Chem 521 Statistical MechanicsFall 2014

Homework1. This file is a list of suggested problems for the entire course.2. I will periodically update this list.3. A subset of these problems will be assigned as homework, please make sure you check themost up to date version of the file for assigned homework. 4. Assignments are due at noonon Fridays, unless otherwise stated. You can drop them in the box in front of my office room248, chem 73 building.5. Some of these problems may be used in the first midterm and the final exam. Assuch I strongly encourage you to also try to solve problems that have not been assigned ashomework.6. I will only provide solutions for the problems that are assigned. Feel free to stop by theoffice hours to ask questions about the rest of them.7. If you notice a typo or a mistake, please let me know.

1. Reif 1.9

2. Reif 1.10

3. Reif 1.13

4. Reif 1.16

5. Reif 1.17

6. Reif 1.22

7. Reif 1.23

8. Reif 1.29 hint: 0

sinx2

x2dx =

pi

2

9. Evaluation of the integral0exxndx:

(a) Show that0exxndx = n

0exxn1dx = n!

(b) Define: (n) =0exxn1dx = (n 1)! , show that (1) = 1 and (1

2) =pi

(Hint: use x = y2 and use the Gaussian integral)

10. Using a programing language of your choice, for N=10,100 and p = 12, 13, 23;

(a) Generate a number N of trajectories, calculate m for those trajectories and gen-erate a histogram of PN(m)

(b) Discuss the accuracy of PN(m) as you increase N and N .

(c) Fit the data to a Gaussian function f(m) = ae(x)2/22 . Calculate and as

a function of N . How does this compare with your expectation of these valuesbased on the probability p?

11. Repeat the problem above with N = 100 and p = 120, 1

200. Does a Gaussian fit still

work? What about a Poisson function?

12. Assume that the probability density of a process is given by a Gaussian function(x)dx e(x)2/22dx.(a) Show that < x >= and

< (x )2 > = . Make sure that you start with a

normalized probability function.

(b) In a random walk, = N(p q)l and = 2Npql, where N is the number ofsteps, p is the probability of moving forward and q = 1 p. Using a program ofyour choice (Matlab, Wolfram, Maple) plot this function for a p > 1

2and three

different N values. Space N in a logarithmic fashion (e.g. 1000, 10000, 100000)to visualize the evolution of the function. Try this with p = 1

2

13. Consider a container that can be divided by an imaginary partition into two partswith volumes V1 and V2, respectively. There are N molecules in the gas state in thecontainer.

(a) On average how many molecules are there in partition 1? How many are there inpartition 2?

(b) What is the probability that N1 of the molecules are in partition 1 and N2 =N N1 are in partition 2?

(c) Using Stirlings formula to derive an expression for the probability above whenV2 V1 and N2 N1. Assume that N2 1 but do not make the sameassumption about N1. Let both N2 and V2 approach infinity in such a way thattheir ratio approaches a limiting value equal to n. (Credit: Liu)

14. Reif 2.1

15. Reif 2.2

16. Reif 2.4

17. Reif 2.5

18. Reif 2.6

19. Reif 2.7

20. (a) A classical particle is free to move in one dimension in a box with walls located atx = 0 and x = L and has a known energy that lie between E and E + E. Drawthe classical phase space of the particle, indicating regions of the space that isavailable to the particle. What is the probability that the particle is found withina distance L0 from x = 0? (L0 < L)

(b) Consider the same box now with two weakly interacting particles, each of massm and free to move in one dimension. Denote the respective position coordinatesof the two particles by x1 and x2 and their respective momenta by p1 and p2. The

total energy of the system lie between E and E + E. The phase space of thissystem is a four-dimensional one. Plot three projections of this phase space. Theone involving x1 and x2, the one involving p1 and p2 and a last one involving x1and p1. Indicates in each regions accessible to the system. What is the probabilitythat a single particle lie within a distance L0 from x = 0? What is the probabilityof finding both particles in that space?

