40a Final Exam, Fall 2006 Data: P 0 0 5 Pa, R = 8.34 0 3 J/kmol K = N A k, N A = 6.02 0 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? /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
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
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
. An ideal gas of N particles has energy U = 2fNkT, for some constant f. The gas is initially in a box of volume V, at temperature T. A valve opens, and the gas undergoes free expansion, to fill a larger volume V 2 > V. No work is done in this process, and no heat is added or removed. What is the change in entropy of this process? For full credit, an explanation of what you re doing, and why, must accompany the calculation. [5 points]. 2. The particles of a monatomic ideal gas initially have v rms,i = 400m/s. The gas then undergoes an adiabatic process, in which the volume increases by a factor of eight, V f /V i = 8. What is the final v rms,f of the particles, after the process? [5 points] 3. An certain gas starts off at temperature T 0, with an entropy S 0. It then undergoes a total process, consisting of the following four steps. First, the system moves along an isotherm, to a state with entropy 2S 0. Next, it moves along an adiabat, to a state with temperature 6T 0. Next, the system then moves along an isotherm, to a state with entropy S 0. Finally, it moves on an adiabat, to a state with temperature T 0. (a) What is the net heat added to the system, for the total process (consisting of the above four steps)? Write it as negative, if appropriate. You don t need to write the separate contributions of each step, if you don t need that to write the net answer. (Just be sure to explain what you re doing, and why!) [5 points] (b) What is the net change in the internal energy of the gas, after the total process? (Same comments as the last part.) [5 points] (c) What is the net work done by the gas, after the total process? (Same comments.) [5 points] 4. The number of states available to a system of N particles, with total energy U, in volume V, is generally written as Ω(N, U, V ). Suppose that a certain system has Ω(N, U, V ) = Ω(N, UV b ), i.e. it only depends on U and V via the variable UV b, where b is some constant. The function Ω(N, UV b ) is a rapidly increasing function of both N and UV b. (a) The system initially has volume V i = 2m 3, and total energy U i = 00J. It then undergoes an adiabatic expansion to volume V f = 4m 3. What is the final total energy U f? (Write your answer in terms of the constant b). [5 points] (b) Derive a general expression for the pressure of the gas, as a function of U and V (and the constant b). Hint: the key to correctly solving this problem is to use the first law 4
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
(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 + 3 2 )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