Worked Solutions to Problems

Transcription

1 Worked Solutions to Problems. Water A. Phase diagram a. The three phases of water coexist in equilibrium at a unique temperature and pressure (called the triple point): T tr 7.6 K 0.0 P tr 6. x 0 bar b. If pressure decreases, boiling point decreases, but melting point increases (slightly). c. Beyond this point, there is no distinction between liquid and vapour phases of water. Put alternatively, it is possible to have liquid to vapour transition by a continuous path going around the critical point. (In contrast, solid-liquid transition is discontinuous.) d. T 00K, P.0 bar : liquid phase T 70K, P.00 bar : solid phase e. Below P 6. x 0 bar, ice heated isobarically will sublimate to vapour. f. If x l and x v are the mole fractions of water in liquid and vapour phases, V Vl V Vv V x x l l V l V V v v x v V V V V 4.6 x xl Vl 0.40 V v l x l l 0 ( x ) l V v Mumbai, India, July 00 5

2 B. lausius lapeyron equation a. dp dt V T V molar enthalpy change in phase transition molar change in volume in phase transition. For ice-liquid water transition : > 0 V < 0, since ice is less dense than water. dp dt < 0 Since V is not large, the P-T curve for this transition is steep, with a negative slope. Thus decrease of pressure increases the melting point slightly. For liquid water - vapour transition û > 0 dp > dt 0 û V < 0 Decrease of pressure decreases the boiling point. b. lausius - lapeyron equation for (solid) liquid - vapour transition is dp dt P RT vap This equation follows from the lapeyron equation under the assumptions:. Vapour follows ideal gas law.. Molar volume of the condensed phase is negligible compared to molar volume of vapour phase.. If further is assumed to be constant (no variation with T), the eq. vap is integrated to give P ln P R vap T T 5 5 Mumbai, India, July 00

3 ere P.0 bar, T 7.5 K T 9.5 K R 8. J K mol - P.0 bar vap kj mol The estimate is based on assumptions, and. c. For ice - liquid water equilibrium, use lapeyron equation At T 7.5 K, P.0 bar. Assume that for a small change in T, is constant. V Integrating the lapeyron equation above P P T 7.95 K, V û (fusion) T ln T 6008 Jmol V x x 0 6 m mol P P 7.0 bar P 8.0 bar The estimate is based on assumption.. Irreversible condensation a. n the P-T plane, this equilibrium state is a solid phase (ice). Water in liquid phase at this temperature and pressure is not an equilibrium state - it is a supercooled state that does not lie on the given P-T plane. b. Treating the metastable state as equilibrium state, we can go from the supercooled liquid state to the solid state at the same temperature and pressure by a sequence of reversible steps.. Supercooled liquid at -.0 to liquid at 0 q number of moles x p (liquid water) x change of temperature Mumbai, India, July 00 5

4 8.5g x 76.JK mol 8.05 g mol x.0 K 445 J. liquid at 0 to ice at 0 q 8.5 g x (.5) J g 9505 J. Ice at 0 to ice at.0 q number of moles x p (liquid water) x change of temp. 8.5 x 7.5 J K mol 8.05 g mol x (.0 K) 705. J q q q q 8765 J Since all the steps are at the constant pressure of.00 bar, q But is independent of the path, i.e., it depends only on the end points. Thus for the irreversible condensation of supercooled liquid to ice q 8765 J c. The actual irreversible path between the two end states of the system is replaced by the sequence of three reversible steps, as above. For each reversible step, S can be calculated. S n T T T p dt n p ln T T S 8.5 g 8.05 g mol 76. J K mol x 7.5 ln J K Mumbai, India, July 00

5 S J K T 7.5 S 8.5 g 8.05 g mol 7.5 J K mol ln J K S system S S S.0 J K S sur q T sur sur JK S univ S system S sur.54 JK The entropy of the universe increases in the irreversible process, as expected by the Second Law of Thermodynamics.. van der Waals gases a. For a van der Waals gas PV b P n a n ý n R T R T V R T V The ratio of the magnitudes of the second and third terms on the right side is : b b PV RT, taking PV nrt up to zeroth order. n a a The ratio of the magnitudes of the fourth and third terms on the right side is : nb V bp RT i. From the ratios above, it follows that at sufficiently high temperature for any given pressure, the second term dominates the third and fourth terms. Therefore, a b R T b P ý > R T For small P, Z nearly equals unity. Mumbai, India, July 00 55

6 ii. At lower temperatures, the third term can be greater (in magnitude) than the second term. It may be greater (in magnitude) than the fourth term also, provided P is not too large. Since the third term has a negative sign, this implies that Z can be less than unity. iii. For a 0 ý b P R T which shows that Z increases linearly with P. b. elium has negligible value of a. Graph () corresponds to e and () corresponds to. c. Above T > T c, only one phase (the gaseous phase) exists, that is the cubic equation in V has only one real root. Thus isotherm () corresponds to T < T c. d. At T T c, the three roots coincide at V V c This is an inflexion point. dp dv V c d P dv V c 0 The first condition gives RTc (V nb) c n a V c () The second condition gives RTc n a 4 (V n b) V These V c c n b equations and For e, Tc c T c give 8a 7b R 5. K () For T, 8K Since, T c ( ) is greater than T c (e), is liquefied more readily than e Mumbai, India, July 00

7 e. W V P dv V V R T V b V a dv V V b RT ln V b 56.7 L bar mol a V V. Rates and reaction mechanisms a. Mechanism : d [I dt Since the first step is fast, there is a pre - equilibrium : K [I d[i dt [I Mechanism : k [I [ k K [I [ k [ [I d[i dt ' K d[i dt k [I [I d [I d k [ ' K [I [ k [ [I Both mechanisms are consistent with the observed rate law. b. i. k E With E a a T A e the Ea RT T k R ln k given 70 kj mol numerical values, Mumbai, India, July 00 57

8 ii. The activation energy is greater than the bond dissociation energy of I. ence the second step is rate determining in both the mechanisms. c. The activation energy E a for the reverse reaction is E ' a E a û kj mol d. i. d [I dt '' K k d [I dt [IAr [I [IAr[Ar [I[Ar '' K k k [I [I [Ar [Ar ii. A possible reason why this is negative is that Ea is positive and less in magnitude than, while o is negative. k k A K '' e we know û G k A E a RT e û6 R e e û þ û* RT (E û þ ) a RT Tû S The activation energy for the overall reactionis E a û þ 4. Enzyme catalysis a. i. The differential rate equations for the Michaelis-Menten mechanism are d [ES k [E [S dt d[p dt k [ES - k ' [ES k [ES () () Mumbai, India, July 00

