Northeastern Illinois University Thermal Physics Interactions and Implications Greg Anderson Department of Physics & Astronomy Northeastern Illinois University Spring 2020 c 2004-2020 G. Anderson Thermal Physics – slide 1 / 47 Northeastern Illinois Overview University Temperature Paramagnetism Thermal Equilibrium & Temperature Pressure The Thermodynamic Identity Diffusive Equilibrium c 2004-2020 G. Anderson Thermal Physics – slide 2 / 47 Northeastern Illinois Statistical Mechanics University Fundamental assumption of statistical mechanics: All accessible microstates of a system are equally likely. Thus, the probability of finding the system in a macrostate q: P ∝ Ω(q) Any large state in thermal equilibrium will be found in the macrostate q of greatest multiplicity Ω(q). Given Boltzmann’s definition of entropy: S ≡ k lnΩ (entropy) The entropy of an isolated system always increases: Thermal Equilibrium: The spontaneous flow of energy stops when a system is at or near its most likely macrostate. c 2004-2020 G. Anderson Thermal Physics – slide 3 / 47 Northeastern Illinois University Statistical Mechanics Temperature Entropy ES Entropy Plot (NA = 150, NB = 150, q = 12) Multiplicities of Einstein Solids Temperature from Entropy Temperature Entropy and Heat Paramagnetism Thermal Equilibrium & Temperature Pressure The Thermodynamic Identity Diffusive Equilibrium c 2004-2020 G. Anderson Thermal Physics – slide 4 / 47 Northeastern Illinois Entropy Plot (NA = 200, NB = 100, q = 12) University S = k lnΩ. In equilib.: ∂S = ∂S = 0. ∂SA = ∂SB = 1 . ∂qA ∂UA ∂UA ∂UB T 60 S 50 k 40 30 B T = A 20 T SA SB Entropy10 in Units of TA <TB TA >TB 0 0 1 2 3 4 5 6 7 8 9101112 qA c 2004-2020 G. Anderson Thermal Physics – slide 5 / 47 Northeastern Illinois Entropy Plot (NA = 150, NB = 150, q = 12) University S = k lnΩ. In equilib.: ∂S = ∂S = 0. ∂SA = ∂SB = 1 . ∂qA ∂UA ∂UA ∂UB T 60 S 50 k B 40 T = A 30 T 20 SA SB Entropy in Units of 10 TA <TB TA >TB 0 0 1 2 3 4 5 6 7 8 9101112 qA c 2004-2020 G. Anderson Thermal Physics – slide 6 / 47 Northeastern Illinois Multiplicities of Einstein Solids University qA ΩA SA/k qB ΩB SB/k Ω S/k 0 1 0 100 2.8 × 1081 187.5 2.8 × 1081 187.5 1 300 5.7 99 9.3 × 1080 186.4 2.8 × 1083 192.1 2 45150 10.7 98 3.1 × 1080 185.3 1.4 × 1085 196.0 . 11 5.3 × 1019 45.4 89 1.1 × 1076 175.1 5.9 × 1095 220.5 12 1.4 × 1021 48.7 88 3.4 × 1075 173.9 4.7 × 1096 222.6 13 3.3 × 1022 51.9 87 1.0 × 1075 172.7 3.5 × 1097 224.6 . 59 2.2 × 1068 157.4 41 3.1 × 1046 107.0 6.7 × 10114 264.4 60 1.3 × 1069 159.1 40 5.3 × 1045 105.3 6.9 × 10114 264.4 61 7.7 × 1069 160.9 39 8.8 × 1044 103.5 6.8 × 10114 264.4 . 