For Noble Gases, Energy is Positive for the Gas Phase, Negative for the Liquid Phase
aa r X i v : . [ c ond - m a t . o t h e r] J un For Noble Gases, Energy is Positive for the Gas Phase, Negative for the Liquid Phase
ASANUMA Nobu-Hiko ∗ No research affiliation
We found from experimental data that for noble gases and H , the energy E is positive for thegas phase, and negative for the liquid, possibly except the small vicinity of the critical point, about(1 − T /T c ) . . E = E c in the supercritical region is found to lie close to the Widomline, where E c is the critical energy. PACS numbers: 64.70.fm, 64.60.Ej, 45.50.JfKeywords: energy, vapor, liquid, bound state, many-body system, critical point
I. INTRODUCTION
What distinguishes the gas phase from the liquidphase? For noble gases, we declare that it’s the signof the energy E : E >
E < . (1)It’s because a gas is an unbound state, so it cannot be E <
0; otherwise, it would condense. Similarly, if
E > t < .
006 along the sat-uration curve (gas-liquid coexistence curve) for 4 noblegases, where t := 1 − T /T c is the reduced temperature, T the temperature, and T c the critical temperature. For theextreme case of H , the violation is only for t < . E c is 0, or equivalently K = | U | at the criticalpoint, where K is the kinetic energy and U the potentialenergy. This strongly indicates a symmetry. Even if it isnot exact, we conjecture that there is a symmetry, andit is weakly broken. More will be discussed in section V.Rule (1) somehow seems to have been unnoticed de-spite its simplicity and decisive power. In the article in2012 titled “What separates a liquid from a gas?”, [2] it ∗ [email protected]; http://z2.skr.jp/phys/ For
T < T c , the internal degrees of freedom of H can be ignored,as is explained in section II C. There is a sole exception, ref 1, but the arguments of this paperdo not make sense. In section 2, they “prove” that the zero ofthe internal energy in thermodynamics is not arbitrary. This isof course absurd. What’s really proven is that it’s not possibleto assign arbitrary zeros to each of subsystems of one entire sys-tem. In spite of this assertion, they do not define the zero ofthe energy, nor does it mention molecule’s internal excitations. is not mentioned. It is not found in recent textbooks ofstatistical physics [3–7] nor in liquid theory textbooks.[8, 9]Looser explanations like “ K ≫ | U | for gases, K ≪ | U | for solids, and K ≈ | U | for liquids” are on the otherhand common. Ref. 10 studied these relations a bitmore further for van der Waals fluid, and heuristicallyobtained the estimate that | U | / ( k B T ) ≈ . k B T ≷ | U | are good estimates for the liquid andgas phases, but not to the precision we give.This letter is organized as follows. In section II, theexperimental data we used and the theory are explained.In section III, the result is stated. In section IV, wetry to extend rule (1) to the supercritical region, and wedraw a qualitative conclusion that the line E = E c liesclose to the “Widom line”,[11] the line of the maxima of C p , the constant-pressure heat capacity. Section V givesdiscussion and outlook. Section VI is the conclusion. II. METHODA. Cited experimental data
As “experimental” data, we rely on NIST ChemistryWebBook data on fluids (hereafter “WebBook”). [12] Infact they are not true experimental data, but the out-put of the program “REFPROP” which computes modelequations. Their parameters are fit to the results of ex-periments done in various conditions, ranging from lowto high temperature and pressure, and near and far fromthe critical point. In addition, models differ from sub-stance to substance. Thus an accurate error estimate isnot available. It is only stated that “These equations arethe most accurate equations available worldwide.” [13]The lower bound of the temperature at which theyprovide data is T tp , the triple point temperature, and forHe, the λ -point temperature. They provide data alongthe saturation curve, in addition to isotherm, isobar, etc. Then in section 4, they assume that the sign of the energy cannotchange within one phase, and concludes that
E > E = 0 at the critical point, etc. They provide data on 75 fluids. All noble gases exceptRn are included.
