ME 501. Exam #2 2 December 2009 Prof. Lucht. Choose two (2) of problems 1, 2, and 3: Problem #1 50 points Problem #2 50 points Problem #3 50 points

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1 1 Name ME 501 Exam # December 009 Prof. Lucht 1. POINT DISTRIBUTION Choose two () of problems 1,, and 3: Problem #1 50 points Problem # 50 points Problem #3 50 points You are required to do two of the problems. Please indicate the problems you have chosen.. EXAM INSTRUCTIONS Write your name on each sheet. This exam is closed book and closed notes. Seven equation sheets are attached. When working the problems, list all assumptions, and begin with the basic equations. If you do not have time to complete evaluation of integrals or of terms numerically, remember that the significant credit on each problem will be given for setting up the problem correctly and/or obtaining the correct analytical solution.

2 ME 501 Exam # Dec 009 Name 1. (50 points) The system shown below has available energy levels of 0, k B, k B, and 3k B units, where = 100 K. The degeneracy of each of the four levels is given by g = 10, ,000. The thermodynamic assembly has 1000 particles (N = 1000) and the temperature of the assembly is 00 K. For this dilute assembly, the population distribution for the most probable macrostate is given by the Boltzmann distribution law, g exp / kbt g exp / kbt N N mp N N g exp / k T Z B (a) Using the Boltzmann distribution law, calculate the most probable macrostate {N 0mp, N 1mp, N mp, N 3mp }. Round the populations to the nearest integer. (b) For corrected Maxwell-Boltzmann statistics, the number of microstates in a particular macrostate {N } is given by N g WCMBenN s N! Use the Stirling approximation ( ln N! N ln N N ) to show g ln WCMBenN s N N ln N (c) What is the entropy (J/K) of the assembly? (d) What is the energy (J) of the assembly? (e) Calculate the number of microstates associated with two macrostates that are very similar to the most probable macrostate. Macrostate A is given by {N 0mp, N 1mp +5, N mp -10, N 3m +5}. Macrostate B is given by {N 0mp, N 1mp -5, N mp +10, N 3m -5}. Comment on the results.

3 3 ME 501 Exam # Dec 009 Name. Diatomic hydrogen gas is contained in a rigid pressure vessel with m. The initial pressure is 1 kpa and the temperature is 50 K. (a) Calculate the amount of heat transfer required to raise the temperature of the gas from 50 K to 100 K. Assume that the H is an ideal gas (translational mode fully excited), a rigid rotator, and a harmonic oscillator (see equation sheets). For H, odd-j rotational levels have a nuclear spin statistical weighting factor (NSSW) of 3, and even-j rotational levels have an NSSW of 1. Do not assume that Z T /. Instead use the combined rotationalnuclear partition function: rot Z NSSW g J rot, nuc rot, J exp J J B rot kt Consider rotational levels with 0 J 4. Recall that P NkBT the next page to be useful in organizing your work.. You may the tables on (b) Calculate the change in entropy S S1as the hydrogen gas is heated from 50 K to 100 K at constant volume. For H : 6339 K, 87.5 K, rot Be cm, e cm The ground electronic level has a degeneracy of 1: Level 0: g 1, / hc 0cm 0 0

4 4 ME 501 Exam # Dec 009 Name T= 50 K J T = 100 K J

5 5 ME 501 Exam # Dec 009 Name 3. A rigid pressure vessel with a volume of 1 m 3 contains 0.1 kmols of diatomic oxygen (O ) at 300K. The gas is heated to 3500 K. How many O-atoms (monatomic oxygen) does the pressure vessel contain at 3500 K, and what is the assembly pressure in Pascals? Hint: Consider the reaction O O. Also, remember that P NkB T. The general solution for quadratic equations is in the equation sheets. Alternatively, you could iteratively solve the equation that you get by assuming for the initial step that N O at 3500 K is the same as the initial value at 300K. Assume that diatomic oxygen is a rigid rotator and harmonic oscillator, neglect any ionization, and use the following characteristic temperature and dissociation energy data: rot (K) (K) (D 0 /hc) (cm -1 ) O ,300 Consider the following electronic levels: O : Level 0: g 3, ( / hc) 0 cm Level 1: g, ( / hc) 7,90 cm 1 1 O: Level 0: g 5, ( / hc) 0 cm Level 1: g 3, ( / hc) 158 cm 0 0 Level : g 1, ( / hc) 6 cm

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12 1 Equation Sheets TdS Equations: du T ds Pdv, h u P v, g h T s, f u T s [ dg] RT [ d(ln f )] ds Q / T T T rev First Law - Closed System: U Q W neglecting KE, PE First Law - Open System: neglecting KE, PE Q W d z z shaft u d au Pvf VdA dt CV CS Specific Heats: h s u s cp T ; cv T T T T T P P v v Schrödinger wave equation: ( rt, ) i ( r, t) V( r, t) ( r, t) t m Time-independent Schrödinger wave equation: ( r) V( r) ( r) ( r) m Normalization Condition: r t r t d three - dimensional form * 1 (, ) (, ) Dynamical Variable r p p p p B( r, p) Operator rop r pop i p op op i t Bop B( r, i ) xop, p i x

