FIRST PUBLIC EXAMINATION SUBJECT 3: PHYSICAL CHEMISTRY

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1 CCHE 4273 FIRST PUBLIC EXAMINATION Trinity Term 2005 Preliminary Examination in Chemistry SUBJECT 3: PHYSICAL CHEMISTRY Wednesday June 8 th 2005, 9:30am Time allowed: 2 ½ hours Candidates should answer ALL questions in Section A and any TWO questions in Section B. Use SEPARATE booklets for your answers to Section A and Section B. The numbers in square brackets indicate the weight that the Examiners expect to assign to each part of the question. Fundamental Constants Molar gas constant, R Planck constant, h Boltzmann constant, k B Speed of light, c Avogadro constant, N A Electron mass, m e Elementary charge, e Faraday constant, F Atomic mass unit, u Vacuum permittivity, ε o Gravitational constant, G = J K 1 mol 1 = J s = J K 1 = m s 1 = mol 1 = kg = C = C mol 1 = kg = J 1 C 2 m 1 = N m 2 kg 2 Other Conventions P o = 1 bar = Pa 1 atm = kpa = 760 Torr Do not open this paper until instructed to do so by an invigilator. 1

2 SECTION A Answer ALL SIX questions in this section. Note that the questions in this section do not all carry the same number of marks. 1. (a) Define the terms (i) rate equation, (ii) half-life and (iii) molecularity, as used in chemical kinetics. [3] Show that the half life for a first order reaction, A B, with rate constant k, is t 1/2 = ln 2 k [2] (e) The radioactive decay of 239 U is a first-order process with a half life of 1410 seconds. Determine the rate constant for the decay. [1] In a given sample, the rate of disintegration of 239 U is 640 s 1. After what period will the rate have fallen to 15 s 1? [2] 239 U decays to form 239 Np, which then decays in a first order process with a half-life of 56 hours to form 239 Pu. 239 U 239 Np 239 Pu Sketch how the relative amounts of 239 U, 239 Np and 239 Pu will vary with time in a sample that initially contains only 239 U. [2] 2. (a) Choose a particle that has been shown experimentally to display wave-particle duality, and briefly describe situations in which it appears to behave as i) a wave and ii) a particle. [4] Diffraction experiments using X-rays or electrons are often used in determining the structures of bulk crystals and surfaces, respectively. Bragg s law of diffraction is nλ = 2d sinθ, where n is the diffraction order, λ the wavelength, d the spacing between atomic planes, and θ the scattering angle. (i) (ii) When X-rays of wavelength 192 pm are used to study a crystal plane of Si, a first order diffraction peak is observed at a scattering angle of 30 degrees. What is the atomic spacing in this plane? [2] The surface structure of Si may be probed using a beam of electrons with the same de Broglie wavelength as the X-rays in i). What is the velocity of such a beam? [2] CCHE

3 3. The Maxwell-Boltzmann distribution of molecular speeds is: f(v) = 4π M 2πRT 3/2 v 2 exp -Mv 2 2RT (a) Show that the most probable speed is given by v mp = 2RT M 1/2 [4] Determine the most probable speeds for H 2 and D 2 at a temperature of 298 K (atomic masses of H and D are and g mol 1, respectively) and sketch, on a single diagram, the form of the Maxwell-Boltzmann distribution for the two gases. [3] 4. A glass bulb of volume 2.0 dm 3 is filled with methane to a pressure of 0.8 bar at a temperature of 298 K. Assume methane behaves as an ideal gas. (a) Use the equipartition theorem of classical thermodynamics to determine the heat capacities at constant volume and constant pressure, C V and C p, of the methane. Ignore vibrational contributions. [2] Determine the amount of heat required to raise the temperature of the gas by 20 K (ignore absorption of heat by the bulb). [2] Calculate the change in entropy of the methane, S, associated with the heating process. Justify the sign of your result. [3] 5. (a) What is meant by the activity of an ion in solution? How is it related to the molality, m? [2] The Debye-Hückel limiting law, relating the activity coefficient γ ± to the ionic strength I, is log γ ± = -A z + z I 1/2 where A = (kg mol 1 ) 1/2 and z + and z - are the ionic charges. State the assumptions on which this law is based. [3] Plot a suitable graph to confirm that the following mean activity coefficients for NaCl solutions at 298 K obey the Debye-Huckel limiting law, and comment on the magnitude of the slope m / mol kg γ ± [4] CCHE Turn over