9 In the steady-state approximation, d[es 0 () dt Eq. () then gives k [E [S [ES (4) k ' k ow [E 0 [E [ES (5) where [E 0 is the total enzyme concentration. Eqs. (4) and (5) gives [E 0[S [ ES Km [S where K m ' k k k is the Michaelis-Menten constant. From eq. (), d[p k [E 0 [S (7) dt K [S m (6) Since the backward rate is ignored, our analysis applies to the initial rate of formation of P and not close to equilibrium. Further, since the enzyme concentration is generally much smaller than the substrate concentration, [S is nearly equal to [S 0 in the initial stage of the reaction. Thus, according to the Michaelis-Menten mechanism, the initial rate versus substrate concentration is described by eq. (7), where [S is replaced by [S 0. For [S << K m, Initialrate k K m [E [S i.e., initial rate varies linearly with [S. o (8) For [S >> K m, Initial rate k [E 0 (9) i.e., for large substrate concentration, initial rate approaches a constant value k [E 0. Thus the indicated features of the graph are consistent with Michaelis-Menten mechanism. Mumbai, India, July 00 59

10 ii. The asymptotic value of initial rate is k [E 0 From the graph, k [E 0.0 x 0-6 M s With [E 0.5 x 0 9 M we get k.0 x 0 s iii. From eq. (7), for [S K m, the initial rate is half the asymptotic value. From the graph, therefore, K m 5.0 x 0 5 M For [S.0 x 0 4 M, using eq. (7) again, 9 [.0 x 0 s x [.5 x 0 M x [.0 x 0 Initial rate 5 4 [5.0 x 0 M [.0 x 0 M 4 M.0 x 0 6 M s k k iv. We have Km k 5.0 x 0 5 M The enzyme equilibrates with the substrate quickly, that is the first step of equilibration between E, S and [ES is very fast. This means that is much greater than k. Therefore, neglecting k above, k x 0 M k The equilibrium constant K for the formation of ES from E and S is, K k 5.0 x 0 k M b. From the graph at the new temperature, k [E x 0 6 M s k i.e., k 6.0 x 0 M s.5 x 0 M x 0 s Mumbai, India, July 00

11 Using Arrhenius relation for temperature dependence of rate constant : k A e E a RT (0) where E a is the molar activation energy. k(t ) k(t ) E a R T T e k(t ) ln i.e. k(t ) E () a R T T k (0) ow.0, R 8. J K mol k (85) E a 0.4 kj mol c. i. The fraction of the enzyme that binds with the substrate is, from eq. (6): [ES [E K 0 m [S [S () where [S is nearly equal to [S 0 in the initial stage of the reaction. ow [S 0.0 x x mol L.0 x 0 M and K m 5.0 x 0 5 M [ES [E 0 (5.0 x x 0 M.0 x 0 )M 0.98 early the whole of the enzyme is bound with the substrate. ii. From eq. (7), Integrating the equation gives, d[s dt k [E K m 0 [S [S Mumbai, India, July 00 6

12 [S K [S [S k [E t m ln 0 0 [S 0 If at t T, [S /[S 0, T k [E 0 Km ln [S 0.0 x 0 mol ere [E 0.0 x 0.0 x 0 L k.0 x 0 s, K m 5.0 x 0 5 M, 9 M () (4) [S 0.0 x 0 M Substituting these values in eq. (4) gives T 84 s Thus 50% of the antibiotic dose is inactivated in 84 s. d. i. The differential rate equations for the situation are : d [ES dt k [E [S k [ES k [ES (5) d k [E [I k [EI dt [EI (6) d [P dt k [ES (7) where k and k enzyme-inhibitor reaction. are the forward and backward rate constants for the Applying steady-state approximation to [ES and [EI, [ES k [E [S k k (8) k [E [I and [EI (9) k ow [E 0 [E [ES [EI (0) 6 6 Mumbai, India, July 00

13 Eliminating [E and [EI from eqs. (8) to (0) gives : [ES [S K m [E 0 [S [I KI (M) () d[p dt [S k K m [E 0 [S [I KI (M) () k ere, K I(M) k to E and I. is the equilibrium constant for the dissociation of EI The degree of inhibition is i r r 0 Km [I KI (M) Using eq. (), i () [I [S Km KI (M) For fixed [I, i decreases with increase in [S (competitive inhibition). and for large [S, i 0, i.e., the inhibitor ceases to play any role. ii. For small [S i [I K (M) I [I If r r0, i 4 4 i.e., [I K I x (M).5 x 0 4 M The inhibitor concentration required to reduce the rate of inactivation by a factor of 4 is.5 x 0 4 M; i.e., 0.5 µmol in a volume of.00 ml. Mumbai, India, July 00 6

14 5. Schrödinger equation a. i. ne-dimensional Schrödinger equation for a free particle of mass m:! d % E%! m dx h Œ where E stands for the energy of the particle and ψ its wave function. ii. The boundary conditions are : ψ (0) ψ(l) 0 nly % n (x) sin n Œ L x satisfies the required boundary conditions. ther functions are not possible wave functions of the electron in a one-dimensional rigid box. iii.! m d dx sin n Œ x L! Œ n ml sin nœ x L En! Œ ml n h n 8mL iv. Ground state (n ) Œ x % (x) sin L First excited state (n ) % (x) Œ x sin L Second excited state (n ) % (x) Œ x sin L umber of nodes in points. % n ψ (x) 0 ψ (x) 0 ψ (x) 0 n, apart from the nodes at the end x x L/ x L/ L/ L L L Mumbai, India, July 00

15 v. % % (x) (x) b. In the example sin sin L L sin L Œ[ Œ[ Œ[ L m m The first three energy levels are: 0 L % ( x) L L dx L dx L 0 ( is chosen cos to be real Œ x L ) dx E h. 8mL 0 9 J E 4 E J E 9 E J In the ground state, the four electrons will occupy the levels E and E, each with two electrons. E E E E E E Ground state Lowest excited state Mumbai, India, July 00 65