100 1.7 × 1096 221.6 0 1 0 1.7 × 1096 221.6 Table 1: Macrostates, multiplicities, and entropies of a system of two Einstein solids, one with 300 oscillators and the other with 200, sharing a total of 100 units of energy. c 2004-2020 G. Anderson Thermal Physics – slide 7 / 47 Northeastern Illinois Temperature from Entropy University A theoretical definition of temperature: ∂S −1 1 ∂S T ≡ or ≡ ∂U T ∂U N,V Example: Einstein solid at large temperature U = ǫq: ∼ N S/k = lnΩ ∼ (q + N)ln q +(q + N) q − N ln N − q ln q ∼ N ln q/N + N + O(N 2/q) ∼ qe ∼ N ln N ∼ Ue e ∼ N ln Nǫ = N ln U + N ln Nǫ Active Learning: Show that U = NkT c 2004-2020 G. Anderson Thermal Physics – slide 8 / 47 Northeastern Illinois Entropy and Heat University At constant V , N: dU d¯Q dT dS = = = C T T V T The increase in entropy: Tf Tf dT Tf ∆S = dS = C ∼ C ln V T V T ZTi ZTi i c 2004-2020 G. Anderson Thermal Physics – slide 9 / 47 Northeastern Illinois University Statistical Mechanics Temperature Paramagnetism Paramagnetism Two State Paramagnet Two State Paramagnet (s = 1/2) Entropy: Two Paramagnetism State Paramagnet Temperature of a Paramagnet Plot: S(U) Negative Temperature Plot: S(T ) Plot: U(T ) Plot: CB (T ) Plot: M(T ) Plot: M(T ) II Plot: S(U) Negative Temperature Plot: S(T ) Plot: U(T ) Plot: CB (T ) Curie’s c 2004-2020 Law G. Anderson Thermal Physics – slide 10 / 47 Northeastern Illinois Paramagnetism University Paramagnet: a material in which constituent particles behave like tiny compass needles which line up parallel to an externally applied magnetic field. c 2004-2020 G. Anderson Thermal Physics – slide 11 / 47 Northeastern Illinois Two State Paramagnet University Consider N, identical spin-1/2 dipoles, e.g. electrons: Electron Spin: ↓ ↑ ↑ N↑ = spin up, N↓ = spin down ↑ ↑ ↓ N electrons: ↑ ↓ ↑ N = N↑ + N↓ The total energy of the system is a function of N↑. Multiplicity of states: N N! N! Ω= = = N N !(N − N )! N !N ! ↑ ↑ ↑ ↑ ↓ c 2004-2020 G. Anderson Thermal Physics – slide 12 / 47 Northeastern Illinois Two State Paramagnet (s =1/2) University +µB B ↑ ↓ ↓ ↑ ↑ ↑ ↓ ↓ ↑··· U −µB N = N↑ + N↓ Total Energy| {z } U = µB (N↓ − N↑)= µB (N − 2N↑)= −M · B Magnetization, M: U M = µ (N − N )= − ↑ ↓ B Multiplicity N N! N! Ω= = = N N !(N − N )! N !N ! ↑ ↑ ↑ ↑ ↓ c 2004-2020 G. Anderson Thermal Physics – slide 13 / 47 Northeastern Illinois Entropy: Two State Paramagnet University Multiplicity N N! N! Ω(N )= = = ↑ N N !(N − N )! N !N ! ↑ ↑ ↑ ↑ ↓ Using Stirling’s approximation and S = k lnΩ: S ≈ N ln N − N↑ ln N↑ − (N − N↑)ln(N − N↑) − N + N↑ + N↓ ≈ N ln N − N↑ ln N↑ − (N − N↑)ln(N − N↑) c 2004-2020 G. Anderson Thermal Physics – slide 14 / 47 Northeastern Illinois Entropy: Two State Paramagnet University Using Stirling’s approximation and S = k lnΩ: S ≈ N ln N − N↑ ln N↑ − (N − N↑)ln(N − N↑) − N + N↑ + N↓ ≈ N ln N − N↑ ln N↑ − (N − N↑)ln(N − N↑) The energy of a two state paramagnet is: U = µB (N↓ − N↑)= µB (N − 2N↑) From which we can write: 1 U 1 U