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Fit function for the vapour pressure of 3He
Volume 149, number 4
PHYSICS LETTERS A
24 September 1990
Fit function for the vapour pressure of 3He Albrecht Eisner Max-Planck-Institut für Plasmaphysik, Garching, FRG Received 4 December 1989; revised manuscript received 17 July 1990; accepted for publication 18 July 1990 Communicated by J.P. Vigier
The 1962 3He scale of temperatures, as well as the later EPT-76 (3He) scale, is defined by a fit function forthe vapour pressure of 3He. At very low temperatures and above 3 K each of the two fit functions proves to be thermodynamically inconsistent. The correct formulation of a fit function can be used for determining the evaporation energy of 3He and its temperature behaviour at absolute zero, yielding ~k for the energy (~ being in units of K, and k being the Boltzmann constant) and a scaling as T’” with rn~1 for the temperature derivative. At the critical point, a fit function has to satisfy the condition d2 ln p1 (d In T)2 = 0. The two fit functions, however, do not obey this constraint and produce values forthe temperature derivatives ofthe vapour pressure d2p/ dT2 and d3p/dT3 which are too high above 3 K.
1. Introduction Both the 1962 3He [1] and the EPT-76 (3He) [21 scales of temperatures are defined by fit functions for the vapour pressure which were accepted by the International Committee on Weights and Measures. Since these two scales do not qualitatively differ (a vapour pressure value from ref. [1] is, for equal temperature, only slightly higher than one from ref. [21), all statements on the 1962 3He scale in the following are, mutatis mutandis, also valid for the EPT76 (3He) scale. When the vapour pressure equation is used for constructing the phase diagram of 3He, there are numerical inconsistencies between various thermodynamic parameters of state in the temperature ranges around absolute zero and the critical
pressure data well above absolute zero. This means that a correctly formulated fit function, though unimportant for determining the temperature in the very low range owing to the immeasurably low Vapour pressure of 3He, can nevertheless be used for determining the evaporation energy at absolute zero, giving e= &,~+ ~2 Tm with rn> 1 and thus making it possible to eliminate the deficiency of the non-vanishing temperature derivatives de/dT= const present in refs. [1] and [21. Secondly, the fit function has to be modified in such a way that the function d in (pIT) /d ln T constitutes a strictly monotonically decreasing temperature function. The strict monotonicity of d in(piT) /d ln T is derived from a relation between the vapour pressure p, evaporation energy e and volume energy (v~ v 5 )p (v~ v5 volume difference between the vapour and liquid). The insufficiently exact reproduction of the vapour pressure values by the fit functions of refs. [11 and [2] above 3 K is manifested by theagain fact increase that the as functions dln(p/T)/din Tobtained the temperature after passing through a minimum at 3 K. This can be avoided by observing the monoto—
—
point. It is thus necessaryto make two corrections to the fit function, which according to refs. [1] and [2] is of the form 3. lnp=a1 in T+ p=2 ~ a~T~ Firstly, it is thermodynamically demonstrated that the parameter a 1 preceding the series term in Thas to take the value 1 so that the evaporation energy e and its temperature dependence in the vicinity ofabsolute zero can be correctly obtained from vapour 184
nicity condition.
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Elsevier Science Publishers B.V. (North-Holland)
Volume 149, number 4
PHYSICS LETTERS A
2. Fit function for the vapour pressure With the abbreviations a for the evaporation energy U.,. u5 and 9~for the volume energy (v~ V5 )p one obtains from the Carnot—Clausius—Clapeyron relation —
—
—
d(p/T) d(i/T)
a
u~—u5 =
~
~P,
—~,
the thermodynamic relation for the vapour pressure: d ln (piT) d in T
=
(1)
—.
Here p is expressed in terms of the ratio of the measurable heats a and (Dand hence, according to Carnot, is associated with the thermodynamic temperature T. The vapour pressure is therefore a primary thermometer. The two functions a and ~ have the property of being order parameters, i.e. for 0< T< T~they are positive temperature functions and vanish when the critical point is approached, in accordance with Int
lim
T—.T,lfl(T~T) =
mci,
.
=
hm
a) /dT= (~~/a) [din ((Die) /dln( T~—T) I dln( T~—T) / dT=(~~/e)[ln~/ln(T~—T)—ln e/ln(T~—T)]/(T T~)= (ç/e ) 0/1 = 0.) According to eq. (1) this —
yields dinT >0 dTdln(p/T)
(5)
Eq. (5) states that the function dln T/dln(p/T) for T< T~is a strictly monotonically increasing temperature function and its reciprocal value d in(p/T) / din T a strictly monotonically decreasing temperature function, in each case with vanishing temperature derivative at T= T~. According to eq. (1) the fit function for the vapour pressure p is essentially determined by the fit function for e/q. As the ratio dci’ in the vicinity of absolute zero scales as 1 / T in accordance with eq. (3) and at the critical point tends towards the fixed positive valuedin T/d ln (piT) T. accordancewith eqs. (1) and (2), one may attempt to represent experimental data for p and T by the fit function lnp=lnT+~a~T~’,
_______
(6)
-
T_.T~lfl(T~—T)
lim ln(v~—v5) mfl. In ( T~ T)
T-. T~
24 September 1990
where the a,, are the fit parameters. (2)
—
Here /J is ‘he usual critical exponent of the density difference between the liquid and vapour. When absolute zero is approached, a remains finite while ~ scales as kT (k Boltzmann constant):
It now has to be shown that expansion (6) is in accord with eq. (1), and the physical meaning ofthe first four parameters a1—a4 has to be stated. Integration of eq. (1) yields directly
J
in T
lnpln T+lnp. —in T~ +
—
dln T,
(7)
in Tr
lime=a0>0,
limci~=klimT.