16 The lowest excitation energy E E J c. The condition that ψ(φ) is single valued demands that (φ) (φ π) % % e iλφ e iλ(φπ) e i πλ i.e. λ m, where m 0, ±, ±, ±,. This shows that angular momentum projection (L z ) cannot be an arbitrary real number but can have only discrete values: m!, where m is a positive or negative integer (including zero). 6. Atomic and molecular orbitals A. Atomic orbitals a. i. % s e r a o % s dv 4 Œ a o 4 Œ x [ ŒD o a o 4 Œ a o ( chosen to be real) % s ao [ Œ a e o r ii. Probability of finding an electron between r and r dr 4Œ r x r a [ ŒD e 0 0 dr This is a maximum at r r max, given by d dr r e r a 0 r rmax Mumbai, India, July 00

17 This gives r max a 0 The s electron is most likely to be found in the neighborhood of r a 0. b. % s 0 at r a0 odal surface is a sphere of radius a 0 0 at % p z Œ odal surface is the xy plane. % d z 0 at cos 0, i.e., cos ± odal surfaces are cones with these values of half-angle, one above the xy plane and the other below it. (ote: all three wave functions vanish as r. At r 0, ψls does not vanish, but the other two wave functions vanish.) c. Each electron in n shell of helium atom has energy Z eff.6 ev elium ground state energy Z eff 7. ev Energy of e ground state ev Ionization energy ( 54.4 Z eff 7.) ev 4.46 ev This gives Z eff.70 B. Molecular orbitals a. % and % are bonding orbitals % ~ and are antibonding orbitals % ~ Bonding orbital o nodal surface between the nuclei. Electronic energy has a minimum at a certain internuclear distance. Qualitative reason: electron has considerable probability of being between the nuclei and thus has attractive potential energy due to both the nuclei. Mumbai, India, July 00 67

18 Antibonding orbital odal surface between the nuclei. Electronic energy decreases monotonically with internuclear distance. ence bound state is not possible. b. R e. 0 0 m D.6 (5.6).76 ev c. It will dissociate to a hydrogen atom in s state and a bare hydrogen nucleus (proton). d. The two electrons occupy the same molecular orbital with the lowest energy. By Pauli s principle, their spins must be antiparallel. ence the total electronic spin is zero. e. In the first excited state of, one electron is in ψ (bonding orbital) and the other in ψ (antibonding orbital). It will dissociate into two hydrogen atoms. f. Using the aufbau principle, in the ground state two electrons of e are in ψ (bonding orbital) and two in ψ (antibonding orbital). The bond order is ½ ( ) 0 Therefore, bound e is unstable and difficult to detect. owever, if one or more electrons are elevated from the antibonding orbital to (higher energy) bonding orbitals, the bond order becomes greater than zero. This is why it is possible to observe e in excited states. 7. Fission a. 5 9 U n 94 8 Sr Xe n 5 9 U n 4 56 Ba 9 6 Kr n b. The net nuclear reaction is 5 9 U n Zr e n 6e (Q) The energy released is Q [m ( 5 U) m ( 94 Zr) m ( 40 e) m n 6m e c Mumbai, India, July 00

19 where the small energy of the initial thermal neutron has been ignored. (m denotes the nuclear mass.) ow m ( 5 U) m ( 5 U) 9m e ignoring the small electronic binding energies compared to rest mass energies. Similarly for other nuclear masses. Q [m ( 5 U) m ( 94 Zr) m ( 40 e) m n c Using the given data, Q. MeV c. MWd 0 6 Js x 4 x 600 s 8.64 x 0 0 J 8.64 x 0 0 o. of atoms of 5 U fissioned. x.60 x 0.5 x 0 x 5 Mass of 5 U fissioned 0.99 g 6.0 x 0.5 x 0 Mass of 5 U in kg uranium removed from the reactor g Abundance of 5 U is 0.6 % 8. Radioactive decay a. µi.7 x 0 4 disintegrations per second (dps). Initial β activity.7 x 0 6 dps d o dt t 0 where o constant..7 0 dps is the number of atoms of 6 0 Bi at t 0 and is its decay Mumbai, India, July 00 69

20 Intial mass of 0 Bi g g b. umber of atoms of 0 Bi at time t is given by 0 e t The number of atoms of 0 Po,, is given by equation d dt where is the decay constant of 0 Po. d dt 0 e -t Using the integrating factor t e t d dt e t 0 e ( )t d dt ( e t Integrating ) 0 e ( )t t e e To calculate, use the condition that at t 0, 0 This gives 0 0 ( )t 0 (e t e t ) 70Mumbai, India, July 00 70

21 The time t T when is maximum is given by the condition d dt t T 0 which gives T ln 4.9 d At t T, can be calculated from above..04 x 0 Mass of 0 Po at t T, 7. x 0 0 g c. α-disintegration rate of 0 Po at t T.8 x 0 5 dps At t T β - disintegration rate of 0 Bi α-disintegration rate of 0 Po.8 x 0 5 dps 9. Redox reactions a. i. ver-all reaction Sn Fe Sn 4 Fe E 0.67 V ûg nfe FE V 9 KJ Mumbai, India, July 00 7

22 ii. E n logk log K K ( 0.67) b. Before the equivalence point, E of the cell is given by following equation E cell ox E o S..E red E o Sn 4 /Sn log [Sn [Sn log [Sn [Sn 4 i. The addition of 5.00 ml of Fe converts 5.00/0.00 of the Sn to Sn 4 ; thus [Sn [Sn 4 5.0/ / E cell 0.0 V. ii. At the equivalence point, add the two expressions corresponding to Sn 4 /Sn and Fe /Fe. E cell ox E S..E red E Sn 4 /Sn [Sn log [Sn 4 E cell ox E S..E red E Fe /Fe [Fe log [Fe to get E E E E log [Sn [Sn [Fe ox red red 4 cell S..E 4 Sn /Sn Fe /Fe [Fe 7Mumbai, India, July 00 7