B. Definition of E = 0 To define the zero of the energy, for noble gases wesafely ignore internal states, i.e. thermal excitation ofelectrons. (The first excitation energy of, for example, Heis about 20eV, and that of Xe is 8.3eV.) In dilute limit,all fluids become an ideal gas. Therefore we naturallydefine that E = 3 / N k B T in dilute limit. Here, N is thenumber of atoms.WebBook provides the data on various thermodynamicproperties, and we in particular need those on the inter-nal energy. The zero of the internal energy is arbitrary,and in WebBook it depends on the kind of fluid. Tointerpret the energy of WebBook, we determine the zeroof E from the value at T = T c , p = 0Mpa for each fluid,where p is the pressure. (WebBook indeed provides datadown to 0MPa, probably extrapolated.) This choice of T is arbitrary, and does not matter. At these points, | C v / (3 / N k B ) − | < − for all available substances,where C v is the constant-volume heat capacity. So theycan be reliably thought as dilute limit. C. Inclusion of H We also examine the behavior of H because for T ≤ T c internal excitations are almost “frozen” and can beignored. (According to WebBook, | C v / (3 / N k B ) − | =3 × − at T = T c , P = 0MPa.) It’s because hydrogenis an exceptional molecule by having the large momentsof inertia. (This is not true even for D , deuterium, forwhich C v ( T c ) / (3 / N k B ) = 1 .
12 at 0MPa.)From WebBook it’s not clear if it is true equilibriumhydrogen, or “normal hydrogen”, i.e. the 3:1 mixture oforthohydrogen and parahydrogen. If it is normal hydro-gen, an orthohydrogen molecule should be considered asstable, not an excited state of parahydrogen. So still thezero of the energy is determined as E = 3 / N k B T at p = 0MPa, where N is the number of molecules. D. Other Comments
Helium is to some extent a quantum fluid on the sat-uration curve, since λ ρ = 0 . ∼ λ is the thermal de Broglie wavelength and ρ is thenumber density. But it’s common to both classical andquantum mechanics that boundness is determined by the For most fluids it says that the origin of E is taken at T =273 .
15K for “saturated liquid”, but for not few fluids it is in thesupercritical region, and this explanation is dubious.
FIG. 1. (Color online) The energy of 5 noble gases and H onthe saturation curve. The upper curve is of the gas phase, andthe lower of the liquid. Horizontal dashed line is for E = 0 . The curves are so scaled that E = − E c and T c for these 6 fluids,and the line is for E = − . Nk B T . sign of the energy, so it is not necessary to modify rule(1) for this case. Rule (1) should apply not only to pure substances butalso to mixtures, as long as there is the natural definitionof the origin of the energy, namely E = 3 / N k B T indilute limit. Normal hydrogen falls into this category. III. RESULT
In figure 1 we show the WebBook data of the energy ofgas and liquid on the saturation curve, for 5 noble gasesand H . The energy in the plot is so scaled that theenergy of liquid at the triple point is − . As it can be seen, rule (1) is satisfied except the neigh-borhood of the critical point. For He, the violation ofrule (1) happens for t < . , for Ar, Kr, and for Xe t < . , and for Ne t < . , it is only for t < . . Considering the inherent uncertainty of WebBook dataand experimental difficulty, this agreement is remarkableand cannot be accidental. We conclude that rule (1) is:“Correct, possibly except very narrow regions near thecritical point.” However, we cannot determine quantita-tively the region where rule (1) does not hold, because ofthe lack of the error estimate in WebBook.Possibly except an area close to the critical point, weare sure that rule (1) is correct not only on the saturation In quantum mechanics, bound states with positive energy ispossible for systems. See for example ref. 14, sec. 10.4. Butit is only for potentials which satisfy special conditions, and weignore such cases. curve, but in a very wide range of p when T tp < T < T c .It’s because heat capacity is positive, and on isotherm ∂E/∂p < T < T tp forwhich WebBook doesn’t provide data.We also note that E is always > E c for the gas phase,and < E c for the liquid phase on the saturation curveaccording to WebBook. A. Critical energy
To assess E c , we also plot E c and T c of the same 6fluids in the inset of figure 1. (Remember an error baris not available.) We also draw the line E = − . N k B T, which is simply “fit by eye.” The agreement of this linewith the experimental data looks good, so we’re temptedto say that E c is indeed ≈ − . N k B T c = 0, but we avoidto draw any conclusion. IV. THE LINE E = E c IN THE SUPERCRITICALREGION AND THE WIDOM LINEA. Introduction
