VIEWS: 8 PAGES: 21 POSTED ON: 7/18/2010
PHY203: Thermal Physics Topic 6: Statistics, Entropy & Equilibrium • The Einstein Solid • Interacting Einstein solids • What entropy really means • Thermal equilibrium • Links between statistical mechanics and thermodynamics Einstein Model of a Solid •Essentially, this is the familiar “balls and springs” model: each atom represented by a hard sphere connected to its neighbours by springs •Each atom undergoes simple harmonic oscillation about its equilibrium position •The Einstein model treats the atoms as quantum mechanical SHO’s, all with the same frequency •Energy levels given by: En = (n + ½)h •All oscillators independent •Each atom can oscillate in 3 independent directions •So, a solid containing N atoms behaves like 3N independent oscillators Pictorial representation of QM oscillator •In this treatment we’ll represent each oscillator as shown opposite (parabolic potential energy with “ladder” of discrete quantised energy levels) n=3 •Measure energies relative to ground state n=2 •So each oscillator can have energies in n=1 discrete multiples of h, call this one unit. n=0 •Represent the energy state of the oscillator by placing an arrow next to the Oscillator in 2nd excited state, appropriate energy level with 2 units of energy •Remember (for later), that although we’re treating the ground state as the zero of energy, it really has E = h/2 Microstates of a tiny Einstein solid (3 oscillators!) Total energy = 0 units n=3 n=3 n=3 n=2 n=2 n=2 n=1 n=1 n=1 n=0 n=0 n=0 (0) = 1 Microstates of a tiny Einstein solid (3 oscillators!) Total energy = 1 unit n=3 n=3 n=3 n=2 n=2 n=2 n=1 n=1 n=1 n=0 n=0 n=0 n=3 n=2 n=3 n=2 n=3 n=2 (1) = 3 n=1 n=1 n=1 n=0 n=0 n=0 n=3 n=3 n=3 n=2 n=2 n=2 n=1 n=1 n=1 n=0 n=0 n=0 Microstates of a tiny Einstein solid (3 oscillators!) Total energy = 2 units n=3 n=3 n=3 n=3 n=3 n=3 n=2 n=2 n=2 n=2 n=2 n=2 n=1 n=1 n=1 n=1 n=1 n=1 n=0 n=0 n=0 n=0 n=0 n=0 n=3 n=3 n=3 n=3 n=3 n=3 n=2 n=2 n=2 n=2 n=2 n=2 n=1 n=1 n=1 n=1 n=1 n=1 n=0 n=0 n=0 n=0 n=0 n=0 n=3 n=3 n=3 n=3 n=3 n=3 n=2 n=2 n=2 n=2 n=2 n=2 n=1 n=1 n=1 n=1 n=1 n=1 n=0 n=0 n=0 n=0 n=0 n=0 (2) = 6 Calculation of for bigger numbers New Representation n=3 n=3 n=3 n=3 n=3 n=3 n=2 n=2 n=2 n=2 n=2 n=2 n=1 n=1 n=1 n=1 n=1 n=1 n=0 n=0 n=0 n=0 n=0 n=0 Number of oscillators = N, number of lines (|) = N-1 Number of energy units = number of dots (●) = q Total number of symbols = q + (N-1) General formula for for Einstein solid Total number of microstates for an Einstein solid with N oscillators and q energy units = number of ways of arranging lines and dots: (q N 1)! (N , q ) q !(N 1)! Example: Calculate the multiplicity (number of microstates) for an Einstein solid with 30 oscillators and 30 units of energy………. Interacting systems Solid A Solid B Energy UA, NA, qA, A UB, NB, qB, B Utotal = UA + UB = constant total= A B Interacting systems Consider first 2 tiny interacting Einstein solids, 3 oscillators in each, sharing a total of 6 units of energy: 100 q q tota 80 A B A B l 60 0 6 1 28 28 total 40 1 5 3 21 63 20 2 4 6 15 90 0 3 3 10 10 100 0 1 2 3 4 5 6 qA 4 2 15 6 90 Probability of most likely macrostate 5 1 21 3 63 (qA=qB=3) = 100/462 6 0 28 1 28 Probability of least likely macrostate (qA or qB=0) 7 possible macrostates = 28/462 Interacting systems •The calculations show that if, for example, the 2 solids are initially in the qA=1, qB=5 macrostate, at some later time the system is more likely to be in the qA=2, qB=4 (or qA=3, qB=3) macrostate than in the qA=0, qB=6 macrostate. 