Thermodynamic Evidence for Water as a Quantum Mechanical Liquid

0.1 MB·English

by  A. Widom

Thermodynamic Evidence for Water as a Quantum Mechanical Liquid
A. Widom
Physics Department, Northeastern University, Boston, MA USA
S. Sivasubramanian
NSF Nanoscale Science & Engineering Center for High-rate Nanomanufacturing,
Northeastern University, Boston MA USA
D. Drosdoff
0 Physics Department, University of South Florida, Tampa, FL USA
1
0
Y.N. Srivastava
2
Physics Department and & INFN, University of Perugia, Perugia IT

n Physics Department, Northeastern University, Boston, MA USA
a
J We consider general theoretical models of water and in particular the nature of the motions

2 of the hydrogen nuclei. If the motion of hydrogen nuclei is classical, then the thermodynamic
2 pressureequationofstateforheavywaterwherein thehydrogennucleiaredeuteronsisidenticalto
the pressure equation of state for light water wherein the hydrogen nuclei are protons. Since the

] experimentalthermodynamicphasediagramforlightwaterisclearlymeasurablydifferentfromthe
r
e experimental thermodynamic phase diagram for heavy water, one may deduce that the motions of
h hydrogen nuclei are quantum mechanical in nature. This conclusion is in physical agreement with
t a recent analysis of X-ray,neutron and deep inelastic neutron scattering data.
o
.
t PACSnumbers: 47.37.+q,61.25.Em,61.20.-p,61.20.Gy
a
m
- I. INTRODUCTION
z z 1 z
d V =e2 a b + − a , (1)
n  R r |r −R |
ab ij j a
o The study of the applications of quantum mechanics Xa<b Xi<j Xaj 
c to water is of long standing interest. Recent important whereintheindicesa,b,...refertothenucleiofhydrogen
[ studies[1–4]ofthe momentum distributions andposition and oxygenand the indices i,j,... refer to the electrons.
1 correlations[5] in light water H2O and in heavy water We also presume that the hydrogennuclei obey classical
v D2O have provided strong evidence that the proton mo- mechanics to a sufficient degree of accuracy; i.e.
0 tions in light water and deuteron motions in heavy wa-
9 ter arein reality quantum mechanical. The experimental P2 = 3k T (Classical Hydrogen Nucleus). (2)
0 data arises from X-ray and neutron scattering experi- 2M 2 B
(cid:28) (cid:29)
4
ments. In particular, from deep inelastic neutron scat- In Sec.II we prove the following:
.
1 teringfromprotons,onemaydeducethemomentumdis- Theorem: Under the presumptions of classical motions
0 tributionofprotonswithinwateranditisfoundtodiffer for the hydrogen nuclei and with the free energy per
0
substantially from the Maxwellian distribution expected molecule f(v,T) obeying
1
from classical proton motions. In fact, quantum fluc-
: df =−Pdv−sdT, (3)
v tuations in the proton velocity dominate the sometimes
Xi presumed classical thermal fluctuations. the pressure equation of state P(v,T) will be identical
Our purpose here is to provide experimental thermo- for light water with proton hydrogen nuclei and heavy
r
a dynamic evidence that the motions of hydrogen atoms water with deuteron hydrogennuclei. Here, v represents
in liquid water cannot be adequately described by clas- the volume per molecule. As a consequence, light water
sical mechanics. The thermodynamic data is hardly of and heavy water are predicted to have identical phase
recent vintage. However, the thermal data has not been diagrams. These conclusions remain valid for more fully
previouslyanalyzedindetailforthe purpose ofassessing quantumelectrodynamicmicroscopicmodelsasshownin
the contribution of quantum mechanics to the thermo- Sec.III.
dynamic properties of water. As a theoretical model, we The experimental evidence[6] is that the theorem fails
at first presume a Hamiltonian which consists of kinetic inthe laboratory. For example,atatmosphericpressure,
energy plus Coulomb potential energy; It is the melting temperature of light ice into light water is
T ≈ 0 oC. At atmospheric pressure the melting tem-
m1
H=K+V, peratureofheavyiceinto heavywaterisT ≈4 oC. In
m2
P2 p2 the concluding Sec.IV, the modifications to the incorrect
K= a + j , presumption of classical motion required by thermody-
 2M 2m
Xa a Xj  namic experiments is discussed.
 