T-.O
T-.O
(3) if an
T-.O
arbitrary point on the vapour pressure curve is
The sum a + ~i is the evaporation enthalpy. As the temperature increases, the contribution of the volume energy to the evaporation enthalpy increases monotonically in relation to the contribution of the evaporation energy, i.e. it holds that
denoted by Tr and Pr p( Tr). According to what was said about the functions a and çs the fit function
d
can be proposed for the ratio a/(D. With this ansatz eq. (7) can be rewritten:
d
~, ~
~
a
(4)
Hence it immediately follows that d((D/e)/dT~0, the equality sign being taken at the critical point owing to eq. (2). (To prove limT. ~0d( co/a) /dT= 0, the l’Hospital rule is applied, yielding ~/a=d~/da=d~/ d[~ln(p/T)/dln T] =dln T/dln(p/T)~0 and d(~/
—
ci’
=
~
(v—3)a,,T”’
(8)
V=2
lnp=lnT+ln(pr/Tr)+
~ a,,(T”’—T~’). (9)
In this equation the variable ln T only occurs once and has the prefactor a1 1. Eqs. (6) and (9) are identical if 185
Volume 149, number 4
PHYSICS LETTERS A
24 September 1990
(15) thus does not correctly describe the vapour a,mln(pr/Tr)_a2/Tr_ ~ ~
(10)
The values of a and a, are thus fixed and hence do not constitute fit parameters in the series (6). If the critical point (pa, T~)is taken as reference point (Pr, Tr),calculation. the boundary condition included~1 ini the It then follows(5) thatcan d Edbelnp/din
pressure in the vicinity of absolute zero. The vapour pressure formula (15) was conceived Ofl the lines ofthe equation of state for the ideal gas. As is known, the entropy of N non-interacting monatomic particles in the volume V is2/2itrnkT)”2, given by s~= h k{~ + lnPlanck [(V/N)constant, } (where (h atomic mass). is the and rnvu is the /Vq]
dTIT
d(e/~)/dT1 r. = 0 or, according to eq. (8), 2a,,T~’=0. (11) ~ (v—3) 0
=
2
This relation was used to calculate the entropy of the liquid s5 by means of the Carnot—Clausius—Clapeyron equation (s~—s 5)i(v.,—v5)=dp/dT. If then in the thermodynamic relation s5 = s,, (~+a) / T the term s.~is replaced by and in S5 the term V/N by V.,, one obtains V \ +~ st=k~+ln—)_~_. (16) —
This is an equation for determining T~.Adding the two equations (10) and (11) yields a further expression for a3: 3. 9=5 (v—2) (v—4) a,,T~ a,=ln(p~/T~)+~ (i2)
S5,
In order to reach limT.osQ=O, it must be assumed that the relations e/kT= ~in (e x 2~~rnkT/h2) + In v,,
Hence the parameter a 3 can be given in terms of the critical value (p~~ T~)and the parameters a5, a~, which govern the vapour pressure value in the hightemperature range. The significance of the fit parameter a2 is made immediately obvious by combining the three equations (1), (3) and (6). It is found that 2=—ka lim a~e0=klim ~ (v—3)a,,T~ 2. ...,
T-.O
T—.O v=2
(13) The fit parameter a2 thus represents the condensation energy at absolute zero in units of the Boltzmann constant: a2 = t0/k. The evaporation energy in the vicinity of absolute zero according to eq. (13) is —
(14)
In the vapour pressure formula of refs. [1] and [2], ~np —a1 I
*—
*1
tfl
‘rj ~-r ~ ~‘
186
—
—
‘.‘-‘
—
~.
—
>
tion for the real vapour ~ s~ = (q~ + a) / T is taken as a basis and, since s5=s~—(~+a)/T, the limit lim~.os~ = 0 is readily reached: limss=!~_~~~=0.
(17)
In particular, because of eqs. (1) and (3), it also holds that Iim~,0din(p/T)/dlnT= (ao/k)lim~.0 T It is thus not possible to justify introducing a prefactorotherthan 1 infrontofthesenestermlnT in the vapour pressure formula. Three further examples discussed below may show that a1 = 1, obtamed tion andfrom fromthe theCarnot—Clausius—Clapeyron temperature behaviour of theequareal gas in the vicinity of absolute zero, excludes all other possibilities a ~# 1 coming from the said model calculation approximation or purely numerical fitting: —‘.