23 At the equivalence point, [Fe [Sn and [Fe [Sn 4 Thus, E cell E $ ox S..E red E $ Sn 4 /Sn red E $ Fe / Fe 0.4 ()(0.54) V Beyond the equivalence point, E of the cell is given by following equation o o [Fe E ox ES..E rede log Fe /Fe [Fe cell When 0 ml of Fe is added, 0 ml of Fe is in excess. i.e. [Fe [Fe E cell 0.5 V c. i. û G - RT ln K 68.7 sp K J $ û G E - n F E - $, V n ii. u I ui (s) E V u e u E 0.5 V The overall reaction for reduction of u by I is u I e ui (s) E 0.86 V The E value for the reduction of u by I can now be calculated x u I e ui (s) E 0.86 V I e I E 0.55 V Mumbai, India, July 00 7

24 The over-all reaction is u 4I ui (s) I E 0.5 V The positive value of effective E indicates that the reduction reaction is spontaneous. This has come about since in this reaction, I is not only a reducing agent, but is also a precipitating agent. Precipitation of u as ui is the key step of the reaction, as it practically removes the product u from the solution, driving the reaction in the forward direction. iii. o û G - n F E o ere n, E o 0.5V o û G -. kj o û G - RT lnk logk 5.47 K Solubility of sparingly soluble salts a. Ag 4 (s) Ag 4 - The solubility product Ksp is given by K sp [Ag [ 4 If S is the solubility of Ag 4 [Ag S () The total oxalate concentration, denoted by ox, is ox S [ 4 [ 4 [ 4 () The dissociation reactions are: 4 4 K () 4-4 K (4) 74Mumbai, India, July 00 74

25 Eqs. (), () and (4) give ox S [ - 4 [ - 4 K [ [ 4 KK [ [ - 4 α ox α S where α K K [ K[ KK (5) At p 7, [ 0 7 and α K sp 4S.5 0 At p 5.0, [ 0 5 From the values of K, K and [, we get α 0.86 (6) K sp [S [αs S K sp 4..7 x 0-4 b. [ 0.00 At p 0.8, [ Eq. (5) implies α i.e ox S [ 4 - (7) The total silver ion in the solution is given by Ag S [Ag [Ag [Ag( ) (8) Mumbai, India, July 00 75

26 The complex formation reactions are Ag Ag Ag Ag( ) K.59 0 (9) K (0) From eqs. (8), (9) and (0) Ag S [Ag { K [ K K 4 [ } [Ag β x Ag β x S where Using the values of K, K 4 and [, β. x 0-4 K sp - [Ag [4 [ S [S S K ( sp ) 4 K [ K K 4[ 5.47 x 0 -. Spectrophotometry a. Denote the molar absorptivity of Mn 4 at 440 nm and 545 nm by ε and ε and that of r 7 by and 4 : 95 Lmol cm, 70 Lmol cm, 50 Lmol cm 4 Lmol cm The absorbance A is related to % transmittance T by A - log T From the values given for the sample solution A log A log Mumbai, India, July 00 76

27 ow if one denotes the molar concentrations of Mn 4 - and r 7 - in the steel sample solution by and respectively, we have A 440 x x X x A 545 x x 4 X x Using the given data, we get M 0.00 M Amount of Mn in 00 ml solution moll gmol - 0. L g x 00 % Mn in steel sample.74 0.% Amount of r present in 00 ml solution 0.00 mol L - x x 5.00 g mol - x 0. L 0.08 g 0.08 x 00 % r in steel sample % b. In solution, since all the ligand is consumed in the formation of the complex, [ol x x 0 5 Absorptivity of the complex ol is x 0 L mol cm x 0 mol L.0 cm Mumbai, India, July 00 77

28 If the concentration of the complex ol in solution is, x 0 L mol cm x.0 cm. x 0-5 M [o [o total - [ol x x x 0-5 Similarly, [L [L total - [ol 7 x x. x The complex formation reaction is o L [ol The stability constant K is given by [ol [o [L K.08 x 7 0. Reactions in buffer medium R 4 4e R Ac Ac K i.e a pk a [ [Ac [ Ac [Ac p log [Ac - 78Mumbai, India, July 00 78

29 [ Ac log [Ac - [Ac [Ac - [Ac [Ac [Ac 0.8 [Ac mmoles of acetate (Ac ) present initially in 00 ml 0.8 x mmoles of acetic acid (Ac ) present initially in 00 ml 0.88 x mmoles of R reduced 00 x From the stoichiometry of the equation, mmoles of R will consume moles of for reduction. The is obtained from dissociation of Ac. n complete electrolytic reduction of R, mmoles of Ac mmoles of Ac p p log Mumbai, India, July 00 79

30 . Identification of an inorganic compound a. The white gelatinous precipitate in group (III) obtained by qualitative analysis of solution B indicates the presence of Al ions. The white precipitate with Ag indicates the presence of l ions. From the above data the compound A must be a dimer of aluminium chloride Al l 6. b. The reactions are as follows Al - l6 [Al.6 6l 6l 6Ag 6Agl(s) 6 Agl (s) 4 (aq) Ag( ) or Ag( ) l - Al 4(aq) Al() 4 (s) Al() (s) a (aq) [Al() 4 - a [Al() Al() - 4 (s) - excess of Li All6 Li (Al ) n Li[Al4 4. Ionic and metallic structures a. i. The lattice of al consist of interpenetrating fcc lattices of a and l - ii. iii. The co-ordination number of sodium is six since, it is surrounded by six nearest chloride ions. For al, the number of a ions is: twelve at the edge centres shared equally by four unit cells thereby effectively contributing x /4 80Mumbai, India, July 00 80

31 a ions per unit cell and one at body center. Thus, a total of 4 a ions per unit cell. umber of l - ions is: six at the center of the faces shared equally by two unit cells, thereby effectively contributing 6 / l ions per unit cell and eight at the corners of the unit cell shared equally by eight unit cells thereby effectively contributing 8 /8 l - ion per unit cell. Thus, a total of 4 l - ions per unit cell. ence, the number of formula units of al per unit cell 4a 4l 4al iv. The face diagonal of the cube is equal to times 'a' the lattice constant for al. The anions/anions touch each other along the face diagonal. The anion/cations touch each other along the cell edge. Thus, a (r a r l - ) Face diagonal a 4 r l -..()..() Substituting for 'a' from () into () we get : (r a r l - ) 4 r l - from which, the limiting radius ratio r a / r - l 0.44 v. The chloride ion array is expanded to make the octahedral holes large enough to accommodate the sodium ions since, the r - a / r l ratio of is larger than the ideal limiting value of 0.44 for octahedral six coordination number. vi. As the cation radius is progressively increased, the anions will no longer touch each other and the structure becomes progressively less stable. There is insufficient room for more anions till the cation / anion radius ratio equals 0.7 when, eight anions can just be grouped around the cation resulting in a cubic eight coordination number as in sl. Mumbai, India, July 00 8