We can not tell if E c is exactly = 0, but the questionif the line E = E c is still meaningful in the supercriticalregion is natural, possibly representing a crossover, di-viding liquid-like and gas-like behavior. In fact lines ofsuch crossover are already proposed, dubbed the “Frenkelline”[15] and the “Widom line”.[11] Actually we feel thatthe arguments on the Frenkel line are more convincingthan those on the Widom line, but we here compare the E = E c line with the Widom line because of the dataavailability.The Widom line is defined as the line of sharp maxi-mum of C p , the constant pressure heat capacity, in the su-percritical region, starting from the critical point. Moreprecisely, the C p divergence at the critical point does notform a round peak, but on each isothermal and isobaricline near the critical point, a sharp C p maximum ex-ists. By connecting those maxima, a “ridge” is formed,and it is the Widom line. It is also characterized as thecollection of maxima of various thermodynamic responsefunctions. Even though the validity of the Widom linenotion is questioned,[16] there is no problem as long aswe consider an area close enough to the critical point. B. Result
We plot in figure 2 the lines E = E c , C p maximum,and also the maximum of C v , the constant volume heatcapacity, for Ne and Xe.Our qualitative conclusion is that the line E = E c runs near the Widom line, in the region of low enough FIG. 2. (Color online) The lines of C v and C p maximumand E = E c in the supercritical region for Ne and Xe. Forhigh enough temperature, the C v maximum lines disappearso they’re not plotted. temperature where the Widom line can be recognizedwithout ambiguity.When the system moves far away from the criticalpoint, C v maximum disappears, and the Widom line maynot be well-defined there. In that region, the line E = E c departs from the C p maximum line.We gave the plots of Ne and Xe, but our result appliesto H , Ar and Kr, too. We cannot assert anything on He:Data close enough to the critical point are not providedby WebBook; for the region with data, the line E = E c and C p maximum do not agree well, and C v maximumcannot be observed. V. DISCUSSION AND OUTLOOK
Rule (1) which we judge almost correct, raises manyquestions. First of all, is it exact? Computer simulationsshould prove it; rather, a disproof will be easier than aproof—experimental verification will be difficult, becauseof finite-size effect and the presence of gravity.[17, 18] Ifcorrect, it must be so for any interactions which havethe critical point and the natural definition of E = 0,independent of dimensionality. (Even though the physicsof noble gases is usually thought to be well described byLennard-Jones potential, the contribution of the three-body forces has to be taken into account to reproducethe third virial coefficient of real noble gases.[19])If rule (1) is not exact, why is its breakdown limited tothe very small region near the critical point? The equa-tion E c = 0 can still be used as the mean-field, zeroth-order value, but how can corrections be calculated? Arethere any system for which exactly E c = 0?If rule (1) is exact, ∂E/∂N → E being = 0 isthe edge of boundness, and is also the point where adimensionful constant vanishes, so it seems to be relatedto the scale invariance of the critical point. However thecondition of E = 0 is not sufficient, since the line ofstates E = 0 does exist in the supercritical region too.Rule (1) also means K = | U | at the critical point. Thisstrongly indicates a symmetry, directly connecting K and U , aside from the scale invariance. Even if not exact, wecan say there must be an approximate symmetry. Whatsymmetry is it precisely? How is it related to the scaleinvariance?As we cautioned, rule (1) is very rough. For example,it completely ignores the formation of atomic clusters. Italso treats the energy from the viewpoint of mechanics,but the energy of a fluid is a thermodynamic quantity,the (canonical) ensemble average, which is not conserved.Definition of boundness is very involved, if ever possible,for many-body systems. At the same time, treatmentin microcanonical, or dynamical system theory may bepossible.Yet, its incisive simpleness allows a clear understand-ing, or new definitions of gas and liquid. For example,consider the solution of solute A and solvent B withoutinternal degrees of freedom. Then it can be said that A isgaseous inside the solution, and B is liquid. Let us writethe Hamiltonian H as: H = K A + K B + U AA + U BB + U AB , (2)where K A is the kinematic energy of A particles, U AB isthe potential between A and B particles, and so on. Nowintegrate out B’s variables. Then we obtain the effectiveHamiltonian H eff which looks like: H eff = K A eff + X i U i , (3)where U i is the i -body effective potential, which is := 0in dilute limit. What rule (1) tells is that h K A eff i + P i ≥ h U i i > , where the bracket is the thermal average,because A is gaseous. In addition since the whole systemis liquid, P i ≥ h U i i < −h K A eff i < . So, −h U i is similarto to the work function of metals, although in the currentcase the temperature is finite. If A and B demixes so thatthe A-rich phase and the B-rich phase coexist, then A isgaseous in B-rich phase and liquid in A-rich one, and soon. It seems almost obvious, in reality a mere heuristicthough, that the critical point of binary fluid consolutionbelongs to the same universality class as gas-liquid’s one.A still easier example is the theory of dilute solu-tions, found in every textbook of thermodynamics. Whenthe author was a student, he felt the appearance of thegas constant R was sudden and absurd. “Interaction isstrong, the