2
II. THERMODYNAMIC EQUATIONS OF Eq.(11) in no waychangesthe pressureequationofstate
STATE P(v,T),
∂∆f
With N molecules in a volume V, the free energy of ∆P =− =0, (12)
water is found from ∂v
(cid:18) (cid:19)T
F =−k T lnTr e−H/kBT , so that the theorem in Sec.I holds true. On the other
B (V,N)
hand,thedifferencebetweenlightwaterandheavywater
dF =−PdV +nµdN −SdoT. (4) ismeasurableinthelaboratory. Thequestionthenarises
as to which presumptions of the theory are incorrect.
Inthethermodynamiclimit,thefreeenergypermolecule
F(V =Nv,N,T)
f(v,T)= lim (5) III. MORE DETAILED INTERACTIONS
N→∞ N
obeys Eq.(3).
In order to consider quantum electrodynamic inter-
Consider the kinetic energy of the hydrogen nuclei
actions, one may include the vector potential in the
N Coulomb gauge,
1
K = P2 wherein N =2N. (6)
H 2M c B(r)=curlA(r) and divA(r)=0. (13)
c=1
X
As an operator, one may consider the Hamiltonian in Eq.(1) must be replaced by
Eq.(1) as a function of the hydrogen nuclear mass M.
H=K+V +H ,
Hence, the identity rad
2
P −(z e/c)A(R )
∂H K= a a a +
KH =−M ∂M . (7) a (cid:0) 2Ma (cid:1)
(cid:18) (cid:19) X
2
p +(e/c)A(r )
By a generaltheoremof statisticalmechanics[7], one has j j ,
2m
j (cid:0) (cid:1)
∂F 3 X
hK i=−M = Nk T, (8)
H B
∂M 2 z z 1 z
(cid:18) (cid:19)V,N,T V =e2 a b + − a ,
 R r |r −R |
wherein Eq.(2) has been invoked. Regarding the proton Xa<b ab Xi<j ij Xaj j a 
orthedeuteronasthenucleusofthehydrogenatom,itis 1
important to note that the proton is a Fermion and the  Hrad = E(r)2+B(r)2 d3r, (14)
8π
deuteron is a Boson. If the nucleus undergoes classical Z
(cid:0) (cid:1)
motion,thenthewholenotionofquantumstatisticsplays wherein the transverse field equal time commutation re-
noroleinthethermalequationsofstate. Thefreeenergy lations read
difference between light water and heavy water then can
i
be found by integrating Eq.(8) from the proton mass to [A (r),E (r′)]=4π∆ (r−r′),
k l kl
¯hc
the deuteron mass; In detail,
kk d3k
∆(r)= 1− eik·r . (15)
M2 ∂F k2 (2π)3
∆F = dM, Z (cid:18) (cid:19) (cid:20) (cid:21)
∂M
ZM1 (cid:18) (cid:19)V,N,T The hydrogen nuclear velocity V is now given by
3 M2 dM 3 M
∆F =− Nk T =− Nk T ln 2 , e
2 B M 2 B M MV=P− A(R), (16)
ZM1 (cid:18) 1(cid:19) c
3
∆F =− NkBTln2=−(3ln2)NkBT. (9) yielding a mean kinetic energy
2
The difference between light water and heavy water 1 3
M V2 ≥ k T,
thereby amounts to 2 2 B
∂F 3 (cid:10) (cid:11)
∆f =−(3ln2)k T. (10) −M ≥ Nk T =3Nk T. (17)
B ∂M 2 B B
(cid:18) (cid:19)V,N,T
If the hydrogen nucleus moves via classical mechanics,
then the only thermodynamic difference between light InEq.(17)forthehydrogennuclearkineticenergy,equal-
and heavy water is the entropy change per molecule ity holds true when the hydrogen nuclear motions are
classicalandtheinequalityholdstruewhenthehydrogen
∂∆f nuclearmotionsarequantummechanical. Theinequality
∆s=− =(3ln2)k . (11)
∂T B in Eq.(17) is proved in Appendix A.
(cid:18) (cid:19)v
3
Only in the classical case will heavy water have the wherein
same pressure equation of state P(v,T) as light water.
It is experimentally[6]clearly the case that the phase di- i ¯h
agram for heavy water differs from that of light water V≡R˙ = [H,R], β = and ℑmζ >0. (A2)
¯h k T
B
providing strong evidence for quantum mechanical hy- The mobility obeys the sum rules
drogen nuclear motions.
∞
2M
ℜe{µ(ω+i0+)}dω =1,