32 vii. Generally, the fcc structure with a six coordination number is stable in the cation/anion radius ratio range 0.44 to 0.7. That is, if 0.44 < r /r - < 0.7 then, the resulting ionic structure will generally be al type fcc. b. i. Bragg's law states λ d hkl Sin(θ) 54 pm d d Sin(5.8 ) 54 pm 54 pm 8 pm Sin(5.8 ) 0.7 Thus, the separation between the (00) planes of al is 8 pm. ii. Length of the unit cell edge, a d 00 d 00 a 8pm 566 pm. iii. Since it is an fcc lattice, cell edge, a (r a r - l ) - radius of sodium ion r a a - r l pm c. i. The difference in an hcp and a ccp arrangement is as follows: The two 'A' layers in a hcp arrangement are oriented in the same direction making the packing of successive layers ABAB.. and the pattern repeats after the second layer whereas, they are oriented in the opposite direction in a ccp arrangement resulting in a ABAB packing pattern which repeats after the third layer. The unit cell in a ccp arrangement is based on a cubic lattice whereas in a hcp arrangement it is based on a hexagonal lattice. ii. Packing fraction Volume occupied by 4 atoms Volume of unit cell 8Mumbai, India, July 00 8

33 Let 'a' be the length of the unit cell edge Since it is an fcc lattice, face diagonal a 4r..() Volume of the unit cell a 4 4 π Packing fraction a r...() Substituting for a from () into (), we get 4 4 ( ) Packing fraction 7 (4r) 0.74 Thus, packing fraction in a ccp arrangement 0.74 r iii. The coordination number() and the packing fraction (0.74) remain the same in a hcp as in a ccp arrangement. d. i. For an fcc, face diagonal a 4r i where a lattice constant r i radius of the nickel atom r i a 5.4 pm 4.6pm 4 4 ii. Volume of unit cell a (.54 Å) 4.76Å iii. Density of ickel, ρi Z M/ V o. of i atoms, Z 4 for an fcc lattice Avogadro constant Z M g mol - ρi V 8.90 g cm cm mol - Mumbai, India, July 00 8

34 5. ompounds of nitrogen a. i. : o. of electrons in the valence shell around nitrogen The Lewis structure for is as shown below : :: :.. : According to VSEPR, the molecule ideally should have linear geometry. owever, this molecule has one single unpaired electron present on nitrogen. Due to the repulsion between the unpaired electron and the other two bonded pairs of electrons, the observed bond angle is less than 80 ( o ). Thus, the shape of the molecule is angular as shown below.. o ii. : o. of electrons in the valence shell around nitrogen (5 ) 8 The Lewis structure is as shown below.... : :: : : : Thus, there are no non-bonded electrons present on nitrogen. The two σ - bonds will prefer to stay at 80 o to minimize repulsion between bonded electron pairs giving a linear geometry (80 o ). The π-bonds do not influence the shape. 84Mumbai, India, July 00 84

35 : o. of electron in the valence shell around nitrogen 5 8 The Lewis structure is as shown below : :: :.. : : 5 o - In case of -, there is a lone pair of electrons present on nitrogen. Due to strong repulsion between the lone pair of electrons and the bonded pairs of electrons the angle between the two bond pairs shrinks from the ideal 0 o to 5 o. b. In case of trimethylamine, the shape of the molecule is pyramidal with a lone pair present on nitrogen. Due to the lone pair Me--Me angle is reduced from 09 o 4 to Si Si Me Me Me Si owever, in case of trisilylamine, d orbital of silicon and p orbital of nitrogen overlaps giving double bond character to the -Si bond. Thus, delocalisation of the lone electron pair of nitrogen takes place and the resultant molecule is planar with 0 bond angle. Si empty d-orbital filled p-orbital Mumbai, India, July 00 85

36 c. Both and B are tricovalent. owever, F is pyramidal in shape. In case of BF, the B-F bond gets double bond character due to the overlapping of p orbitals present on boron and fluorine. The observed bond energy is, therefore, much greater in BF F.. F F F F B F empty p-orbital filled p-orbital d. i. The difference in boiling points of F and is due to hydrogen bonding which is present in ammonia. igh electronegativity of fluorine decreases the basicity of nitrogen in F. Thus, F does not act as a Lewis base. ii. In F, the unshared pair of electrons contributes to a dipole moment in the direction opposite to that of the net dipole moment of the -F bonds. See figure (a)..... F F F F (a) (b) 86Mumbai, India, July 00 86

37 In, the net dipole moment of the - bonds and the dipole moment due to the unshared pair of electrons are in the same direction. See figure (b). e. a 8a( g) 4 a 8a 8g Et aet a Et a is the salt of (yponitrous acid). Structure : or Isomer is: (itramide) 6. Structure elucidation with stereochemistry a. S 4 -oxo-,-pentanedioic acid α - ydroxy carboxylic acids undergo similar reaction. Mumbai, India, July 00 87

38 b. Molecular weight of A 6 0 ml 0.05 M K 8 mg A 000 ml M K 8 g A The acid is dibasic Molecular weight of A 6 80 mg Br 8 mg A 60 gm Br 6 g A A contains one double bond It has anisole ring in the molecule It is formed from It has molecular formula 5 Due to steric hindrance the attachment of the aliphatic portion on the anisole ring will be para with respect to -. ence the structure will be As A forms anhydride the two groups should be on the same side of the double bond. c. Isomers of A ( E ) -( -methoxyphenyl )--pentenedioic acid ( Z ) -( -methoxyphenyl )--pentenedioic acid 88Mumbai, India, July 00 88

39 ( Z ) -( 4-methoxyphenyl )--pentenedioic acid d. Two products are possible when compound A reacts with bromine. Br Br Br Br [ [ Structures and are enantiomers. e. S R Br Br Br Br R S f. B Product obtained by reaction with phenol Product obtained by reaction with resorcinol g. In the formation of compound A from anisole, the attack takes place at the p- position of the group. owever, when compound B is formed from phenol, the attack takes place at the o-position of the group. Steric Mumbai, India, July