*i-v—3
all a ~ values are fit parameters and, in particular, it holds that a~>1 [1,2]. One then obtains result 2 +...]. Nearthe absolute azero = k [it is a~ + (a ~ 1) T+ a T the linear term (ar 1 )kT here that determines the temperature behaviour of a, which is physically not correct. The vapour pressure formula —
and a0 = kT in V., are valid at absolute zero, which may very well be the case. For the pressure it follows in the limit T—~0that p/ T= k/V., = (kivq) exp ( e0/kT) or lim~,0dln(p/T)/dln T=~+(eO/k)limT~.O T ‘=limT.O(eO/k) lim~.0T It has to be noted, however, that the ideal gas model may only be used in the high-temperature range, where the effect of a parameter a~ 1 can of course be numerically ironed out by means of the other fit parameters. Here, on the other hand, without making any assumptions at all, the thermodynamic entropy rela-
Volume 149, number 4
PHYSICS LETTERS A
(1) According to the Thomsen—Berthelot principie [3] it follows directly that a=a mwith 0+a~T rn> 1 for T—+0. (2) The low-temperature expansion of the chem-
24 September 1990
y=u~T/T~+u~[l—(T/T~)2+...]<0, dp/dT=y~iT~2~T/T~+...
ical potential [4], T— TJ
—
d(p/T) dT,
T
dT
follows from the expansion of the integrand, i.e. d(p/T)
=
dT kT+ T2 ...
=
z.~pdln(p/T T d in T 2 + ... a~ +kT+... a2 T =
+d 2 ~ 2+.... Ifone had a
temperatures above 3 K.
and reads u= —a0—a1T+a2T 1 1, this would add the temperature-dependent term (a1 1) Tin T, which would then prevent the ther2p/dT2<0 modynamic conditions dp/dT< 0 and d from being satisfied. (3) Similarly, it can be shown that the temperature derivative of the surface tension, dy/dT, is negative and vanishes for T-+0 only ifa=a 2+.... 0+a2T The case a, ~ 1 would yield dy/dT> 0 in the vicinity of absolute zero and a logarithmic singularity at T= 0 because [4] —
$
T~ ~=
T
a
2+... 0 + (a1 —1 T)T+a2 T
(Vb
g v
2 3
—
5) =
dT
/
-4~{ao+[a 2Tc—eo/Tc+ai Vs~~ 2—
—
l)ln
TC]T
(a
—a2T
1
—
l)Tln T}
.
The values of a2, m limT.O y, p~ lim~.op and the temperature behaviour of a, y and p can now be given: From dy/d T10 =0 it follows that a2 = a0/ T hence (neglecting the contribution of the zero-point energy on p) 2+...]>0, a=e0[1 +2a0 (T/T~) T/T~+ ... ? 0, da/dT= 3>0, Y Yo [1 (Ti T~)2+...] >0, Yo =g0a0 /v~ dy/dT= —2y 0T/T~+...~o, ~,
—
=
_ao <0,
3. Re-determination of the fit function for the vapour pressure Using the vapour pressure values, which according to refs. [1] and [2] are given in the temperature range 1 to 3 K by formula (15) and at the critical point by p~=O.i164±0.0002 J/cm3 and T~= 3.322 ±0.002 K, we shall now determine a fit function for the vapour pressure in the entire temperature range ofliquid 3He using eq. (1) and taking into account relations (3), (5) and (13). On the other hand, it may be noted that critical values, which are different to those of ref. [11, have been published as 3 and well, for example p,~=O.ll46±O.OOO2 J/cm T~=3.309±0.00l K according to ref. [5]. If the function ymln[dln T/dln(p/T)] is represented versus ln T, then because of eq. (5) it is a strictly monotonically increasing function with a horizontal tangent at ln T~.The values of y known from ref. [2] and represented in fig. 1 as a dashed curve can be extrapolated from the temperature range below 3 K to give the solid curve with the ordinate value limT,T~y=yCat in T~=x~. At very low temperatures the temperature dependence of—a the vapour pressure is given by dln(p/T)/dln T= 2/T. As the value of a 2 according to eq. (13) is negative, it decreasing values ln T the function y becomesand linear, holds that din T/din(p/T)=T/(—a2)>0, for in T—ln(—a2). Again, the values of y are extrapolated from the temperature range above 1 K, yielding —ln ~ for the ordinate value —ln(—a2) in accord187
Volume 149, number 4
PHYSICS LETTERS A I
I
I
24 September 1990 In T
I
r dinT
--
lnp=ln T+lnp~—lnT~+
‘He 0
T~=3.3190
j
(19)
e~I1T~
InTc
P
Yc=
90.1150
/
F-P OH
/
-o
0.86539
7
In fig. 2 the vapour pressure and the first three temperature derivatives obtained according to eqs. (15) and (18) are represented versus the temperature as, respectively, dotted and solid lines. The cal-
I
I-
‘1
/‘
l’
>.~
4
2
0
______________________________ ‘0.0
0.2
0.4
0.6
x
=
ri
0.8
1.0
1.2
rameters a~ from ref. [2] and, in the other, with the boundary values T~=3.3l90 culations were performed, in oneK, case,x~=l.l99663, with the 76)= pa0.114988 J/cm3. y=l.l48762 y~=—O.86S39, The main differences and p~~p~( between the two calculations are the deviations in d2p/dT2 and d3p/dT3. In fig. 3 all differences are shown as relative deviations 2(p*_p) and 2(drnp*/dTrn_drnpi dTrn)/(drnp*idTrn+drnp/dTrn) (rn=1, 2, 3), the values marked with an asterisk being taken from the
T
Fig. 1. Function y=ln [dIn T/d ln(p/T)] versus x=ln Tfor 3He according to eq. (15) using the parameters a~ (EPT-76 ‘He scale) (dashed line) and eq. (18) (solid line).
