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On the 100Th Anniversary of the Sackur–Tetrode Equation On the 100th Anniversary of the Sackur–Tetrode Equation Walter Grimus (University of Vienna) Seminar Particle Physics March 8, 2012, University of Vienna Walter Grimus (University of Vienna) On the 100th Anniversary of the Sackur–Tetrode Equation Sackur–Tetrode equation Entropy of a monoatomic ideal gas: 3 E V S(E, V , N)= kN ln + ln + s 2 N N 0 1912: Otto Sackur and Hugo Tetrode independently determined 3 4πm 5 s = ln + 0 2 3h2 2 Sackur–Tetrode equation = absolute entropy of a monoatomic ideal gas Walter Grimus (University of Vienna) On the 100th Anniversary of the Sackur–Tetrode Equation Contents 1 Motivation 2 Tetrode’s derivation 3 Sackur’s derivation 4 Test of the Sackur–Tetrode equation 5 Concluding remarks Walter Grimus (University of Vienna) On the 100th Anniversary of the Sackur–Tetrode Equation Motivation Absolute entropy: Boltzmann (1875), Planck (1900): S = k ln W + const. Argument: Nernst’s heat theorem (1906) (third law of thermodynamics) S should be calculable without any additive constant ⇒ Massive particles: phase space volume of “elementary cells” unknown Sackur (1911): Entropy of a monoatomic ideal gas as a function of the volume of elementary cell Walter Grimus (University of Vienna) On the 100th Anniversary of the Sackur–Tetrode Equation Tetrode’s derivation Ansatz: n degrees of freedom ⇒ n Elementary domains of volume σ = δq δp δqn δpn = (zh) 1 1 · · · N identical particles ⇒ W ′ S = k ln N! W ′ = number of configurations in phase space Total admissible volume in phase space space given E, V , N: 3 3 3 3 (E, V , N)= d x1 d p1 d xN d pN V Z Z · · · Z Z with 1 p 2 + + p 2 E 2m 1 · · · N ≤ Walter Grimus (University of Vienna) On the 100th Anniversary of the Sackur–Tetrode Equation Tetrode’s derivation Dilute gas: 3N N (E, V , N) (2πmE) 2 V S = k ln V with (E, V , N)= (zh)3N N! 3N V Γ 2 + 1 Stirling’s formula: ln n! n(ln n 1) ≃ − Tetrode’s result: 3 E V 3 4πm 5 S(E, V , N)= kN ln + ln + ln + 2 N N 2 3(zh)2 2 Walter Grimus (University of Vienna) On the 100th Anniversary of the Sackur–Tetrode Equation Sackur’s derivation Following derivation in paper of October 1912 Partioning of energy space into cubes: nk ∆ε εk < (nk + 1)∆ε (k = x, y, z) ≤ Energy associated with i-th cube: εi = (nx + ny + nz )∆ε Observation time τ during which collisions are negligible Ansatz: 3 Ni = Nf (εi ) (τ∆ε) N atoms distributed over r cubes: N! W = with N = N1 + N2 + + Nr N !N ! Nr ! · · · 1 2 · · · Walter Grimus (University of Vienna) On the 100th Anniversary of the Sackur–Tetrode Equation Sackur’s derivation Conditions: 3 i Ni = i Nf (εi ) (τ∆ε) = N 3 Pi Ni εi = Pi Nf (εi ) (τ∆ε) εi = E P P Stationary point of S = k ln W : −βεi Ni = Nαe and S = 3kN ln(τ∆ε) kN ln α + kβE − − Referring to Sommerfeld (1911): “smallest action that can take place in nature is given by Planck’s constant” τ∆ε = h Integration over energy space: V τ 3dε dε dε = dp dp dp x y z N x y z Walter Grimus (University of Vienna) On the 100th Anniversary of the Sackur–Tetrode Equation Sackur’s derivation Conditions: summation integration → ⇒ 3N N 3N 3/2 β = and α = 2E V 4πmE Sackur’s result: 3 E V 3 4πm 3 S(E, V , N)= kN ln + ln + ln + 2 N N 2 3h2 2 Sackur misses one kN in entropy! (in previous paper correct number 5/2) Walter Grimus (University of Vienna) On the 100th Anniversary of the Sackur–Tetrode Equation Sackur’s derivation Remarks: Sackur’s derivation resembles derivation with canonical partition function Z N Z = 1 N! 2 1 3 3 p V h Z1 = d x d p exp β = with λ = h3 ZV Z − 2m λ3 √2πmkT λ = thermal de Broglie wave length ⇒ V 5 S(T , V , N)= k(ln Z + βE)= kN ln + λ3N 2 3 is identical with Tetrode’s result upon substitution E = 2 NkT 3 Sackur’s α related to Z1: α = N/(h Z1) Walter Grimus (University of Vienna) On the 100th Anniversary of the Sackur–Tetrode Equation Test of the Sackur–Tetrode equation Planck’s constant h known from black-body radiation Sackur: Test of δq δp = h Tetrode: Determination of z in δq δp = zh Usage of data on phase transition liquid–vapor of mercury Idea: L(T ) = T svapor(T , p¯(T )) sliquid(T ) − exp. ST equation exp.