Equations: q trans = 2 mkt h 2. , Q = q N, Q = qn N! , < P > = kt P = , C v = < E > V 2. e 1 e h /kt vib = h k = h k, rot = h2.

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1 Constants: R = J mol -1 K -1 = L atm mol -1 K -1 k B = cm -1 /K = 1.38 x J/K 1 a.m.u. = x kg 1 atm = x 10 5 Nm -2 = 760 Torr h = x Js For H 2 O = 2.1 x 10-4 K -1, = 49.6 x 10-6 atm -1, vap H = kj/mol, fus H = kj/mol, C p = 75 J/mol-K Equations: q trans = 2 mkt h 2 P = q vib = P = P 0 exp S = k B T ln Q T h /2kT e 1 e h /kt vib = h k = h k, rot = h2 8 Ik, = h 2 2 mkt Mgh RT dp = P H trans, dp = H trans dt RT 2 dt V trans + k B ln Q = V 1 V T P 2 = P 1 + trsh m trs V m ln T 2 T 1, ln P 2 P 1 ln K 2 K 1 S = R = H m R N i = 1 3/2 V, qrot = 8 2 IkT h 2 RT V b a ln Q 2, < P > = kt V V, = V 1 V P = trsh m R 1 T 1, P = P 1 T 0 exp Mgh 2 RT 1 T 1 1 T 2, = T P H, H = C P T x i ln x i where N is the number of components = mgh, P = gh, ds q T, Q = q N, Q = qn N!, < E > = ln Q, dw = PdV, dh = du + PdV S = R ln, du = w, C mono V = 3 2 nr, C mono P = 5 2 nr, S = C Vln T 2, T 1, C v = < E > T S = nr ln V 2 V 1, S = C p ln T 2 T 1, T 2 T 1 = V 1 V 2 nr/cv, H = C p T Name 1

2 NORTH CAROLINA STATE UNIVERSITY Department of Chemistry Name CH 431 Physical Chemistry Practice Final Exam Please answer all questions. Please sign each page. The honor pledge states: This is my work. I have not used information from other students or any notes during theis exam period. Please sign the bottom of each page of the exam. I. Short Answer 1. Describe the partition function for a two-state system. The energy difference between the states is E. A. Sketch the two level system and label the energy difference. What is the magnitude (numerical value) of the partition function at B. absolute zero? C. at infinite temperature? D. At any temperature T (20 points) A. Description of the partition function (labeled sketch of the energy levels). B. T = 0, q = C. T =, q = D. q(t) = Name 2

3 2. Describe the significance of the a and b parameters in the van der Waal's equation of state. A. Physical origin of parameter a (5 points). B. Physical origin of parameter b (5 points). 3. If you have data on the folded, F and unfolded, U states of a protein (i.e. the equilibrium U F) as a function of temperature, describe how you would determine the A. free energy of binding, B. the enthalpy and C. the entropy as a function of temperature. Assume that you have a spectroscopic probe of the concentration of unfolded [U] and folded [F] protein at each temperature (25 points). A. Description of how G o is obtained from experiment. B. Description of how H o is obtained from experiment by the van t Hoff equation. C. Description of how S o is obtained from experiment. Name 3

4 4. We have used a combined first and second law expression for the change in internal energy du. This expression is the total derivative of U in terms of it natural variables (entropy and volume). Please write this expression and explain in terms of the first and second law (30 points). A. First Law. du = Explain physically what the terms in the first law mean. B. Second law. ds = Explain physically what this expression of the second law means. In particular, explain the use of the equals sign in the expression above. C. Combined First and Second law. du = What are the natural variables for the internal energy change? Answer: du = TdS PdV. The natural variables of internal energy are entropy, S and volume, V. D. What is a third law entropy? Write down the general equation used to calculate the entropy of a substance in the liquid phase as a third law entropy. Include the relevant term for the phase transition. E. What is residual entropy? You may use an example or demonstrate using an equation. Name 4

5 5. The kinetic theory of gases is used to describe the average properties of a collection of molecules in an ideal gas. Please write an expression for the internal energy and average speed of the molecules in an ideal gas based on the kinetic theory of gases (20 points). A. Molar internal energy U m =. B. Root-mean-square Speed sqrt(<u 2 >) =. 6. What is osmotic pressure? Use a diagram and an equation to show how you would measure the osmotic pressure of a 1 M NaCl solution (10 points). = 7. What is ideal solubility? How does it relate to colligative properties. Please an equation if possible to describe ideal solubility in terms of the mole fraction of the solute (10 points). A. Describe the chemical potential of a solute in solution in equilibrium with the solid (in the bottom of a beaker). B. Solve for the mole fraction. D. What is fus G*? Include this in the equation and solve. fus G* =. Name 5