ance with refs. [1] and [2]. At very low temperatures the vapour pressure is thus described by d ln (Ps” T) /d in T=~T For the evaporation energy at absolute zero it then follows that a 0= —a2k=~k(a2 or ~ in units of K); this corresponds to the value 20.78 J/mol, which is between the values 20.72 J/mol of ref. [1] and 20.86 J/mol of ref. [2]. The function y=y(x) with x~ln T represented in fig. I by the solid line is now given in analyticalform. It is assumed here that y=x—ln( —a2), and the solution for the region around the critical point, y(x~)=y~with dy/dx1~=0. The interpolation formula then reads —~.
calculation according to formula (15). It is evident that formula (15) yields values in the temperature range below 0.2 K which are too high. According to refs. [1] and [2], of course, formula (15) is only applicable in the temperature range above 0.2 K. It is assumed that the vapour pressure values p~$~ of ref. [2] are well known in the temperature range from 1 to 2.8 K. Hence a criterion for determining the
I
I
I
I
‘He
/
y=x—ln(—a2)—yexp[(x—x~)/y] xmln T,
The temperature derivative, dy/dx= 1 —exp[ (x— x~)iy], decreases monotonically with increasing temperature from 1 at absolute zero to 0 at the critical point.
188
/
.
ad
. °
°
m=3
2
1
.
...
0
.
1
This fit function y=y(ln T) allows one to determine the vapour pressure in thetoentire temperature 3He. According eq. (7) it is given range of liquid by
/
.
,
y~x~—y~—ln(—a2), —a2=~. (18)
/
..-.
/
I_0
2°
2 temperature
3
K
Fig.‘He 2. Vapor pressure and(18) the first three derivatives for calculated frompeq. (solid lines) and fromd”p/dT’” eq. (15) (dotted linea).
Volume 149, number 4 ‘1
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PHYSICS LETTERS A I
I
I
I
.
.
No
/
d
m=0
F
/
—
0
1
The task of finding a fit function for the vapour
I
‘He
----
/
2 -ON
24 September 1990
pressure of ‘He is accomplishedwith eqs. (18) and (19). This can also be done, however, by representing eq. (19) as a fit function on the lines of eq. (6), lnpmlnT—~T’+a,+ 9=4 ~ a,,T~3,
II
I
(21)
where a
to
3 is fixed by relation (12) and a4 is equal to a2/ T~.In this case one obtains the parameter values a5, ..., a~from a least-squares-fit calculation of 2 exp[y(TiT~)”~’]—1— (T/T~) n—4 =~ ~ (v+1)a,,~ 2. (22) 4T~’ 9=1 —
‘I
—--H.---
0
a .2 0
/
o ‘0
“
,
___________________________________________ 1
2 temperature
K
Fig. 3. Relative deviations 2(p*_p)/(p*+p) ~ 3) calculated from eqs. (18) and (15).
(rn=0) and (rn= 1,2,
As a consequence of the ansatz (8) for the ratio a/ ~thepower T~~2 appears in the series (22), which does not allow the left-hand side of eq. (22) to be fitted well. The fit parameters a,, calculated are therefore not presented here. The vapour pressure of 4He is used to define the 4He scale (up to 5.2 K) [6]. For temperatures above the triple-point temperature of H 2 (13.81 K) the International Committee on Weights and Measures have recommended that the internationally 1958
boundary values
y~and
x~may be deduced. This
means that Ye and x~ = in T~can be calculated by minimizing the value of the integral 2.8K
C 1K
I~ P (Ye, x~)I dT pp~,
x~)
(see fig. 3), leading to the numerical values just stated. The selected critical values can be tested to be flumerically correct if the corresponding relations at the critical point are sufficiently satisfied. These 2 ln p/ (din T) ~, rela= 0, tionsread follow from eq. (5), i.e. d and T2
T~ :p dT2 dT dlnpfdlnp \ dlnp = din T~,,dlnT 1). dinT1, -~--~:
for T= T~.
agreed practical temperature scale, IPTS-68, be used [7] in conjunction with the provisional 0.5 to 30 K temperature scale, EPT-76 [2]. A new IPTS-90 is under preparation. Candidates for defining a ternperature scale from 5.2 to 13.81 K are the gas thermometer [8] and the platinum and germanium resistance thermometers. At temperatures below the useful the ‘He vapour thermometer, i.e.range in theofmillikelvin range,pressure temperature scales are proposed which are based on measurements of the specific heat of liquid 3He at melting density [9] and of the magnetization of metallic samples [10].
Acknowledgement (20)
With p,, = 0.1150 J/cm3, T~ = 3.319 K and d in p/ d in T 1~—1 + exp(0.86539) (see fig. 1) one has3 dp/dT1~ = (p~/T~) d in p/d ln T1~=0.1169 J/K cm and d2p/dT~~ = (dp/dT 1~)[dlnp/dln T1~—1 ]/T~= 2 cm3.
