question

i) Explain what is meant by the variational theorem as applied to the solution of the Schrödinger wave equation. Indicate whether energies calculated using the variational method are upper or lower bounds to the true energy of the system. (4 marks)

ii) Explain what is meant by electron correlation (2 marks)

iii) Construct a Slater determinant for H2 and show that it satisfies the Pauli exclusion principle. (4 marks)

iv) Explain with reference to a Molecular Orbital (MO) energy diagram for H2, what s is meant by the terms ground and excited state. (2 ½ marks)

b) Answer ALL parts of this question

For 22 CHCHCHCHCHCHCHCH =−=−=−=

i) Construct the Huckel secular determinant identifying the approximations used (3 ½ marks)

ii) Find the Huckel MO energies for the HOMO and LUMO (4 marks)

iii) Find the Huckel MO’s for the HOMO and LUMO and qualitatively draw their forms (5 marks )

c) Answer ALL parts of this question

For the first row homonuclear diatomics , i.e. Li2, Be2 etc.

i) Quantitatively draw and label Molecular Orbital (MO) diagrams for the complete series from Li2 to Ne2. Exclude the MO’s formed from the 1S atomic orbitals.

(8 marks)

ii) Explain the origin of the paramagnetism of O2. Indicate the rule you are using to determine the occupancy of the MO’s.

(2 marks)

iii) For Hydrogen Fluoride, draw a qualitative MO diagram indicating the various overlaps between atomic orbitals

(2 ½ marks)

Q5 Electronic Properties of Solids Answer part (a) and EITHER part (b) OR part (c)

(a) Answer ALL parts

(i) Explain with the aid of simple diagrams, the difference between a metal, an insulator and a semiconductor.

(6 marks) (ii) Explain the basic difference between a direct gap material and an indirect

gap material and how this influences the optical properties of the solid. (6 marks) (b) Answer ALL parts

(i) Write down the Fermi-Dirac distribution and define all terms.

(2 marks) (ii) Use the Fermi-Dirac distribution to show how the probability of electron

occupation in a solid changes with energy at a temperature of 0K and at finite temperatures.

(5 marks) (iii) For an intrinsic semiconductor such as gallium arsenide (band gap, Eg = 1.43 eV) use the Fermi-Dirac distribution to show that the probability of finding an electron in the conduction band at any temperature T can be given approximately by;

f(E) ~ exp (-Eg/2kBT)

Use this expression to calculate the number of conduction electrons in a sample of gallium arsenide at 100 K and 400 K given that A = 1 x 1025 m-3. Comment on the values obtained (6 marks)

(c) Answer ALL parts

(i) Define the Fermi energy of a metal at 0 K. (1 mark)

(ii) Use the free electron theory of metals to show that the wavector of the electrons at the Fermi level, kf, is given by;

( ) 3/123 nk f π= where n is the electron density of the metal. (6 marks)

(iii) The density and relative atomic mass of zinc are 7133 kg m-3 and 65.39 respectively. Calculate n and kf for copper and use these to determine the Fermi energy of copper at 0 K.

(6 marks)