4 6. (a) Explain what is meant by the Born interpretation in quantum mechanics. [1] State and justify the conditions that a wavefunction must satisfy as a consequence of the Born interpretation. [2] Based on your answer to b), explain which of the following functions could be suitable wavefunctions for a quantum mechanical particle confined within the stated limits (N is a normalisation constant in each case). (i) ψ 1 (x) = N tan(kx) x > 0 (ii) ψ 2(x) = N ekx x -L x L (iii) ψ 3(x) = N sin(nπx/l) -L x L (iv) ψ 4(x) = -Nx ; -L x < 0 Nx ; 0 x L [4] Determine the normalisation constant for wavefunction ψ 3 above. [2] CCHE

5 SECTION B Answer any TWO questions from this section all questions carry equal marks. 7. (a) What is meant by the steady state approximation (SSA) in reaction kinetics? Why is it useful? [3] A possible ion-molecule reaction mechanism for synthesis of ammonia in interstellar gas clouds is shown below. N + + H 2 NH + + H k 1 NH + + H 2 NH H k 2 NH H 2 NH H k 3 NH H 2 NH H k 4 NH e - NH 3 + H k 5 NH e - NH 2 +2H k 6 Use the steady state approximation to derive equations for the concentrations of the intermediates NH +, NH 2 +, NH 3 + and NH 4 + in terms of the reactant concentrations [N + ], [H 2 ] and [e - ]. Treat the electrons as you would any other reactant. [12] Show that the overall rate of production of NH 3 is given by d[nh 3 ] dt = k 1k 5 k 5 +k 6 [N + ][H 2 ] [4] What is the origin of the activation energy in a chemical reaction? [2] (e) The rates of many ion-molecule reactions show virtually no dependence on temperature. (i) What does this imply about their activation energy? [2] (ii) What relevance does this have to reactions occurring in the interstellar medium? [2] CCHE Turn over

6 8. A mass of 0.8 g of methanol was burned in excess dry oxygen in a sealed container. The combustion was carried out under adiabatic conditions. The initial temperature of the reactants was K. As a result of reaction, the temperature rose by 0.73 K. The total heat released by the combustion was determined to be kj. (a) What is meant by adiabatic conditions? [2] Write a balanced equation for the reaction between methanol and oxygen. [2] Calculate the mass of H 2 O formed when all of the methanol initially present had been consumed. [m H = g mol 1, m C = g mol -1, m O = 16.0 g mol 1 ] [2] (e) The vapour pressure of water near 298 K is 3.33 x 10 3 Pa. When the reaction was complete, the vapour phase was found to be just saturated with water vapour, but no liquid was present. What was the volume of the container in which the reaction was carried out? Assume water behaves as an ideal gas. [4] Calculate the molar internal energy change and the molar enthalpy change for the combustion of methanol under the conditions of the experiment. Assume that all reactants and products are in the gas phase. [6] (f) The standard molar enthalpies of formation at 298 K of carbon dioxide and gaseous water are kj mol 1, and kj mol 1, respectively. Estimate the standard molar enthalpy of formation of methanol. [4] (g) Use the Clausius-Clapeyron equation, together with the vapour pressure of water at 298 K and at its boiling point, to calculate the enthalpy of vapourization of water. Assume H vap is independent of temperature over this range. [5] CCHE

