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CYLI10: SEMESTER II 2007-2008, MAJOR EXAM

Date: May 1, 2008 Time: 2 hour/Marks 200

IMPORTANT: 1. Note that there is a choice in Q. 1.

2. Answer ALL parts of a question TOGETHER.

Q 1. This question concerns many-electron atoms. .(a) Write the Hamiltonian for many-electron atom with nuclear charge +Ze

and N electrons. (10)(b) Explain the approximation used in order to separate the many-electron

Hamiltonian into a sum of one-electron Hamiltonians and define the termatomic orbitf;ll. (10)

(c) Two types of functions frequently used to represent atomic orbitals areSlater orbitals and Gaussian orbitals. A Slater orbital is of the form

n+ 1 .

(2~) 2 rn-le-i;r ym( 8 cjJ) , while a Gaussian orbital is of the form~(2n)! I' .Nnrn-le-aryr(8,cjJ). Discuss, with reference to the appropriate eigenvalueequations, if these wavefunctions correctly represent: .'

(i) the behaviour as the electron approaches very close to the nucleus. .(5)

(ii) the behaviour as the electron moves far from the nucleus.. (5)

(iii) the angular distribhtion of the electrons.(iv) or is there something missing?

(10)(10)

OR

Ql. Using the LCAO-MO approach, a trial wavefunction for Hz+ may be writtenas cjJ=Cll/lls+C2l/l1s ,where l/lls and l/llS are h y dro g enic atomic orbitalsA 8 A 8centered on nuclei A and B, respectively. Answer the following questions:

(a) Write secular determinant explaining each term in the determinant. (5)(b) Express matrix elements in your secular determinant using Coulomb

integral, exchange integraL overlap integral and energy of hydrogenicatomic orbital. (10)

(c) Solve the secular equation and represent the energies in terms of CoulombintegraL exchange integraL overlap integral and .energy of hydrogenic

atomic orbital. (10)(d) Obtain the energies of Hz"'-relative to the completely dissociated species,

H+ and H. (5)(e) Plot the two energies obtained in part (d) versus the internuclear

separation, R. (5)(D Plot the separate contributions of the Coulomb integral and the exchange

integral to the stability of Hz+versus the internuclear separation, R. (5)(g) Based on your plots in part (fL explain the reason of chemical bonding in

Hz+. _:: (10) --,/''

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~ molefractionofB Y ~ C.dJ 11 0 1-J--.!;{ -f~ 'L

b I~). 8 '1 2:

Q2. Consider a porphyrin molecule having 18 IT electrons which can bemodeled either as a square (length 1000 pm) or a circle (diameter 1000 pm).The experimental lowest energy absorption transition of this porphyrin occurs at770 nm. (h = 6.626x10-34 Js, C = 2.998x108 ms-I, me = 9.109x10--31 kg, 1 amu =1.661 X 10-27 kg)

(a) For the square and circular models draw (no derivation required) the

energy level diagrams up to the first unoccupied state. (10+10)(b) Using appropriate and complete mathematical calculations determine thelowest energy absorption transition for the two models and suggest which(and why) would be a better model for this porphyrin. (15+15)

Q3. A wavefunction for hydrogen atom is1

(

1

)

3/2

(

r) (

r)

q;(r,e,cf- - 2-- exp --./32 Tl ao ao 2ao

(a) Assign n, 1, ml and name the atomic orbital. (5)(b) Calculate the most probable distance of this electron from the nucleus in

terms of aD. (10)(c) Calculate the average distance of this electron from the nucleus in terms

of aD. (10)(d) Calculate the most probable point for this electron in terms of aD. (10)(e) Sketch the radial wave function and the radial distribution function versus

r for this orbital indicating most probable distance, average distance, andIX) I

most probable point. (Given: [rne-ar dr= a~~l );....(5 + 5 + 5)

Q4. (a) Decide whether the following statements are true or false giving

reasoning. (5+5)i. For a system composed of C components with chemical potentials Pi and n;the number of moles of the i-th component, Lf Pini will always have itssmallest possible value. .

ii. A body in equilibrium and in thermal and mechanieal contact with areservoir at constant temperature and pressure will have the lowestpossible value of the Gibbs energy for that body.

(b) One mole of a pure substance at constant pressureis found to melt at Tm and boil at Tv. For atemperature range Tlow