The author is grateful to the referees for their valuable criticisms, which helped to clarify the text and were responsible for the insertion of examples 1 to 3.
0.0837 J/K 189
Volume 149, number 4
PHYSICS LETTERS A
References [l]T.R. Roberts, R.H. Sherman, S.G. Sydoriak and F.G. 3He scale of temperatures, in: Brickwedde, The 1962 Progress in low temperature physics, Vol. IV, ed. C.J. Gorier (North-Holland, Amsterdam, 1964) pp. 480—5 14. [2] M. Durieux and R.L. Rusby, Metrologia 19 (1983) 67. [3] H.B. Callen, Thermodynamics (Wiley, New York, 1960) ch. 10.3. [4] A. Eisner, Phys. Lett. A 130 (1988) 225. [5] R.P. Behringer, T. Doiron and H. Meyer, J. Low Temp. Phys. 24(1976)315.
190
24 September 1990
[6] H. Van Dijk and M. Durieux, The temperature scale in the liquid helium region, in: Progress in low temperature physics, Vol. lIed. C.J. Gorier (North-Holland, Amsterdam, 1957) pp.431—462. [7] M. Durieux, The international practical temperature scale of 1968, in: Progress in low temperature physics, Vol. VI, ed. C.J. Gorier (North-Holland, Amsterdam, 1970) pp. 405—425. [8] D.N. Astrov, L.B. Beliansky, Y.A. Dedikov, S.P. Polunin and A.A. Zakharov, Metrologia 26 (1989) 151. [91D.S. Greywall, 18th mt. Conf. on Low temperature physics, LT-18 (1987) pp. 2082—2083. [l0]G.Eska,Lowlemp.Phys.73(1988)207.
PHYSICS LETTERS A
24 September 1990
Fit function for the vapour pressure of 3He Albrecht Eisner Max-Planck-Institut für Plasmaphysik, Garching, FRG Received 4 December 1989; revised manuscript received 17 July 1990; accepted for publication 18 July 1990 Communicated by J.P. Vigier
The 1962 3He scale of temperatures, as well as the later EPT-76 (3He) scale, is defined by a fit function forthe vapour pressure of 3He. At very low temperatures and above 3 K each of the two fit functions proves to be thermodynamically inconsistent. The correct formulation of a fit function can be used for determining the evaporation energy of 3He and its temperature behaviour at absolute zero, yielding ~k for the energy (~ being in units of K, and k being the Boltzmann constant) and a scaling as T’” with rn~1 for the temperature derivative. At the critical point, a fit function has to satisfy the condition d2 ln p1 (d In T)2 = 0. The two fit functions, however, do not obey this constraint and produce values forthe temperature derivatives ofthe vapour pressure d2p/ dT2 and d3p/dT3 which are too high above 3 K.
1. Introduction Both the 1962 3He [1] and the EPT-76 (3He) [21 scales of temperatures are defined by fit functions for the vapour pressure which were accepted by the International Committee on Weights and Measures. Since these two scales do not qualitatively differ (a vapour pressure value from ref. [1] is, for equal temperature, only slightly higher than one from ref. [21), all statements on the 1962 3He scale in the following are, mutatis mutandis, also valid for the EPT76 (3He) scale. When the vapour pressure equation is used for constructing the phase diagram of 3He, there are numerical inconsistencies between various thermodynamic parameters of state in the temperature ranges around absolute zero and the critical
pressure data well above absolute zero. This means that a correctly formulated fit function, though unimportant for determining the temperature in the very low range owing to the immeasurably low Vapour pressure of 3He, can nevertheless be used for determining the evaporation energy at absolute zero, giving e= &,~+ ~2 Tm with rn> 1 and thus making it possible to eliminate the deficiency of the non-vanishing temperature derivatives de/dT= const present in refs. [1] and [21. Secondly, the fit function has to be modified in such a way that the function d in (pIT) /d ln T constitutes a strictly monotonically decreasing temperature function. The strict monotonicity of d in(piT) /d ln T is derived from a relation between the vapour pressure p, evaporation energy e and volume energy (v~ v 5 )p (v~ v5 volume difference between the vapour and liquid). The insufficiently exact reproduction of the vapour pressure values by the fit functions of refs. [11 and [2] above 3 K is manifested by theagain fact increase that the as functions dln(p/T)/din Tobtained the temperature after passing through a minimum at 3 K. This can be avoided by observing the monoto—
—
point. It is thus necessaryto make two corrections to the fit function, which according to refs. [1] and [2] is of the form 3. lnp=a1 in T+ p=2 ~ a~T~ Firstly, it is thermodynamically demonstrated that the parameter a 1 preceding the series term in Thas to take the value 1 so that the evaporation energy e and its temperature dependence in the vicinity ofabsolute zero can be correctly obtained from vapour 184
nicity condition.