ideal gas has nothing to do here, no?” It is theconsequence of thermal average, but we have an alterna-tive view. In fact, solute is a gas trapped in the solvent, Thermodynamics of dilute solution can be derived within pure and in dilute limit, it becomes an ideal gas, because theinteraction between solute molecules can be neglected.The mean free path of the bare solute molecule does notmatter.Rule (1) imposes a limit on the spinodal curve, too.The spinodal curve is difficult to define theoretically. Intextbooks, it is often explained mean-field theoreticallyas “the” inflection point of (the metastable branch of) thefree energy. (See for example ref. 6, section 8.7.3, or ref.21, section 3.4.3.) More careful definition is as the occur-rence of negative compressibility for all wavelengths,[22]but it still suffers from the fact that it may not be welldefined due to metastability. We know however that thesupercooling of gas and superheating of liquid cannot ex-ceed the line E = 0 . It is only a necessary condition, butthe energy of the system is always defined. At the veryleast it explains the existence of spinodal curves in gas-liquid transition.We do not know how to extend rule (1) for molecu-lar fluids. Molecules have internal degrees of freedom,namely ro-vibrational modes. Intermolecular interac-tions depend on the internal states, or in other words,they mix and it is not possible to define the quantity U separately from internal states. In rule (1) translationaldegrees of freedom are concerned, so to promote it tomolecular fluids, we have to extract and separate themfrom internal degrees of freedom. It must be possible,since critical points exist also for molecular fluids, butwe are clueless how to do it.Rule (1) also hints at something on the notion of clusterand percolation in lattice and off-lattice systems, whichis easiest to describe from the standpoint of Monte Carlosimulation. (For an introduction see for example ref. 23,section 5.1) “Cluster algorithms” in general update allvariables in a group, called cluster, but we call it “up-dating cluster” (UC). There is another cluster, whichpercolates at the critical point, which we call PC. PCis used to locate the critical point in “invaded clusteralgorithm”[24]. In Ising model, PC is the set of paral-lel spins which are connected. It is also generalized forexample to Widom-Rowlinson model,[25, 26] but not forgeneral fluids. UC is a subset of PC, and it has to sat-isfy detailed-balance. It is usually chosen to make thealgorithm most efficient, but it is not necessary. Becausepercolation is deeply connected to criticality, the currentsituation where PC is lacking for general systems is un-satisfactory. Our questions are, how to define PC forgeneral systems, and does UC have a physical meaningbeyond a mere computational utility? Is it possible to de-fine an analogue of the kinetic energy for lattice systems?By answering them, it may be possible to obtain moreinsight on the opaque relations between the lattice-gasmodels and fluids. thermodynamics, without the need of statistical mechanics. Seefor example ref. 20, chapter 7. FIG. 3. (Color online) The energy difference of evaporation∆ E ( t ) of 75 fluids [12] along the saturation curve, normalizedto 1 at the triple point T = T tp . The solid line means t . with the same normalization. The data of He, H , and D arerepresented by specially thick dots, which substantially differfrom others. In physics, models, even toy models, have served tomake various advances, and we inevitably pose this ques-tion: Is there any one-particle, central force system, clas-sical or quantum, which has a phase transition at T = T c , and for T ≷ T c , E ≷
0? For classical cases, natural orderparameters are h /r i and h U i . Rule (1) also suggests that the energy may be an or-der parameter. What we have discovered recently [27]is that the energy difference ∆ E ( t ) of evaporation along the saturation curve is universal, by being well approx-imated by the power law ∝ t a , where a ≈ . includ-ing molecular fluids . Figure 3 shows ∆ E for 75 fluids ofwhich data is provided by WebBook, This is surprisingand uncanny, because ro-vibrational modes are diverseamong substances. We also found that T ∆ S ∝ t . .They are not critical phenomena; these two power lawsapply to the entire saturation curve except an area nearthe critical point , but instead down to the triple point .The critical exponents of them and of p ∆ V = − ∆ F are β . Here ∆ E and T ∆ S are per particle; contrary to ourmotivation, we couldn’t find anything conclusive on the spatial energy density.We pointed out that E is a quantity that can be definedpurely in mechanics, without thermodynamics. But notonly ∆ E, but also ∆1 /V, the density difference, is anorder parameter along the saturation curve, as knownvery well, and V is a pure mechanical quantity, too. Somemysterious truth seems to be still hidden. VI. CONCLUSION
We found from experimental data that for noble gasesand H , the energy is positive for the gas phase, andnegative for the liquid phase. According to the used data,this rule dose not hold for 1 − T /T c < .
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