(c) Solve part (a) for a quantum mechanical particle with an energy that lie on oneof the eigenvalues of the system En. What is the answer for the limit of reallylarge n?

21. Reif 3.2

22. Reif 3.5

23. Reif 3.6

24. Reif 4.3

25. Consider an ideal gas of N noninteracting particles in d dimensions (d is an arbitrarypositive integer) contained in a (hyper) volume V . From the partition function, cal-culate F (free energy), P , S, E (energy), and = Cp/Cv and notice which quantitiesdepend on d. Numerical factors in the partition function are unimportant. (Credit:Kamien)

26. A system with temperature-independent heat capacity C is initially at temperature T1and a system with temperature-independent heat capacity 2C is initially at tempera-ture T2, where T2 < T1.

(a) The systems are placed in thermal contact and exchange heat until they are inthermal equilibrium. What is the final temperature of those two bodies?

(b) A heat engine is run in infinitesimal cycles between the two bodies leading toa slow reversible approach to a final common temperature. What is the finaltemperature?

(c) How much work is extracted in this process? (credit: Liu)

27. For a pure substance in an external magnetic field H the Helmholtz free energy canbe written as

dF = SdT PdV VMdHIn most practical cases where the substance is liquid or solid the term PdV is negligi-ble because the substance is highly incompressible. We may define the two heat capaci-ties, one at constant field and the other at constant magnetization M as CH = T

(ST

)H

and CM = T(ST

)M

show that (credit: Liu)

CH CM = TV(M

T

)H

(H

T

)M

28. The work done on a rubber band when it is stretched can be written as dL were isthe tension and L is the length of the rubber band.

(a) Neglecting PV terms, setup an expression for dE for a rubber band. Bt a legendretransformation obtain an expression for dF where F is the Helmholtz free energy.

(b) Derive two Maxwell relation from the above equations.

(c) Experiments show that a rubber band heats up when it is stretched adiabatically.From this fact determine whether a rubber band will contract or expand when it iscooled at constant tension. (Hint: Use considerations of thermal and mechanicalstability to show that certain thermodynamic derivatives are positive.) (credit:Liu)

29. Chandler 1.16

30. Chandler 1.17

31. When a particle with spin 1/2 is in a magnetic field H, it has two energy levels H.Consider a system of N such particles in a magnetic field H. Assume that a fractionx of particles are in the spin up ( = H ) state.(a) What is the total number of available states of the system?

(b) What is the entropy of the system? Plot the entropy as a function of x.

(c) Calculate the averate temperature T as a function of x. Plot the temperature asa function of x. Explain your results.

(d) Using these results calculate average magnetization M per spin, for the system asa function of H and T . Sketch your result as a function of T for fixed H. Sketchyour results as a function of H for constant T . Discuss the results. (credit: Liu)

32. Reif 5.1

33. Reif 5.2

34. Reif 5.4

35. Reif 5.6

36. Reif 5.7

37. Reif 5.15

38. Reif 5.19

39. Reif 5.20

40. Reif 5.23

41. Reif 5.26

42. The Stirling Engine: An Stirling engine toy as shown in the figure is placed on a coffeemug that contains 1/4 liters of boiling water. Assume that the mugs opening is 10cmin diameter and the mug is well insulated. (The only heat loss is through contact withthe engine.)

http://www.howstuffworks.com/stirling-engine.htm

(a) Explain why a full cycle of the Strilings engine can be described by the followingcycle:

http://en.wikipedia.org/wiki/Stirling_engine

(b) Calculate the work done in one cycle of the engine and show that it is equal to

W = nR lnV1V2

(TH TC).

Where V1 and V2 are the initial and final volumes of the engine, and TH and TCare the temperatures of the hot and the cold surfaces of the engine.

(c) Calculate the head lost in the heat reservoir (the mug) and the temperature changein the mug as a result.

(d) Using realistic values calculate the efficiency of this engine in one cycle.