0375-9601 7907$ 03.50 © 1990
—
Elsevier Science Publishers B.V. (North-Holland)
Volume 149, number 4
PHYSICS LETTERS A
2. Fit function for the vapour pressure With the abbreviations a for the evaporation energy U.,. u5 and 9~for the volume energy (v~ V5 )p one obtains from the Carnot—Clausius—Clapeyron relation —
—
—
d(p/T) d(i/T)
a
u~—u5 =
~
~P,
—~,
the thermodynamic relation for the vapour pressure: d ln (piT) d in T
=
(1)
—.
Here p is expressed in terms of the ratio of the measurable heats a and (Dand hence, according to Carnot, is associated with the thermodynamic temperature T. The vapour pressure is therefore a primary thermometer. The two functions a and ~ have the property of being order parameters, i.e. for 0< T< T~they are positive temperature functions and vanish when the critical point is approached, in accordance with Int
lim
T—.T,lfl(T~T) =
mci,
.
=
hm
a) /dT= (~~/a) [din ((Die) /dln( T~—T) I dln( T~—T) / dT=(~~/e)[ln~/ln(T~—T)—ln e/ln(T~—T)]/(T T~)= (ç/e ) 0/1 = 0.) According to eq. (1) this —
yields dinT >0 dTdln(p/T)
(5)
Eq. (5) states that the function dln T/dln(p/T) for T< T~is a strictly monotonically increasing temperature function and its reciprocal value d in(p/T) / din T a strictly monotonically decreasing temperature function, in each case with vanishing temperature derivative at T= T~. According to eq. (1) the fit function for the vapour pressure p is essentially determined by the fit function for e/q. As the ratio dci’ in the vicinity of absolute zero scales as 1 / T in accordance with eq. (3) and at the critical point tends towards the fixed positive valuedin T/d ln (piT) T. accordancewith eqs. (1) and (2), one may attempt to represent experimental data for p and T by the fit function lnp=lnT+~a~T~’,
_______
(6)
-
T_.T~lfl(T~—T)
lim ln(v~—v5) mfl. In ( T~ T)
T-. T~
24 September 1990
where the a,, are the fit parameters. (2)
—
Here /J is ‘he usual critical exponent of the density difference between the liquid and vapour. When absolute zero is approached, a remains finite while ~ scales as kT (k Boltzmann constant):
It now has to be shown that expansion (6) is in accord with eq. (1), and the physical meaning ofthe first four parameters a1—a4 has to be stated. Integration of eq. (1) yields directly
J
in T
lnpln T+lnp. —in T~ +
—
dln T,
(7)
in Tr
lime=a0>0,
limci~=klimT.
T-.O
T-.O
(3) if an
T-.O
arbitrary point on the vapour pressure curve is
The sum a + ~i is the evaporation enthalpy. As the temperature increases, the contribution of the volume energy to the evaporation enthalpy increases monotonically in relation to the contribution of the evaporation energy, i.e. it holds that
denoted by Tr and Pr p( Tr). According to what was said about the functions a and çs the fit function
d
can be proposed for the ratio a/(D. With this ansatz eq. (7) can be rewritten:
d
~, ~
~
a
(4)
Hence it immediately follows that d((D/e)/dT~0, the equality sign being taken at the critical point owing to eq. (2). (To prove limT. ~0d( co/a) /dT= 0, the l’Hospital rule is applied, yielding ~/a=d~/da=d~/ d[~ln(p/T)/dln T] =dln T/dln(p/T)~0 and d(~/
—
ci’
=
~
(v—3)a,,T”’
(8)
V=2
lnp=lnT+ln(pr/Tr)+
~ a,,(T”’—T~’). (9)
In this equation the variable ln T only occurs once and has the prefactor a1 1. Eqs. (6) and (9) are identical if 185
Volume 149, number 4
PHYSICS LETTERS A
24 September 1990
(15) thus does not correctly describe the vapour a,mln(pr/Tr)_a2/Tr_ ~ ~
(10)
The values of a and a, are thus fixed and hence do not constitute fit parameters in the series (6). If the critical point (pa, T~)is taken as reference point (Pr, Tr),calculation. the boundary condition included~1 ini the It then follows(5) thatcan d Edbelnp/din
pressure in the vicinity of absolute zero. The vapour pressure formula (15) was conceived Ofl the lines ofthe equation of state for the ideal gas. As is known, the entropy of N non-interacting monatomic particles in the volume V is2/2itrnkT)”2, given by s~= h k{~ + lnPlanck [(V/N)constant, } (where (h atomic mass). is the and rnvu is the /Vq]
dTIT
d(e/~)/dT1 r. = 0 or, according to eq. (8), 2a,,T~’=0. (11) ~ (v—3) 0
=
2
This relation was used to calculate the entropy of the liquid s5 by means of the Carnot—Clausius—Clapeyron equation (s~—s 5)i(v.,—v5)=dp/dT. If then in the thermodynamic relation s5 = s,, (~+a) / T the term s.~is replaced by and in S5 the term V/N by V.,, one obtains V \ +~ st=k~+ln—)_~_. (16) —
This is an equation for determining T~.Adding the two equations (10) and (11) yields a further expression for a3: 3. 9=5 (v—2) (v—4) a,,T~ a,=ln(p~/T~)+~ (i2)
S5,
In order to reach limT.osQ=O, it must be assumed that the relations e/kT= ~in (e x 2~~rnkT/h2) + In v,,
Hence the parameter a 3 can be given in terms of the critical value (p~~ T~)and the parameters a5, a~, which govern the vapour pressure value in the hightemperature range. The significance of the fit parameter a2 is made immediately obvious by combining the three equations (1), (3) and (6). It is found that 2=—ka lim a~e0=klim ~ (v—3)a,,T~ 2. ...,
T-.O
T—.O v=2
(13) The fit parameter a2 thus represents the condensation energy at absolute zero in units of the Boltzmann constant: a2 = t0/k. The evaporation energy in the vicinity of absolute zero according to eq. (13) is —
(14)
In the vapour pressure formula of refs. [1] and [2], ~np —a1 I
*—
*1
tfl
‘rj ~-r ~ ~‘
186
—
—
‘.‘-‘
—
~.
