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