(e) One cycle of this engine takes time T to complete (period). Assuming that theengine has negligible friction, it is safe to assume that the period remains constant.After an elapsed time t T the temperature of the water in the mug has changedto T H . Show that

T H = C exp

{(nR ln V1

V2

CV (Water)+ CV (gas)

)t

}+

CV TC

CV (gas)nR lnV1V2

In this equation C is a constant. Hint: To show this assume infinitesimal changesin the heat, and find a differential equation based on the derivative of the tem-perature as a function of time.

(f) Based on reasonable assumption about the initial and final temperatures of water,find the total number of cycles that the engine will rotate before stopping.

(g) What is the efficiency of one cycle at some intermediate temperature T H?

(h) What is the total efficiency of the engine after it stops?

43. Reif 6.1

44. Reif 6.2

45. Reif 6.3

46. Reif 6.6

47. Reif 6.10

48. Reif 6.11

49. Reif 6.13

50. Suppose N atoms are arranged regularly to form a perfect crystal. If one moves natoms among them, where 1 n N , from lattice sites to interstitial sites one formsn defects called Frenkel defects. Suppose there are N interstitial sites an atom canenter. Let w be the energy necessary to remove an atom from a lattice site to aninterstitial site. Consider the system held at temperature T such that w kBT .

(a) Construct the canonical partition function in terms of a sum over n. (Hint: Startfrom calculating the degeneracy)

(b) What is the average number of defects at this temperature? (credit: Liu)

51. A simple one dimensional quantum oscillator has energy levels given by

En =

(n+

1

2

)~

where is the characteristic frequency of the oscillator and the quantum number nhas possible integral values n = 1, 2, ...

(a) Derive an expression for the canonical partition function of a single quantumoscillator.

(b) Using the results of part a, find the average energy E for the system of N oscilla-tors. Plot as E/(N~) as function of kBT/(~). This is known as Planck resultfor the thermal average energy of photons in a single mode of frequency .

(c) Find the Helmholts free energy, specific heat and entropy of the N oscillatorsystem. Plot F/(N~), C/(NkB) and S/(NkB) as a function of kBT/(~). Whatis the high-temperature limit of the heat capacity? (credit: Liu)

52. Reif 7.1

53. Reif 7.3

54. Reif 7.5

55. Reif 7.6

56. Reif 7.10

57. The rotational energy levels of a diatomic molecule are given by Ej = j(j + 1) wherej is a non negative integer. The degeneracy of each rotational level is 2j+ 1. ConsiderN diatomic molecules at temperature T .

(a) Write the canonical partition function Z for the rotational states of one moleculein terms of a sum over the quantum number j.

(b) Evaluate Z approximately for high temperature by converting the sum to anintegral.

(c) Evaluate Z approximately for low temperature by truncating the sum after thesecond term.

(d) Find expressions for the energy E and the specific heat C in both limits. Sketchthe behavior of E/NkBT and C/NkBT as a function of /kBT . (credit:Liu)

58. In the grand canonical ensemble, with boundary conditions of (V, , ) (constant vol-ume, constant temperature and constant average number of particles) the probabilityof finding a particle in the state N, j, where N is the number of particles and j is thestate of energy the assembly of particles are taking is given by

PN,j(V, , ) =eEN,jeN

N

j eEN,jeN

where the sum is over all possible number of particles and all possible states of energyand = where is the chemical potential. Using this probability function showthat

(a) E(V, , ) = ( lnZG

)V,

(b) P (V, , ) = 1

( lnZGV

),

(c) N(V, , ) = ( lnZG

)V,

= 1

( lnZG

)V,

where ZG =

N

j eEN,jeN is the grand canonical partition function.

59. consider the function f = lnZG, based on the findings of previous question,

(a) Show that df = Ed Nd + PdV .(b) Using the relationship between the extensive and intensive variables, write the

derivative of a new function dA such that A = A(E, V,N). What is this function?