—
>
tion for the real vapour ~ s~ = (q~ + a) / T is taken as a basis and, since s5=s~—(~+a)/T, the limit lim~.os~ = 0 is readily reached: limss=!~_~~~=0.
(17)
In particular, because of eqs. (1) and (3), it also holds that Iim~,0din(p/T)/dlnT= (ao/k)lim~.0 T It is thus not possible to justify introducing a prefactorotherthan 1 infrontofthesenestermlnT in the vapour pressure formula. Three further examples discussed below may show that a1 = 1, obtamed tion andfrom fromthe theCarnot—Clausius—Clapeyron temperature behaviour of theequareal gas in the vicinity of absolute zero, excludes all other possibilities a ~# 1 coming from the said model calculation approximation or purely numerical fitting: —‘.
*i-v—3
all a ~ values are fit parameters and, in particular, it holds that a~>1 [1,2]. One then obtains result 2 +...]. Nearthe absolute azero = k [it is a~ + (a ~ 1) T+ a T the linear term (ar 1 )kT here that determines the temperature behaviour of a, which is physically not correct. The vapour pressure formula —
and a0 = kT in V., are valid at absolute zero, which may very well be the case. For the pressure it follows in the limit T—~0that p/ T= k/V., = (kivq) exp ( e0/kT) or lim~,0dln(p/T)/dln T=~+(eO/k)limT~.O T ‘=limT.O(eO/k) lim~.0T It has to be noted, however, that the ideal gas model may only be used in the high-temperature range, where the effect of a parameter a~ 1 can of course be numerically ironed out by means of the other fit parameters. Here, on the other hand, without making any assumptions at all, the thermodynamic entropy rela-
Volume 149, number 4
PHYSICS LETTERS A
(1) According to the Thomsen—Berthelot principie [3] it follows directly that a=a mwith 0+a~T rn> 1 for T—+0. (2) The low-temperature expansion of the chem-
24 September 1990
y=u~T/T~+u~[l—(T/T~)2+...]<0, dp/dT=y~iT~2~T/T~+...
ical potential [4], T— TJ
—
d(p/T) dT,
T
dT
follows from the expansion of the integrand, i.e. d(p/T)
=
dT kT+ T2 ...
=
z.~pdln(p/T T d in T 2 + ... a~ +kT+... a2 T =
+d 2 ~ 2+.... Ifone had a
temperatures above 3 K.
and reads u= —a0—a1T+a2T 1 1, this would add the temperature-dependent term (a1 1) Tin T, which would then prevent the ther2p/dT2<0 modynamic conditions dp/dT< 0 and d from being satisfied. (3) Similarly, it can be shown that the temperature derivative of the surface tension, dy/dT, is negative and vanishes for T-+0 only ifa=a 2+.... 0+a2T The case a, ~ 1 would yield dy/dT> 0 in the vicinity of absolute zero and a logarithmic singularity at T= 0 because [4] —
$
T~ ~=
T
a
2+... 0 + (a1 —1 T)T+a2 T
(Vb
g v
2 3
—
5) =
dT
/
-4~{ao+[a 2Tc—eo/Tc+ai Vs~~ 2—
—
l)ln
TC]T
(a
—a2T
1
—
l)Tln T}
.