40 hindrance of group favours the attack at the para position. Steric hindrance of the group is comparatively less. Thus, the attack is possible at the ortho or para positions. owever, addition at ortho position is favoured as it leads to cyclization of the intermediate acid to stable B. h. Phenol has only one group on the phenyl ring whereas resorcinol has two groups on the phenyl ring at the m-positions. ence, position 4 is considerably more activated (i.e, electron rich) in the case of resorcinol. Phenol Resorcinol Therefore, under identical reaction conditions, the yield of compound is much higher than that of B. 7. rganic spectroscopy and structure determination a. The given Molecular formula is 6 0. Therefore, the possible structures are: I II III IV V VI The MR spectrum of compound A shows a single peak which indicates that all the protons in A are equivalent. This holds true only for structure I. The IUPA name of this compound is -propanone Mumbai, India, July 00

41 The MR spectrum of compound B shows four sets of peaks, which indicate the presence of four non-equivalent protons. This holds true for structures III and IV. owever, for structure IV, no singlet peak (see peak at δ ) will be observed. So, compound B must have structure III. The IUPA name is - methoxyethene. b. b c a Three doublets of doublets centred at 6.5 ppm,.9 ppm,.5 ppm are seen in the spectrum. The assignments in the spectrum are a : 6.5 ppm b :.5 ppm c :.9 ppm Due to the presence of electron donating, the trans proton b has higher electron density and thus more shielded than c. Thus, b appears upfield as compared to c. There is also a singlet line at δ. This corresponds to the in. c. oupling constants a :, 6 z J ( a, b ) z J ( a, c ) 6 z b : 8, z J ( a, b ) z J ( b, c ) 8 z c : 8, 6 z J ( b, c ) 8 z J ( c, a ) 6 z ote: J (difference in two lines in ppm) x (Instrument frequency) Geminal coupling < cis-vicinal coupling < trans-vicinal coupling Mumbai, India, July

42 d. Peak positions in z (for 400 Mz instrument) Peak positions in z (for 600 Mz instrument) e. ompound A will react with malonic acid in the following manner Addition ompound A Malonic Acid - Meldrum's acid ( ) The structure of Meldrum s acid is consistent with the -MR and IR data. The peak in the IR spectrum at cm - is because of the stretching. The presence of peaks only between 0 7 δ in the -MR spectrum indicates that the compound doesn t have any acidic group like or. If compound B reacts, the only possibility is that it will add across the double bond giving a product with molecular formula equal to This molecular formula does not match with the one stated in the problem. 9 9 Mumbai, India, July 00

43 ompound B Malonic Acid f. The increased acidity is due to active group of Meldrum's acid flanked by two groups. The carbanion formed at will be stabilised by these groups, which are coplanar. Meldrum's acid ( ) g. The condensation product of Meldrum's acid with an aromatic aldehyde has the structure Ar 8. Polymer synthesis a. Ag cat / 50 Ethylene P Q b. P Q R Mumbai, India, July

44 Sl l l R S l l alc. K S T c. BS peroxide l 4 Br K Et/ Br / / or i) a / ii) / / p-xylene dimethyl benzene-,4-bis(acetate) d. Three signals (three singlets for -, and aromatic protons) e. Structure of polymer n f. Polymer I) K / / II) / Polymer LiAl 4 g. With Glycerol (being a triol), cross-links between the polymer chains involving Mumbai, India, July 00

45 the secondary hydroxyl group will form giving a three-dimensional network polymer is possible. Glycerol The polymer is unsuitable for drawing fibers because of its cross-linked, resinlike property. 9. rganic synthesis involving regioselection a. The product obtained in the presence of catalyst SbF 6 is m-bromophenol. From the mass spectra given in the problem, direct bromination of phenol gives o/p bromo derivatives as group present in phenol is o/p- directing. b. SbF 6 ( at ) ( ) S 4 Br a Br Phenol A Mg / TF / toluene Br B MgBr ompound B may undergo nucleophilic reaction at the carbon bearing bromine. ompound contains a carbanion and hence functions as a Mumbai, India, July

46 nucleophile and will attack an electrophile. Thus, reactivity of B is reversed on its conversion to (umpolung). c. yclohexanone D MgBr D Tramadol d. Ph ( ) Ph ( ) Ph ( ) Ph ( ) Tramadol has two asymmetric carbon atoms. It has two pairs of enantiomers. 0. arbon acids a. The molecular formula of the keto ester is 5 8. Since X and Y are keto esters, they must have the following unitsketo group ester group This accounts for 4.The remaining part comprises of 8. Thus, only two types of ester groups are possible, methyl or ethyl Mumbai, India, July 00

47 For a methyl ester: will be a part of the ester moiety. This leaves 5 to be attached. For an ethyl ester: will be a part of the ester group. Therefore unit needs to be accounted for. Therefore, possible structures of the keto esters are: Structure I Structure II Structure III b. Reaction sequence for keto esters * Ph Br a Structure I Ph Keto acid ( 4 ) LDA ( equiv. ) I Ph / Ph Keto acid ( 4 ) * a Ph Br Structure Ill Ph LDA ( equiv. ) I Ph / Ph Keto acid ( ) Mumbai, India, July

48 Structure ll ame Ph Br LDA ( equiv. ) MeI Ph LDA ( equvi. ) / MeI Ph Ph / / Ph Ph Bl - Keto acid B l - Keto acid - - Ph ( 4 ) Ph ( 4 ) Structure I gives a keto acid with molecular formula 4 which matches with the formula of the keto acid obtained from Y. Structure I is Y. Structure II gives a neutral compound with molecular formula 4 that matches with the molecular formula of the neutral acid stated for X. Structure II is X. Structure III gives a keto acid with molecular formula that also does not match with any given molecular formula Mumbai, India, July 00

49 ence the two keto esters are : ompound Y ompound X (Structure I) (Structure II) α-keto ester β-keto ester c. The β-keto ester gives on hydrolysis a β-keto acid. This acid readily undergoes decarboxylation involving a 6-membered transition state, giving a neutral product ( Ketone ). Ph Ph Ph d. i. When equivalent of LDA is used compound X produces a carbanion (monoanion) as shown below. LDA ( equiv. ) - Ph Ph arbanion ompound X ( monoanion ) ii. Use of equivalents of LDA leads to the formation of a dianion. LDA ( equiv. ) - - Ph ompound X Ph Dianion Mumbai, India, July