(c) Show that the entropy of a grand canonical ensemble, under the condition ofconstant (V, T, ) is given by S = kB(lnZG + E N). Compare you answerwith the entropy obtained for the canonical ensemble.

(d) Calculate the Helmholtz free energy based on the partition function of a grandcanonical ensemble.

60. The Gibbs free energy is defined as G(P, T,N) = E + pV TS. Show that G = Nand therefore pV = kBT lnZG. Note that in the Gibbs free energy for a system withconstant number of particles N is a function of p and T .

61. (E, V,N) is the density of states with energy E in a container of volume V and numberof particles N . Entropy can be defined as S = kB ln (E, V,N) in micro-canonical en-semble. We showed how minimizing the Helmholtz free energy F (, V,N) = kB lnZwhere Z =

E (E, V,N)e

E is identical to maximizing entropy, if one assumes thefunction (E, V,N)eE is only non-zero close to its maximum. Similarly we calculatedthe Helmholtz free energy F (, V, ) through another Lagrange multiplier that relatesthe chemical potential to the average number of particles and showed = F

N. Use

similar methods to define the ishothermal-isobaric partition function = (N, T, P )such that G(N,P, T ) = kB ln . Show that the new multiplier is in the form of pkBTwhere p is the average pressure.

62. Obtain the pressure of a classical ideal gas as a function of N, V and T using thegrand canonical partition function.

63. Reif 8.12

64. Reif 8.13

65. Reif 8.14

66. Reif 8.15

67. Reif 8.16

68. Reif 8.19

69. When water containing a small amount of surfactant is placed in contact with air, thesurfactant molecules adsorb to the air-water interface, forming a Langmuir monolayer.To a good approximation the surfactant solution is an ideal gas of noninteractingsurfactant molecules. The following system is a simple model of Langmuir trough atsubmonolayer coverage. Consider an adsorbent surface of N sites, each of which canadsorb one surfactant molecule. Suppose that the surface is in contact with an idealgas of surfactants with chemical potential . Assuming that an adsorbed molecule hasenergy 0 compared to one in the free state, determine the covering ration (the ratioof adsorbed molecules to adsorbing sites). use the grand canonical ensemble. (Credit:Liu)

70. Consider a gas of hydrogen molecules, enclosed in a box of volume V and in thermalequilibrium with a heat reservoir at temperature T.

(a) Calculate the partition function for the gas, assuming ideal gas conditions.

(b) What is the probability that the rotational quantum number is J at temperatureT?

(c) What is the most probable value for J at 300K and 1000K?

(d) What is the probability that the vibrational quantum number is n at temperatureT?

(e) What is the most probable value for n at 300K and 1000K?

(f) Using the results of the above calculations, calculate the dissociation rate constantfor hydrogen molecule H2.

(g) At 1000K what percent of the molecules are dissociated compared to room tem-perature?

(h) What is the percentage of dissociated molecules at 10000K?

(i) Compare your results from part 70h with the percentage of hydrogen atoms thatare dissociated at this temperature into an electron and a proton.

Note that the bond dissociation energy of hydrogen molecule is about 100Kcal/mol.The dissociation energy for electron in a hydrogen molecule is given by 0 =

e2

2a0where a0 is the Bohr radius.

(j) Calculate the dissociation rate constant for the molecule HD.

(k) Calculate the rotational contribution to the entropy of HD at 20K and 300K

71. Show that the entropy of an ideal gas in thermal equilibrium is given by the formula

S = KB

[n + 1 ln n + 1 n ln n]

in the case of bosons and by the formula

S = KB

[n 1 ln n 1+ n ln n]

in the case of fermions. (The sum is over all allowed energy levels .) Verify that theseresults are consistent with the general formula

S = KB

[n

P(n) lnP(n)

]

where P(n) is the probability that there are exactly n particles in the energy state (credit: Kamien)

72. Reif 9.1

73. Reif 9.4

74. Reif 9.7

75. Reif 9.17

76. Reif 9.23