+theory | {z } L ........... molar latent| heat{z } | {z } s ...... absolute molar entropy p¯ ..............vapor pressure Walter Grimus (University of Vienna) On the 100th Anniversary of the Sackur–Tetrode Equation Test of the Sackur–Tetrode equation Sackur–Tetrode equation: R = kNA, λ = thermal de Broglie wave length kT 5 svapor = R ln + p¯(T ) λ3(T ) 2 Entropy of liquid: In good approximation entropy and heat capacity of liquids pressure-independent T ′ ′ cp(T ) sliquid(T )= dT ′ Z0 T Kirchhoff’s equation: ∆cp = difference of molar heat capcity across coexistence curve dL 5 liquid ∆cp with ∆cp = R c dT ≃ 2 − p Walter Grimus (University of Vienna) On the 100th Anniversary of the Sackur–Tetrode Equation Test of the Sackur–Tetrode equation Vapor pressure: L(T ) (2πm)3/2(kT )5/2 5 T c (T ′) p T T ′ p ln ¯( )= + ln 3 + d ′ − RT h 2 − Z0 RT Test of Sackur–Tetrode equation: Suitable temperature intervall [T1, T2] corresponding to vapor pressure interval [¯p1, p¯2] liquid Experimental input: L(T0), cp andp ¯ in intervall Experimental + theoretical: T ′ 0 ′ cp(T ) dT ′ Z0 T Walter Grimus (University of Vienna) On the 100th Anniversary of the Sackur–Tetrode Equation Test of the Sackur–Tetrode equation Entropy of a liquid (part 1): c ∂v ds(T , p)= p dT dp T − ∂T p Reference point (T0, p0) at coexistence curve Third law of thermodynamics ⇒ T0 ′ ′ cp(T , p0) sliquid(T0, p0)= dT ′ Z0 T ′ Tm solid T0 liquid ′ ′ cp (T , p0) Lm(Tm) ′ cp (T , p0) = dT ′ + + dT ′ Z0 T Tm ZTm T T →0 csolid from model by Nernst (1911), csolid 0 exponentially p p −→ Walter Grimus (University of Vienna) On the 100th Anniversary of the Sackur–Tetrode Equation Test of the Sackur–Tetrode equation Entropy of a liquid (part 2): sliquid at (T , p¯) by (T , p ) (T , p¯) along coexistence curve 0 0 → liquid cp 1 ∂v ds = dT αv d¯p with α = T − v ∂T p α = isobaric expansion coefficient Numerical estimation: −4 −1 3 liquid −1 −1 α 1.8 10 K , v/NA 10 A˚ , cp 28 J mol K ≃ × ∼ ◦ ≃ ◦ Temperature intervall: T1 = Tm = 39 C, T2 200 C −− ∼ Vapor pressure intervall:p ¯ 3 10 8 bar,p ¯ 0.2 bar 1 ≃ × 2 ∼ liquid T2 One has to compare cp ln with αv(¯p2 p¯1) T1 − αv 1 bar 1 10−4 J mol−1 !!! × ∼ × Walter Grimus (University of Vienna) On the 100th Anniversary of the Sackur–Tetrode Equation Test of the Sackur–Tetrode equation Test with modern data: CODATA Key Values for Thermodynamics Standard State: T0 = 298.15 K, p0 = 1 bar L (Hg) = 61.38 0.04 kJ mol−1, s (Hg) = 75.90 0.12 J K−1 mol−1 0 ± 0 ± CRC Handbook of Chemistry and Physics: liquid Vapor pressure and cp data between T = 38.84 ◦C and T = 200 ◦C 1 − 2 CODATA: h = 6.62606957(29) 10−34 Js × Fit of z to mercury data: h z (mean value of h) → × z = 1.003 0.004(L ) 0.005(s ) ± 0 ± 0 Walter Grimus (University of Vienna) On the 100th Anniversary of the Sackur–Tetrode Equation Concluding remarks Papers: i O. Sackur, The application of the kinetic theory of gases to chemical problems (received October 6, 1911): S of a monoatomic ideal gas as a function of the size of the elementary cell ii O. Sackur, The meaning of the elementary quantum of action for gas theory and the computation of the chemical constant (no “received date”, must have been written in spring 1912): Postulate: size of elementary cell is hn Absolute entropy S of a monoatomic ideal gas Vapor pressure over a solid Comparison with data on neon and argon numerical results not really satisfying and conclusive Walter Grimus (University of Vienna) On the 100th Anniversary of the Sackur–Tetrode Equation Concluding remarks iii H. Tetrode, The chemical constant and the elementary quantum of action (received March 18, 1912): Derivation of S, assuming size (zh)n of elementary cell Fit of z by using data on the vapor pressure of liquid mercury, due to some numerical mistakes z 0.07 ≈ iv H. Tetrode, erratum to The chemical constant and the elementary quantum of action (received July 17, 1912): Correction of formulas and numerics, z 1 Reference to the papers of Sackur by noting∼ that the formula for S has been developed by both of them at the same time v O. Sackur, The universal meaning of the so-called elementary quantum of action (received October 19, 1912): Good agreement ( 30%) with data on Hg vapor pressure Comments on paper± by Tetrode Walter Grimus (University of Vienna) On the 100th Anniversary of the Sackur–Tetrode Equation Concluding remarks Historical context: Thermodynamics, Planck’s constant and early applications 1900 Planck: black body radiation, Plank’s constant 1905 Einstein: photoelectric effect 1906 Nernst: Third law of thermodynamics 1907 Einstein: Einstein model of heat capacity of solids 1907 Stark: minimal wavelength of X-rays 1911 Sommerfeld: “action in pure molecular processes” 1912 Debye: Debye model of solid Walter Grimus (University of Vienna) On the 100th Anniversary of the Sackur–Tetrode Equation Concluding remarks Subatomic physics: X-rays (Roentgen (1895)) radioactivity (Becquerel (1896)) electron (J.J.
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