50 . Amino acids and enzymes a. The protonated amino group has an electron withdrawing effect. This enhances the release of proton from the neighboring, by stabilizing the conjugate base -. This effect is greater when the - is physically closer to. As group is present on the α-carbon, the effect is greater on α- than on the γ-. So the pka value of α- is lower than that of γ-. b. The ratio of ionized to unionized γ- group is obtained by using enderson-asselbalch equation, p pk a [ log [ The p 6. and pka of γ- group is 4.. Substituting these values in the above equation we get, [ log [ [ % c. Glutamic acid has two pka values lower than 7.0 and one pka value higher than 7.0. Thus, the isoelectric point (pi) for glutamic acid will lie between the two acidic pka values. at pi (. 4.)/.5 p 6. At p.5, net charge on glutamic acid will be zero since this p coincides with pi of glutamic acid. ence, glutamic acid will be stationary at p Mumbai, India, July 00

51 hbcse rd International Page 0 hemistry lympiad Preparatory 05/0/0 Problems d. In the hydrolysis of the glycosidic bond, the glycosidic bridge oxygen goes with 4 of the sugar B. n cleavage, 8 from water will be found on of sugar A. A 4 B n 8 8 A 4 B TE: The reaction proceeds with a carbonium ion stabilized on the of sugar A. e. Most glycosidases contain two carboxylates at the active site that are catalytically important. Lysozyme is active only when one carboxylate is protonated and the other is deprotonated. A descending limb on the alkaline side of the p profile is due to ionization of -. An ascending limb on the acidic side is due to protonation of - -. Thus the enzyme activity drops sharply on either side of the optimum p. The ideal state of ionization at p 5 will be, Lysozyme (Asp-5) (Glu-5) - Mumbai, India, July

52 TE: It is desirable to study the amino acid side chains (R-groups) and their ionization properties. The pka values of these groups significantly determine the p dependence of enzyme activity. f. Answers and 4 are correct. Ionization of leads to generation of a negatively charged species,. This charged species is poorly stabilized by diminished polarity and enhanced negative charge. ence ionization of group is suppressed and the pka is elevated. g. The ratios of pseudo-first order rate constant (at M ) in (a) to the first order rate constants in (b) and (c) provide the effective local concentrations. For example, () (0.4) / (0.00) 00 i.e the effective concentration 00 M () (0) / (0.00) 0,000 i.e. the effective concentration 0,000 M h. In addition to the enhanced local concentration effect, the group in () is better oriented to act in catalysis. The double bond restricts the motion of and thus reduces the number of unsuitable orientation of, thereby enhancing the reaction rate.. oenzyme chemistry a. Step : Schiff base formation - - : 0 0 Mumbai, India, July 00

53 hbcse rd International Page 0 hemistry lympiad Preparatory 05/0/0 Problems Step : Proton abstraction : B - (-) - - : : : Base (-) Step : Reprotonation (-) - * - : : B* Step 4: ydrolysis - : - b. From the information stated in the problem, the following conclusions can be drawn: Structure : Removal of the phosphate group does not hamper the activity. This indicates that the phosphate is not critical for the activity of PLP. Mumbai, India, July

54 Similarly, Structure : - is not critical. Structure 4: Phenolic is needed in the free form. Structure 5:, a well-known electron withdrawing group, causes benzaldehyde to become activated. ence positively charged nitrogen in structure must be also important for its electron withdrawing effect. Structure 6: Electron withdrawing effect of is only effective from the para position. Introduction of this group at meta position leads to an inactive analog. c. Role of metal ion: The metal ion is involved in a chelation, as shown below, and provides an explanation for the critical role of the phenolic. The planar structure formed due to chelation assists in the electron flow. M - d. Step : Schiff base formation and decarboxlyation : : Mumbai, India, July 00

55 hbcse rd International Page 05 hemistry lympiad Preparatory 05/0/0 Problems Step : Tautomerization - - : :.. Step : ydrolysis - : - GABA e. Step : Schiff base formation followed by carbon- carbon bond scission. B ( X ) Mumbai, India, July

56 Step : Tautomerization followed by hydrolysis Protein folding a. The planar amide group, that is, α,, and the next α are in a single plane - is stabilized by resonance. The - bond of the amide assumes partial double bond character and the overlap between p orbitals of, and is maximized. The α s across this partial double bond can assume cis or trans arrangement... - b. With nineteen of the amino acids, the trans arrangement is sterically favoured (i. e. it is comparatively less crowded). In the case of proline, cis and trans arrangements are almost equally crowded. trans amide cis - amide Mumbai, India, July 00

57 hbcse rd International Page 07 hemistry lympiad Preparatory 05/0/0 Problems trans amide cis - amide c. ote about Ramchandran diagram: In a polypeptide, the amide units are planar (partial double bond character across the - bond) but the bonds connecting and α, and the carbonyl carbon and α are free to rotate. These rotational angles are defined as φ and ψ, respectively. The conformation of the main chain is completely defined by these angles. nly some combinations of these angles are allowed while others are disallowed due to steric hindrance. The allowed range of φ and ψ angles are visualised as a steric contour diagram, shown below, known as the Ramachandran diagram. For nineteen amino acids, the conformational choice is largely restricted to the so-called α and β regions on left half of the Ramachandran diagram (Panel A). This is due to the L - chiral nature of amino acids and the steric effects of their R groups. Glycine is an achiral residue with as the R group. Therefore, much larger conformational regions on both left and right halves of Ramachandran diagram are accessible to this residue (Panel B). Panel A Panel B Mumbai, India, July

58 d. onsecutive residues in α conformation form the α-helix. Similarly, consecutive residues in β conformation form the β-sheet. Both α-helix and β- sheet structures feature extensive networks of hydrogen bonds which stabilise them. Thus random combinations of α and β conformations are rarely found. α - helix β - sheet e. For a polypeptide to fold in an aqueous environment, nearly half the R groups should be nonpolar (water hating) and the other half polar (water loving). Upon folding to form a globular protein, the nonpolar R groups are packed inside (away from water) while the polar groups are positioned on the surface (in contact with water). The phenomenon is similar to the hydrophobic aggregation of a micellar structure in water. If all the R groups are either polar or non-polar, no hydrophobic segregation is possible, and no folding will occur Mumbai, India, July 00