N = N − , N = N + ↑ 2 µB ↓ 2 µB Entropy in terms of U: 2N U N − (U/µB) S/k ≈ N ln + ln 2 − 2 2µB N +(U/µB) N (U/µB) ! c 2004-2020 G. Anderson Thermal Physics – slide 14 / 47 p Northeastern Illinois Entropy: Two State Paramagnet University Entropy in terms of U: 2N U N − (U/µB) S/k ≈ N ln + ln 2 − 2 2µB N +(U/µB) N (U/µB) ! From the definitionp of temperature: ∂S k N − U/µB T −1 = = ln ∂U 2µB N + U/µB V,N c 2004-2020 G. Anderson Thermal Physics – slide 14 / 47 Northeastern Illinois Entropy: Two State Paramagnet University From the definition of temperature: ∂S k N − U/µB T −1 = = ln ∂U 2µB N + U/µB V,N Starting with the previous expression 1 1 N − U/µB = ln kT 2µB N + U/µB Multiply by 2µB and exponentiate to find N − U/µB e2µB/kT = N + U/µB Solving for U 1 − e2µB/kT µB U = NµB = −NµB tanh 1+ e2µB/kT kT c 2004-2020 G. Anderson Thermal Physics – slide 14 / 47 Northeastern Illinois Temperature of a Paramagnet University Entropy S ≈ N ln N − N↑ ln N↑ − (N − N↑)ln(N − N↑) Temperature 1 ∂S ∂N ∂S 1 ∂S ≈ = ↑ = − T ∂U ∂U ∂N 2µB ∂N N,B ↑ ↑ In terms of U 1 k N − U/µB = ln T 2µB N + U/µB Solving for U µB U = −NµB tanh kT c 2004-2020 G. Anderson Thermal Physics – slide 15 / 47 Northeastern Illinois Entropy of a Paramagnet S(U) University T −1 = ∂S/∂U S/k ∂U/∂S = T −N 0 N U/µB 2N U N − (U/µB) S/k ∼ N ln + ln 2 − 2 2µB N +(U/µB) N (U/µB) ! c 2004-2020 G. Anderson p Thermal Physics – slide 16 / 47 Northeastern Illinois Negative Temperature University Negative temperature requires: • Thermal equilibrium • Requires a finite upper bound on energy spectrum. e.g., nuclear paramagnets. • System must be energetically isolated from pos. temp. states. Understanding negative temperature • From Inverted population e.g. N↓ >N↑. • Negative temps. correspond to higher energies. They are “hotter” than positive temperatures. Entropy is more fundamental than temperature. c 2004-2020 G. Anderson Thermal Physics – slide 17 / 47 Northeastern Illinois Entropy of a Paramagnet S(T ) University 1 S Nk 0 0 1 2 3 4 x = µB/kT S/(Nk) = ln2(cosh x) − x tanh x c 2004-2020 G. Anderson Thermal Physics – slide 18 / 47 Northeastern Illinois Energy of a Paramagnet U(T ) University µB U = −NµB tanh kT 1 0 U NµB -1 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 τ = kT/µB c 2004-2020 G. Anderson Thermal Physics – slide 19 / 47 Northeastern Illinois Heat capacity of a spin-1/2 paramagnet University Heat capacity at constant magnetic field: ∂U ∂NµB tanh(µB/kT ) (µB/kT )2 CB = = − = Nk ∂T ∂T cosh2(µB/kT ) B B 0.4 Schottky anomaly 0.3 CB 0.2 0.1 0 01234567 τ = kT/µB c 2004-2020 G. Anderson Thermal Physics – slide 20 / 47 Northeastern µB Illinois Magnetization M = Nµ tanh University kT 1 ) M/Nµ 0 Curie’s Law (T ≫ µB/k) ∼ µB M ∼ Nµ kT Magnetization ( -1 -3 -2 -1 0 1 2 3 x = µB/kT c 2004-2020 G.