100 80 60 total 40 20 0 0 1 2 3 4 5 6 qA •Implies a “preferred” direction for the energy transfer process, reminiscent of 2nd law of thermodynamics A bigger system Now let solid A contain 300 oscillators, solid B 200 oscillators, and let the total energy be 100 units: •As expected, probability distribution “sharpens up” as numbers get bigger, becomes increasingly unlikely to find deviations from equilibrium, or to get energy flowing in the “wrong” direction •Gaussian distribution for physically realistic numbers (>~1022): see SPS derivation & Schroeder2.3 2nd law revisited •When our two bodies are placed in contact, energy is exchanged between them until they find themselves in the most probable macrostate (ie the one with the greatest multiplicity or number of microstates ) •Once they have arrived at this macrostate, spontaneous flow of energy between the bodies ceases •Essentially, then, behind the apparent mystique of the 2nd law we have a simple fact: “Things do what they’re most likely to do according to the standard rules of probability and statistics”! •We can restate the 2nd law along the lines of: “Any large system in equilibrium will be found in the macrostate with the largest ” Or…….” tends to increase” (Schroeder p. 74) Entropy and •The entropy form of the 2nd law says (more or less) that “the entropy of the universe tends to increase” •We have deduced from our statistical arguments that “ tends to increase” •Suggests that entropy is somehow related to •However, if we have 2 systems, A and B, then Stotal=SA + SB, but total=AB •So, S cant be simply proportional to Everything’s OK if we have: S ln S k B ln Boltzmann’s Constant S k B ln Boltzmann’s tombstone (Central Cemetery, Vienna) This equation provides the essential link between thermodynamics and statistics……………….. Conditions for equilibrium Consider 2 systems, isolated from the rest of the universe, separated by a partition UT = U1 + U2 = constant System 1 System 2 VT = V1 + V2= constant U1, V1, N1 U2, V2, N2 NT = N1 + N2 = constant ST = S1 + S2 General principle: total entropy ST is maximised at equilibrium Thermal equilibrium •If the partition is rigid, impermeable, but thermally conducting, then the systems can only interact by exchange of heat. •Therefore only U1 and U2 can vary as equilibrium is approached (but remember U1+U2 = constant) •Choose U1 as single independent variable Q Maximise ST by setting: System 1 System 2 ST U1, V1, N1 U2, V2, N2 0 U 1 Thermal equilibrium ST S 1 S 2 S 1 S 2 U 2 0 U 1 U 1 U 1 U 1 U 2 U 1 U 2 U 2 UT U 1 , therefore 1 U 1 S 1 S 2 U 1 U 2 We know that at equilibriu m, T1 T2, , suggesting that S T is related to U To see how, consider dependence of S on U for our interacting Einstein solids…………………………………… A bigger system Now let solid A contain 300 oscillators, solid B 200 oscillators, and let the total energy be 100 units: •As expected, probability distribution “sharpens up” as numbers get bigger, becomes increasingly unlikely to find deviations from equilibrium, or to get energy flowing in the wrong direction •Gaussian distribution for physically realistic numbers (>~1022): see SPS derivation Entropy vs energy for solids A & B •UA = hqA ; UB = hqB •Note that axes for qA and qB go in opposite directions •So, in indicated region, energy of A increases spontaneously as equilibrium is approached •Energy of B decreases as equilibrium is approached •So, in region indicated, TA<TB S A S B U A U B 1 S T U V ,N Very Important Equations S k B ln 1 S (fundamental definition T U V ,N of absolute temperature)