6 Final answer (partial credit given for correct answer even if derivation is incomplete): x 2 =. II. Numerical 1. Calculate the total entropy change for the following processes including the the entropy of the system and the entropy of the surroundings. You may assume that the gas is ideal (20 points). a. reversible isothermal expansion of 1 mole of gas from 0.05 L to 1.0 L at 300 K. S =. b. reversible adiabatic expansion of 1 mole of gas from 0.05 L to 1.0 L. S =. c. constant pressure expansion of 1 moles of gas against 1 atm of pressure at 300 K. The initial pressure is 24.6 atm. Please give your answer in Joules/Kelvin. S =. Name 6

7 d. heating 1 moles of a monatomic ideal gas reversibly from 300 K to 500 K at constant volume. S =. 2. Thermophilic bacteria live in near boiling water in thermal vents. Their proteins can withstand very temperatures that would denature proteins from other organisms. In deep ocean vents the boiling temperature of water is increased due to the high pressure on the ocean floor. Calculate the boiling point of water on ocean floor in the middle of the Pacific Ocean at a depth of 1000 m. For ease of calculation you may assume that the gravitation acceleration g = 10 m/s 2 (20 points). T boil = in the thermal vent. Name 7

8 3. Given an initial temperature of T 1 = 280 K, what is the final temperature in an adiabatic expansion of monatomic ideal gas from V 1 = 1 L to V 2 = 10 L? Please include a derivation for this calculation. (10 points) T 2 =. 4. The data the reaction 2 N 2 (g) + O 2 (g) = 2 N 2 O (g) at 25 o C is provided in the table below. (a) Calculate r G o for this reaction as written at 25 o C (b) calculate K for this reaction (c) given initial partial pressures P(N 2 ) = 0.5 bar, P(O 2 ) = 1.0 bar and P(N 2 O) = 0.5 bar calculate r G for this reaction. (d) make a table showing the mole fraction of each gas at equilibrium in terms of the extent of reaction x (YOU DO NOT NEED TO SOLVE FOR x) (e) Calculate r G o at 1000 K. (30 points) N 2 (g) O 2 (g) N 2 O (g) f G o (kj/mol) S o (J/mol-K) o C p (J/mol-K) (a) r G o = (units) (b) K =. (c) r G = (units) Name 8

9 (d) Table 2 N 2 O 2 (g) 2 N 2 O Total Initial moles ½ 1 ½ 3 Equilibrium moles Equilibrium Mole fraction (e) r G o at 1000 K = (units) 5. Calculate the equilibrium temperature reached when a 1 mole of ice is added to 4 moles of water at 25 o C. Please account for the enthalpy of phase transition of the ice. What is the entropy change for the ice melting including both the process of melting and thermal equilibration with the water? (15 points) T eq =. S =. Name 9

10 6. Calculate the equilibrium constant for the dissociation of chlorine (15 points): Cl 2 (g) 2 Cl (g) Use the molecular partition functions and the available data given below. The molar mass of chlorine is M(Cl) = a.u. Note that rotational partition functions for homonuclear diatomics should include a symmetry number of 2. You may assume that D o has already been corrected for the zero point energy. Note that the partition functions are most conveniently expressed in terms of the rotational and vibrational temperatures. The translational partition function is conveniently written in terms of the thermal wavelength,, q trans = V/ 3. The thermal wavelength for chlorine is = 1.69 x meters. Note: you may assume that the electronic degeneracy is one (g elec = 1). Species vib (K) rot (K) (m) D o (kj/mol) Cl x A. Analytical expression for the equilibrium constant B. Numerical evaluation of the equilibrium constant at 298 K. K =. Name 10

11 7. The standard free energy for hydrolysis of ATP to ADP is ATP + H2O ADP + Phosphate The standard free energy change for the hydrolysis of ATP is G o = kj/mol at 37 o C and 1 atm. The free energy of this reaction can be used to power the sodium ion pump. For a concentration of ATP = 1.5 mm to ADP = 150 M, what must be the concentration of phosphate to obtain a sufficient chemical potential to pump 10 moles of NaCl across the membrane a membrane with a potential of 69 mv. (10 points) Phosphate concentration =. Name 11

Exam Thermodynamics 12 April 2018

1 Exam Thermodynamics 12 April 2018 Please, hand in your answers to problems 1, 2, 3 and 4 on separate sheets. Put your name and student number on each sheet. The examination time is 12:30 until 15:30.