# 140a Final Exam, Fall 2006., κ T 1 V P. (? = P or V ), γ C P C V H = U + PV, F = U TS G = U + PV TS. T v. v 2 v 1. exp( 2πkT.

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1 40a Final Exam, Fall 2006 Data: P Pa, R = J/kmol K = N A k, N A = particles/kilomole, T C = T K du = TdS PdV + i µ i dn i, U = TS PV + i µ i N i Defs: 2 β ( ) V V T ( ) /dq C? dt P? /dq/t ext S 2 S =, κ T V ( V P 2 ) T /dq R /T, κ S V (? = P or V ), γ C P C V = (f + 2)/f. ( ) V, P S c = C/n, v = V/n, s = S/n, P β κ K T κ T V av V. Ideal gas: PV = nrt = NkT, C P = C V + nr. PV γ = const.. η W / Q 2 T /T 2, ω Q / W. H = U + PV, F = U TS G = U + PV TS. W mech (U T 0 S + P 0 V ), (U T 0 S + P 0 V ) 0. F( v) = dp dt = s 2 s v 2 v = Free particles: Φ = 4 l T v. ( m ) 3/2 exp( 2πkT 2 m v2 /kt), N V v, P = N 3 V mv2. p(x) = σ 2 2π e (x x) /2σ 2, (x x) n p(x) = { π 2 n/2 σ n Γ( ( + n)) for n even 2 0 for n odd

2 0 dt t z e at = Γ(z)a z, 0 x4 e x dx (e x ) = 2 4π4 /5. where Γ(z + ) = zγ(z) and Γ() = (so Γ(n) = (n )! for integer n) and Γ(/2) = π. (p + q) N = i= N N =0 ( N N ) p N q N N. N = Np, N 2 = N2 p 2 + Npq. n! ( n ) n 2πn for n. e n p N i n i p({n i }) = N! N i!, where p i =. g(ɛ)dɛ 4πV 2 (2π h) 3 m3/2 ɛ /2 dɛ ω({n i }) B.E. = i S = k ln Ω k lnω max. ω({n i }) M.B. = i= (for free particle in box) n i= g N i i N i!. (N i + g i )! N i!(g i )! bosons ω({n i }) F.D. = i g i! N i!(g i N i )! fermions. N i g i = where α µ/kt and β /kt. { 0 MB e α βɛ where a = i + a bosons fermions k k lnω(ni i g ie α+βɛ i = kn MB ) + kαn + kβu k i g i ln( e α+βɛ i ) BE k. i g i ln( + e α+βɛ i ) FD Z(T, V ) i g i e βɛ i, ( ) µ MB = kt ln(n/z), U MB = NkT 2 T lnz V F MB = NkT ( + ln(z/n)). Z V ( ) 3/2 2πmkT ideal monatomic gas h 2 2

3 Z d SHO = n=0 e (n+ 2 )hν/kt = e hν/2kt e hν/kt. U SHO = kt 2 T lnz = N(ν)dν = with x m hν m /kt θ D /T. g(ν)dν e hν/kt = C V = 9Nkx 3 m [ 2 hν + hν { 9Nν 3 m e hν/kt ν 2 dν e hν/kt ]. ν ν m 0 ν > ν m. xm 0 x 4 e x (e x ) 2dx, 3

5 of thermodynamics to write the pressure as a certain partial derivative, with something held fixed. Part (a) is also a hint. [5 points] 5. Container # has has n = kilomoles of monatomic ideal gas (γ = 5/3) and is points] at temperature T = 300K. Container #2 has n = 2 kilomoles of diatomic ideal gas (γ = 7/5), and is at temperature T 2 = 600K. The sizes of the containers are unchanging, and the walls do not allow any particle leakage. The two containers are placed in thermal contact for a little while, but are separated before they reach thermal equilibrium. After their separation, container # has temperature T = 400K. (a) What is the temperature T 2 of container #2 after the separation? [5 points] (b) What is the entropy change of the total system (both containers together)? [5 6. A certain particle has energy given by ɛ = c p, where p is the 3d momentum vector and c = the speed of light (this is the relativistic energy of a massless particle). (a) Compute the partition function of such a particle, in a box of volume V. Use the method where the sum over all states is replaced with an integral over the phase space, n d 3 rd 3 p/h 3. For full credit, evaluate the integral (using info in the formulae sheet). [5 points] (b) Compute C V for a gas of N such particles (take the gas to be sufficiently dilute that MB statistics apply). [5 points] (c) Compute the Helmholtz free energy, and use it to find the equation of state of the gas of N such particles. [5 points] 7. At low temperatures, the heat capacity of a certain sample of a solid material is given by C V = bt 3, for some constant b. A cyclic refrigerator cools the sample, from temperature T i = 2K to temperature T f = K. In this process, the refrigerator removes heat Q s = 5J from the sample. The refrigerator also emits some heat Q L into the outside lab, which is at temperature T L = 300K. (a) Using the information given above, find the numerical value of the constant b, in appropriate units. [5 points] (b) What is the change in entropy of the sample, the lab, and the refrigerator itself, in the cooling process? Write your answers as numbers (to the extent possible), in the appropriate units. [9 points (3 points each)] 5

6 (c) According to the laws of thermodynamics, what is the minimum energy which must go into running the refrigerator, for this cooling process? Also, what is the value of Q L in this case? Write your answers as numbers, in appropriate units. [5 points] 8. A certain thermodynamic system has nondegenerate energy levels, with energies 0, ɛ, 3ɛ, 6ɛ, 0ɛ, 5ɛ,.... Suppose that there are four particles, with total energy U = 0ɛ. Identify the possible distribution of particles, and evaluate their ω({n i }) and Ω. Also, compute the average occupation number N 0 of the ground state. (a) When the particles are distinguishable. [5 points] (b) When the particles are gaseous identical bosons. [3 points] (c) When the particles are gaseous identical fermions. [3 points] 9. Consider a modified version of the d simple harmonic oscillator, for which the energy levels are ɛ n = (2n )hν, for n = 0,, 2,.... These energy levels are each nondegenerate. (a) Compute the partition function for such a modified d harmonic oscillator. (For full credit, fully evaluate the mathematical expression for the function.) [5 points] (b) Consider a system of N distinguishable such d harmonic oscillators. Find the specific heat C V of this system, as a function of temperature. [5 points] 6

### 140a Final Exam, Fall 2007., κ T 1 V P. (? = P or V ), γ C P C V H = U + PV, F = U TS G = U + PV TS. T v. v 2 v 1. exp( 2πkT.

4a Final Exam, Fall 27 Data: P 5 Pa, R = 8.34 3 J/kmol K = N A k, N A = 6.2 26 particles/kilomole, T C = T K 273.5. du = TdS PdV + i µ i dn i, U = TS PV + i µ i N i Defs: 2 β ( ) V V T ( ) /dq C? dt P?

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