ME 262A - Physical Gas Dynamics 1996 Final Exam: Open Book Portion. h = 6.62 x J s Energy conversion factor: 1 calorie = 4.

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1 Name: ME 262A - Physical Gas Dynamics 1996 Final Exam: Open Book Portion Useful data and information: k = 1.38 x J/K h = 6.62 x J s Energy conversion factor: 1 calorie = 4.2 J 1. (40 points) A Carbon fullerene is a hollow, spherical (soccer-ball like) molecule made of 60 atoms of carbon (C 60 ) that exists in a gaseous state at a temperature of 600K. The molecule has a diameter of 0.7 x 10-9 m. (a) Using kinetic theory, estimate the upper limit to the specific heat of fullerene vapor, ˆ c p, assuming that the vibrational energy modes are fully excited. Neglect electronic excitation. (b) Estimate the viscosity, thermal conductivity, and the Prandtl number at a pressure of 0.1 Torr and a temperature of 600K. (c) C 60 is partially hydrogenated by reactions with atomic hydrogen: C 60 + H C 60 H + ε o Estimate the equilibrium constant at 0.1 atm and 600K for this reaction, if the energy released in the reaction, εo, is 60 kcal/mole. You may assume that the internal structure of C 60 and C 60 H are approximately equal for the purpose of this calculation. 2. (40 points) A small, insulated spherical tank of diameter, D = 0.1m supplies propellant to a thruster on a spacecraft. The tank is filled with helium at a pressure of p o = 10 2 Pa and T o = 298K when it develops a small hole of diameter d = 10-5 m, allowing the helium to escape into vacuum. (a) Show that the time variation in temperature, T(t) of the helium in the tank is: T T o = c o d 2 t D 3 where c o is the initial mean speed of the atoms in the tank. It is not necessary to rederive expressions for the one way flux or one way energy flux of particles through the hole. (b) What is the pressure in the tank after 10 4 seconds? 2

2 Consider now the case where the tank is instead surrounded by a secondary vessel that is initially evacuated. The secondary vessel has five times the volume of the helium tank. The helium now escapes into this secondary vessel after the small hole is created in the helium tank wall. You may assume that there is no heat loss to the walls of either tank. (c) After a long time, the gas in the tank and secondary vessel establishes an equilibrium. What is the temperature and density of the gas? (d) What is the increase in entropy of the system, comprising the gas in both the tank and the secondary vessel. 3. (20 points) In larger polyatomic molecules, there may be an additional internal energy mode associated with the rotation of functional groups (a smaller collection of atoms bonded to the large molecule) with respect to the rest of the molecule. A consideration of the quantum mechanical problem of the free internal rotator gives the following internal energy levels: with ε jrot = kθ r j 2 j =... 2, 1, 0, 1, 2... g j = 1 Here, j is the rotational quantum number which can take on all positive and negative integer values including zero. The characteristic rotational temperature,θ r, depends on the moment of inertia of the rotating functional group, and in many cases of interest, range from a few to tens of degrees Kelvin. (a) Derive an expression for the rotational partition function, in the limit that the energy spacing between rotational levels is small in comparison to the average rotational energy of the functional group. (b) Using the result of (a), derive an expression for the rotational energy. (c) Derive an expression for the rotational contribution to the entropy.

3 Name: ME 262A - Physical Gas Dynamics 1997 Final Exam: Closed Book and Notes Portion You have 45 minutes to work on this part of the exam. If you finish this part before the allocated time, hand it in, and begin working on the open book portion. Each question is of equal value. 1. You are studying the free molecular flow of a gas around a sphere. The mean free path, λ, for the particular gas of interest is 10-7 m at 1 atm and 298K. The lowest pressure that you can achieve in your facility is 10-7 Torr at 298K. What is the appropriate diameter of your test sphere so that Kn 1? D m 2. The mean speed of O2 (M = 32 g/mole) in a stagnant gas at 273K is 425 m/s. The mean speed of H2 (M = 2g/mole) at 273K is m/s. 3. The temperature of a single component gas is doubled at constant pressure. The collision frequency between molecules in that gas increases by a factor: θ / θ o =. 4. Acetylene (C2H2) is a linear molecule. What is the maximum contribution of the vibrational modes to the total energy of the molecule? e ˆ vib = R ˆ T 5. Write the expression for the one-way kinetic energy flux along the x1 direction. + Λ ke x1 = 6. What is the dimension of the energy distribution function? 7. The intermolecular potential between molecules in water vapor scales as r -3. The molecular collision cross section therefor scales as σ g ( ) ~ g ( ) 8. Consider the endothermic dissociation reaction: M + N 2 + ε o M + N + N. As the temperature is increased, there is a shift in the equilibrium towards (check): products reactants.

4 9. Write down an expression for the enthalpy (mole basis) of species j, in terms of the temperature dependent specific heat and the heat of formation at reference temperature. 10. A cell contains a mole of argon. (a) The temperature of the cell is decreased. The entropy of the argon in the cell increases, decreases, stays the same. (b) An additional 1 mole of argon is added to the cell while maintaining a constant temperature. The entropy of the argon (per atom basis) decreases, increases, stays the same. 11. The bond dissociation energy R ˆ O θ 2 d of the O2 molecule is 495 kj/mole. If the reference temperature is 0K and the reference species are the atoms, then the enthalpy of O 2 at 298K is kj/mole. (Note R ˆ =8.314 J/mole K). 12. Write down a relationship for the Gibbs energy at reference pressure (mole basis) in terms of the sensible enthalpy, the heat of formation, and the entropy for an ideal gas. 13. Consider the general reaction A + B C + D. Write down an expression for the equilibrium constant in terms of the equilibrium constants for the formation reactions of the species A,B,C, and D. 14. State the first postulate of equilibrium statistical mechanics. 15. Consider the translational motion of a molecule confined to a square container. The degeneracy of the lowest translational level is:. The degeneracy of next most energetic level is:.

5 16. Consider the following macrostate of a system comprising of 5 indistinguishable fermions (identified as circles- o) distributed amongst the four translational energy levels shown. ej Cj j=4 o 12e 1 j=3 oo 9e 3 j=2 o 6e 3 j=1 o 3e 1 (i) What is the multiplicity of level 3 (j =3)? W i3 = (ii) What is the multiplicity of the macrostate? W = 17. A system is comprised of three macrostates of multiplicities W1=1, W2=2, W3=2. This system has an entropy S = k (k is the Boltzmann constant). 18. The characteristic vibrational temperature of NO is 2740K. At a temperature of T = 2740K, the fraction of NO molecules in the v = 0 state is Fo =. The fraction of molecules in the v = 1 state is F 1 =. 19. Consider the deuterium-hydrogen diatomic molecule (DH) at a temperature T =10θ r. The fraction of molecules in the l = 1 rotational state, F 1 =. 20. For the following chemical reaction: A + M + ε o B + C + M, complete the expression for the equilibrium constant by filling in the parenthesis: K p/ po ( ) kt ( T) = p o V ( ) exp ( ) ( ) kt