IV. CONCLUSIONS π Z0
2M ∞ 4πne2
ω2ℜe{µ(ω+i0+)}dω =Ω2 ≡ ,
For quantum electrodynamic general theoretical mod- π 3M
Z0
els of water we have shown that the notion of classi-
cally moving Hydrogen nuclei is in conflict with the ex- n= δ(R−r ) . (A3)
j
perimental differences in the thermodynamic phase dia- * +
j
X
grams of light and heavy water. This conclusion is in
physicalagreementwith recent results based ondeep in-
The self diffusion coefficient obeys the Einstein relations
elasticneutronscattering[1–4]andonneutronandX-ray
scattering[5]. The central point is that for a typical mo-
tion frequency Ω, as givenin Eq.(A3)below, a necessary 1
D = lim ℜe{µ(ω+i0+)},
condition for classical motion is that self kBT ω→0
2
¯hΩ D = lim R(t)−R(0) /(6t) . (A4)
self
βΩ≡ ≪1 (Classical Motion) (18) t→∞
kBT D(cid:0) (cid:1) E
whichfailsbyalargemarginforprotonand/ordeuterium The fluctuation-response theorem[8] implies a rigorous
motion in water. expressionforthe kinetic energyofthe hydrogennucleus
Inmorequantitativedetail, supposeonedefines apro-
ton quantum noise temperature T˜ in terms of the mean 1
proton kinetic energy M |V|2 =
2
∞
1M |V|2 ≡ 3k T˜. (19) 3 2M ET(ω)ℜe{µ(ω+i0(cid:10)+)}dω(cid:11) ,
2 2 B 2(cid:26) π Z0 (cid:27)
(cid:10) (cid:11) 1
Ingeneral,T˜ ≥T asprovedinAppendix A. Ifthequan- ET(ω)= n(ω)+ ¯hω,
2
tum noise temperature is equal (to a sufficient degree of (cid:18) (cid:19)
accuracy) to the thermal temperature, T˜ ≈ T, then the E (ω)= ¯hω coth ¯hω , (A5)
T
proton motion may be presumed to be classical. The (cid:18) 2 (cid:19) (cid:18)2kBT(cid:19)
experimental number[2, 3] for light water in the neigh-
borhood of room temperature is whereinE (ω)isthemeanenergyofathermaloscillator
T
of frequency ω. The inequality
T˜ ≈3.7 T at T ≈300 oK. (20)
Thus, the quantum noise in the proton velocity domi- E (ω)≥k T (A6)
T B
natesthepurelyclassicalthermalfluctuationsinthepro-
ton velocity for normal laboratory water samples. The
together with Eqs.(A3) and (A5) imply
large quantum mechanical contribution to the mean ki-
netic energyis due to the quantum uncertaintyprinciple
localization of the proton. 1M |V|2 ≡ 3k T˜,
B
2 2
(cid:10) (cid:11) T˜ ≥T. (A7)
Appendix A: Inequalities
For a particle obeying classical mechanics to a sufficient
The mobility of a hydrogen nucleus may be expressed
degreeofaccuracy,equalityT˜ =T inEq.(A7)holdstrue.
via the Kubo formula[8] as
For a particle obeying quantum mechanics, the quan-
1 β ∞ tum fluctuations yield kinetic energies over and above
µ(ζ)= dλ dt eiζthV(−iλ)·V(t))i, (A1) the classical thermal value T˜ >T.
3h¯
Z0 Z0
4
[1] G.F. Reiter, J. C. Li, J. Mayers, T. Abdul-Redah and P. 065502 (2008).
Platzman, Brazilian J. Phys., 34, 142 (2004). [6] A.H. Harvey and E.W. Lemmon, J. Phys. Chem. Ref.
[2] C. Pantalei, A.Pietropaolo, R. Senesi, S.Imberti,C. An- Data 31, 173 (2002).
dreani,J.Mayers,C.Burnham,andG.Reiter,Phys.Rev. [7] L.D. Landau and E.M. Lifshitz, Statistical Physics, 3rd
Lett. 100, 177801 (2008). Edition,PartIpp50-51,Butterworth-Heinemann,Boston
[3] D. Flammini, M.A. Ricci and F. Bruni, J. Chem. Phys. (1980).
130, 236101 (2009). [8] R.Kubo,M.TodaandN.Hashitsume,Statistical Physics
[4] A.Pietropaolo,Senesi,C.AndreaniandJ.Mayers,Brazil- II,Springer, Berlin (1998).
ian J. Phys., 39, 318 (2009).
[5] A.K. Soper and C.J. Benmore, Phys. Rev. Lett. 101,