7 9. Consider an electron approaching a finite potential barrier of height V 0. The dashed line labelled E indicates the total energy of the electron. (a) Write down the Hamiltonian operators for Regions 1 and 2. [2] For an electron incident on the barrier from the left, show that the following functions are eigenfunctions of the appropriate Hamiltonians for Regions 1 and 2, respectively. ψ 1 = A e ik 1x + B e -ik 1x ψ 2 = C e -k 2x [8] Note: A, B and C are constants proportional to the amplitude of the waves incident on, reflected from, and transmitted through the barrier; k 1 and k 2 are the wavevectors in Regions 1 and 2. Given that ψ and dψ dx must be continuous at each boundary, obtain a set of two equations that the wavefunctions in must satisfy. [4] Use your equations from to show that the transmission probability (the probability of finding the electron inside the barrier) is given by T = C A 2 2 4k 1 = k k [6] 2 (e) How does the behaviour of the quantum mechanical particle differ from that of a classical particle subject to the same potential? Are there situations in which this is relevant to chemistry? [5] CCHE Turn over

8 10. Answer EITHER Part A OR Part B. Part A (a) Describe the type of order present (if any) and its effect on the type of motion occurring at a molecular level for the following states of matter: i) crystalline solid ii) liquid iii) gas [6] Sketch radial distribution functions for a typical solid, liquid and gas, and explain the features present in your sketches. [6] Sketch a typical interaction potential for two molecules separated by a distance r, and describe the types of interaction that may contribute to each region of the potential. [4] The interaction energy associated with dispersion forces between two identical molecules is given by U disp (r) = A r 6 where A = 3 4 α 0 2 I (4πεε 0 ) 2 Repulsive interactions at short range may be modelled by the potential U rep (r) = B C r 12 In the above, α 0 is the molecular polarisability, I is the ionization potential, and A, B and C are constants. Using the following data for gaseous Xe, determine the position of the maximum in the radial distribution function (you may assume that ε = 1). Polarisability volume α = α 0 /(4πε 0 ) = 4.0 Å 3 Ionization potential I = ev B = 6.48 x J C = 0.39 x 10-9 m. [9] CCHE

9 Part B (a) Explain what is meant by the term standard electrode potential. [3] The following standard electrode potentials have been measured at a temperature of 298 K. Ag + + e Ag E o = 0.80 V Ag 2+ + e Ag + E o = 1.98 V Calculate the EMF for a cell in which the cell reaction is the disproportionation 2Ag + Ag + Ag 2+ and hence determine the equilibrium constant, K, for disproportionation. Comment on your result. [6] The EMF of the cell Pt H 2 (g) (p = 1 atm) HBr(aq) (a = 10-4 ) AgBr(s) Ag is V at 298 K. Use this together with data given in to calculate the solubility product of AgBr. [6] The EMF (in Volts) of the Harned cell: Pt H 2 (g) (p = 1 atm) HCl(aq) (a = 1) AgCl(s) Ag is found to vary with temperature as E = A + B(T 273) + C(T 273) 2 where A, B, and C are constants and T is the temperature in Kelvin. Measurements at three different temperatures give the following values: T / K EMF / V Determine the values of A, B and C in the expression for the cell EMF, and hence the values of G o, S o and H o for the cell reaction. [10] END OF PAPER CCHE Turn over

FIRST PUBLIC EXAMINATION. Trinity Term Preliminary Examination in Chemistry SUBJECT 3: PHYSICAL CHEMISTRY. Time allowed: 2 ½ hours

FIRST PUBLIC EXAMINATION. Trinity Term Preliminary Examination in Chemistry SUBJECT 3: PHYSICAL CHEMISTRY. Time allowed: 2 ½ hours FIRST PUBLIC EXAMINATION Trinity Term 004 Preliminary Examination in Chemistry SUBJECT 3: PHYSICAL CHEMISTRY Wednesday, June 9 th 004, 9.30 a.m. to 1 noon Time allowed: ½ hours Candidates should answer