59 hbcse rd International Page 09 hemistry lympiad Preparatory 05/0/0 Problems f. Alternating polar/nonpolar periodicity of R groups favors β-sheets. All the nonpolar groups will face the apolar surface while the polar groups will be exposed to water. So the net folding will be like a β-sheet. n the other hand, a complex periodic pattern of R group polarities is needed in forming the α- helix. ydrophobic Apolar Induced periodicity surface (secondary (primary (interface) structure) sequence) 4. Protein sequencing The sequence of amino acids in a protein or polypeptide is expressed starting from the -terminal amino acid. From Edman degradation method the -terminal amino acid is Asp. In the -terminal fragment generated by trypsin or Br this amino acid should, therefore, be in position. All other peptides generated by Br cleavage will Mumbai, India, July

60 be preceded by Met on their -terminal side. Likewise, all peptides generated by trypsin should be preceded by Arg or Lys. As we proceed from -terminal amino acid to -terminal amino acid, we carefully examine the different amino acids in each position shown in Table(a) and (b) For the first fragment starting from -terminal Asp in position, we look for residues common in each position to Br and trypsin cleaved peptides. This gives Position Residue Asp -Pro/Tyr - Tyr -Val -Ile/Leu -Arg..() At position 6 Arg will render the polypeptide susceptible to trypsin. Therefore, 7 th residue of this Br fragment (Tablea) should be same as residue in another peptide generated by trypsin and 8 th residue of this Br fragment will be same as residue in Table (b). Therefore we get 7 8 Gly/Phe - Tyr..() Since 8 will be Tyr, Pro will be assigned to position of the polypeptide..() Residue 9 in the polypeptide should be at position in the Table(b) and residues 0,,, and 4 should be at positions 4,5,6,7 and 8 respectively in Table(b). The same residues should be in positions onwards in Table(a). one of the residues in position (Tableb) is same as in position in Table (a). owever, positions 4 to 8 in Table (b) have residues common with positions to 5 in Table (a). Further Glu in position (Table a) will be preceded by Met (since it is a part of Br cleaved peptide). And position in Table (b) has Met. Therefore, we get Met- Glu - Thr - Ser - Ilu - Leu Position 5 in the polypeptide can now be firmly assigned to Ilu..(4)..(5) 0 0 Mumbai, India, July 00

61 hbcse rd International Page hemistry lympiad Preparatory 05/0/0 Problems Positions 5 and 6 in the polypeptide will be beyond residue 8 in the trypsin cleaved peptide (not shown here). We now attempt to construct the remaining trypsin or Br fragments. Table (a) shows Arg in position. This will be preceded by a Met. Matching of the unassigned residues in position in Table (a) with those in position in Table (b) and for subsequent positions by the procedure demonstrated earlier that will give. Met - Arg - Tyr - Pro - is - Asn - Trp - Phe - Lys - Gly - ys..(6) (The last two residues are the unassigned residues in position and in Table b) onsidering (), (5) and (6) together it is now possible to firmly assign position 7 on the polypeptide to Gly..(7) a. The amino acid sequence common to the first fragments (-terminal) obtained by Br and trypsin treatments is 4 5 Asp - Pro - Tyr - Val - Ile b. The sequence of the first fragment generated by Br treatment is Asp- Pro - Tyr- Val- Ile -Arg -Gly -Tyr To complete the sequence of the polypeptide we need to construct the sequence of another trypsin fragment. Starting from position 4-(Arg) in Table (a) we get the sequence, Arg-Phe-is-Thr-Ala... (8) At this stage, we again examine the unassigned residues. The Arg in (8) will have to be serially preceded by Asn, Gln, Gly and Met (these are the unassigned residues in respective positions in Table (a). We then get the sequence, Met-Gly-Gln-Asn-Arg-Phe-is-Thr-Ala And following the Ala in (9)..(9) Leu-Ser-ys-Glu..(0) Mumbai, India, July 00

62 From (9) and (0), we get the sequence Met-Gly-Gln-Asn-Arg-Phe-is-Thr-Ala-Leu-Ser-ys-Glu..() Since the smallest fragment is a dipeptide (Table b) and (6) shows that it follows Lys, it follows that this will be at the -terminal end. Therefore, the partial sequence shown in (6) will follow the partial sequence shown in ().Thus, we get Met-Gly-Gln-Asn-Arg-Phe-is-Thr-Ala-Leu-Ser-ys-Glu-Met-Arg-Tyr-Pro-is-Asn- Trp-Phe-Lys-Gly-ys..() There is already a Met in position 9 of the polypeptide. The next Met can only come earliest at position 7 since Br fragment have at least 8 amino acids. Therefore, the starting residues of () can be assigned position 7. This leaves positions 5 and 6 which will be filled by the unassigned residues Val and Ala in the Br fragment at positions 6 and 7 (Table a). c. The final sequence, therefore, will be Trypsin Br Asp - Pro - Tyr - Val - Ile - Arg - Gly - Tyr - Met - Glu - Thr Br Trypsin Ser - Ile - Leu - Val - Ala - Met - Gly - Gln - Asn - Arg - Phe Br Trypsin is - Thr - Ala - Leu - Ser - ys - Glu - Met - Arg - Tyr - Pro Trypsin is - Asn - Trp - Phe - Lys - Gly - ys Arrows ( ) indicate the Br and trypsin-labile sites. d. There are 6 basic amino acid residues in the polypeptide. 6/40 5% e. An α helix has.6 amino acid residues per turn of 5.4Å. Thus, the length of the polypeptide in α helical conformation will be : 40/ Å. Mumbai, India, July 00

63 hbcse rd International Page hemistry lympiad Preparatory 05/0/0 Problems e. The polypeptide has 40 amino acids. Since each amino acid is coded for by a triplet of nucleotides, the total number of nucleotide pairs in the double stranded DA of the exon will be 40 x 0 base pairs. The molecular weight of the DA making the exon 0 x x Da g. If the exon contains 0 base pairs and A and are in equal numbers, there will be 60 A-T pairs and 60 G- pairs. Each A-T pair is held by two -bonds and each G- pair is held by three -bonds. ence the total number of - bonds holding this double helix is : ( 60 x ) ( 60 x ) 00 Mumbai, India, July 00