13 13 xˆ yˆ zˆ Cartesian coord x y z 1 1 eˆ ˆ ˆ r e e spherical coord r r rsin x y z Cartesian coord r r r r sin r sin r sin Spherical coord Expectation Values: B * ( r, t) Bop ( r, t) d three dimensional form ddxdydz Cartesian coord, sin dr dr d d Spherical coord B * ( x, t) Bop ( x, t) dx one - dimensional form B * ( x, t) B op Bop ( x, t) dx one dimensional form Molecular Energy Levels hc 1 1 e exe G(v) v v with zero-point energy included hc G zero-point energy subtracted, G(0) = 0 (v) ev exe v v rot Fv( J) Bv J( J 1) Dv J ( J 1) hc 1 Bv Be ev 4 B De Dv for all 3 e e v Degeneracies: g J 1 g 1 rot

14 14 Rigid Rotator, Harmonic Oscillator rot rot 1 F( J) Be J( J 1) rot J( J 1) G(v) e v hc k hc B Characteristic Temperatures: hc hc hc K rot B k k k cm e e 1 B B B Term Symbols Atomic: S+1L J, J = (L+S), (L+S-1),... L-S Term Symbol: S P D F G... L: Term Symbols Molecular: S+1 Term Symbol ma mb m m A B reduced mass Boltzmann Relation: S kb ln Wtot kbln Wmp Number of Microstates in a Macrostate for large g, N g g ln( WmCMB, ) N ln N N N ln N N g N g N lnwmfd, N ln g ln N g g N g N lnwmbe, N ln g ln N g

15 15 Partition Function, Boltzmann Distribution Relations g exp / kbt Z g exp / kbt, N N, g level degeneracy Z N N, E N Z trans B mk T h 3/ 1 Zrot for rigid rotator, T rot, for homonuclear molecule, rot 1 for heteronuclear molecule, Z exp / T 1 exp / T [with zero-point energy included in G(v)] Z 1 1 exp / T [without zero-point energy included in G(v)] 1 1 Z I I molecule AB, nuclear spins I, nuc A B A I B Rotational Distributions, Homonuclear Molecule N N vj J v Zrot, nuc J 1 NSSW exp rot J J 1 / T rigid rotator Chemical Equilibrium For the general reaction A A B B CC DD C D C D NC ND ZC ZD exp R / kbt ; A B A B N N Z Z A B A B D D D D R A 0A B 0B C 0C D 0D

16 16 Statistical Thermodynamic Property Calculations: CMB Statistics (ln Z) E U NkB T T E Nk B exp / T 1 zero-point energy included E NkB exp / T 1 zero-point energy not included ln Z P NkB T T for an ideal gas: PNk T, Pv R T, R N k ln S k ln W k W B tot B mp B u u Avogadro B Z (ln Z) Z E S NkB ln T 1 NkB ln 1 N T N T Z (ln Z ) Z E Strans NkB ln T 1 NkB ln 1 N T N T tr tr tr tr NkB T P k 3/ 5 m 5/ 5 ln ln ln B h (ln Z) Sint NkB ln Zint T NkB ln Zint T E T int

17 17 Integrals and Derivatives f ( x) g( y) dxdy f( x) dx g( y) dy x y x y x1 y1 x1 y1 r r f () r g() h() drd d f() r dr g() d h() d r11 1 r1 1 1 d sin( ax ) cos( ) acos( ax) d ax asin( ax) dx dx 1 x sin ( ax) dx cos( ax) sin( ax) a z 1 x cos ( ax) dx cos( ax) sin( ax) a z 3 1 sin ( ax) dx cos( ax) csin ( ax) 3a h z sin( ax) cos( ax) dx 1 sin ( ax) a z 1 3 sin( ax) cos ( ax) dx cos ( ax) 3a z 1 3 sin ( ax) cos( ax) dx sin ( ax) 3a z sin( ax) cos ( ax) dx cos ( ax) 4a x sin ( ax) dx x x sin ax cos ax 4 4a 8a x x 1 x cos ax 6 4a 8a 4a 3 x sin ( ax) dx sin 3 ax Quadratic solution For the equation ax bx c 0, x b b 4 ac a

18 18 Constants and Conversion Factors Universal gas constant N m J J Ru ( gmol)( K) ( gmol)( K) ( kmol)( K) 5 N Pressure 1 atm bars x MPa m 8 m 10 cm Speed of light c sec sec Electron charge e coul 31 Electron mass me kg Atomic mass unit amu kg Planck's constant h h J sec ; J sec Dielectric permittivity 0 1 coul J m Avogadro constant NAv gmol or NAv kmol Boltzmann constant J kb K 1 J = 1 kg-m /sec