The values of a2, m limT.O y, p~ lim~.op and the temperature behaviour of a, y and p can now be given: From dy/d T10 =0 it follows that a2 = a0/ T hence (neglecting the contribution of the zero-point energy on p) 2+...]>0, a=e0[1 +2a0 (T/T~) T/T~+ ... ? 0, da/dT= 3>0, Y Yo [1 (Ti T~)2+...] >0, Yo =g0a0 /v~ dy/dT= —2y 0T/T~+...~o, ~,
—
=
_ao <0,
3. Re-determination of the fit function for the vapour pressure Using the vapour pressure values, which according to refs. [1] and [2] are given in the temperature range 1 to 3 K by formula (15) and at the critical point by p~=O.i164±0.0002 J/cm3 and T~= 3.322 ±0.002 K, we shall now determine a fit function for the vapour pressure in the entire temperature range ofliquid 3He using eq. (1) and taking into account relations (3), (5) and (13). On the other hand, it may be noted that critical values, which are different to those of ref. [11, have been published as 3 and well, for example p,~=O.ll46±O.OOO2 J/cm T~=3.309±0.00l K according to ref. [5]. If the function ymln[dln T/dln(p/T)] is represented versus ln T, then because of eq. (5) it is a strictly monotonically increasing function with a horizontal tangent at ln T~.The values of y known from ref. [2] and represented in fig. 1 as a dashed curve can be extrapolated from the temperature range below 3 K to give the solid curve with the ordinate value limT,T~y=yCat in T~=x~. At very low temperatures the temperature dependence of—a the vapour pressure is given by dln(p/T)/dln T= 2/T. As the value of a 2 according to eq. (13) is negative, it decreasing values ln T the function y becomesand linear, holds that din T/din(p/T)=T/(—a2)>0, for in T—ln(—a2). Again, the values of y are extrapolated from the temperature range above 1 K, yielding —ln ~ for the ordinate value —ln(—a2) in accord187
Volume 149, number 4
PHYSICS LETTERS A I
I
I
24 September 1990 In T
I
r dinT
--
lnp=ln T+lnp~—lnT~+
‘He 0
T~=3.3190
j
(19)
e~I1T~
InTc
P
Yc=
90.1150
/
F-P OH
/
-o
0.86539
7
In fig. 2 the vapour pressure and the first three temperature derivatives obtained according to eqs. (15) and (18) are represented versus the temperature as, respectively, dotted and solid lines. The cal-
I
I-
‘1
/‘
l’
>.~
4
2
0
______________________________ ‘0.0
0.2
0.4
0.6
x
=
ri
0.8
1.0
1.2
rameters a~ from ref. [2] and, in the other, with the boundary values T~=3.3l90 culations were performed, in oneK, case,x~=l.l99663, with the 76)= pa0.114988 J/cm3. y=l.l48762 y~=—O.86S39, The main differences and p~~p~( between the two calculations are the deviations in d2p/dT2 and d3p/dT3. In fig. 3 all differences are shown as relative deviations 2(p*_p) and 2(drnp*/dTrn_drnpi dTrn)/(drnp*idTrn+drnp/dTrn) (rn=1, 2, 3), the values marked with an asterisk being taken from the
T
Fig. 1. Function y=ln [dIn T/d ln(p/T)] versus x=ln Tfor 3He according to eq. (15) using the parameters a~ (EPT-76 ‘He scale) (dashed line) and eq. (18) (solid line).
ance with refs. [1] and [2]. At very low temperatures the vapour pressure is thus described by d ln (Ps” T) /d in T=~T For the evaporation energy at absolute zero it then follows that a 0= —a2k=~k(a2 or ~ in units of K); this corresponds to the value 20.78 J/mol, which is between the values 20.72 J/mol of ref. [1] and 20.86 J/mol of ref. [2]. The function y=y(x) with x~ln T represented in fig. I by the solid line is now given in analyticalform. It is assumed here that y=x—ln( —a2), and the solution for the region around the critical point, y(x~)=y~with dy/dx1~=0. The interpolation formula then reads —~.
calculation according to formula (15). It is evident that formula (15) yields values in the temperature range below 0.2 K which are too high. According to refs. [1] and [2], of course, formula (15) is only applicable in the temperature range above 0.2 K. It is assumed that the vapour pressure values p~$~ of ref. [2] are well known in the temperature range from 1 to 2.8 K. Hence a criterion for determining the
I
I
I
I
‘He
/
y=x—ln(—a2)—yexp[(x—x~)/y] xmln T,
The temperature derivative, dy/dx= 1 —exp[ (x— x~)iy], decreases monotonically with increasing temperature from 1 at absolute zero to 0 at the critical point.
188
/
.
ad
. °
°
m=3
2
1
.
...
0
.
1
This fit function y=y(ln T) allows one to determine the vapour pressure in thetoentire temperature 3He. According eq. (7) it is given range of liquid by
/
.
,
y~x~—y~—ln(—a2), —a2=~. (18)
/
..-.
/
I_0
2°
2 temperature
3
K
Fig.‘He 2. Vapor pressure and(18) the first three derivatives for calculated frompeq. (solid lines) and fromd”p/dT’” eq. (15) (dotted linea).
Volume 149, number 4 ‘1
. I
PHYSICS LETTERS A I
I
I
I
.
.
No
/
d
m=0
F
/
—
0
1
The task of finding a fit function for the vapour
I
‘He
----
/
2 -ON
24 September 1990
pressure of ‘He is accomplishedwith eqs. (18) and (19). This can also be done, however, by representing eq. (19) as a fit function on the lines of eq. (6), lnpmlnT—~T’+a,+ 9=4 ~ a,,T~3,
II
I
(21)
where a
to
3 is fixed by relation (12) and a4 is equal to a2/ T~.In this case one obtains the parameter values a5, ..., a~from a least-squares-fit calculation of 2 exp[y(TiT~)”~’]—1— (T/T~) n—4 =~ ~ (v+1)a,,~ 2. (22) 4T~’ 9=1 —
‘I
—--H.---
0
a .2 0
/
o ‘0
“
,
___________________________________________ 1
2 temperature
K
Fig. 3. Relative deviations 2(p*_p)/(p*+p) ~ 3) calculated from eqs. (18) and (15).
(rn=0) and (rn= 1,2,
As a consequence of the ansatz (8) for the ratio a/ ~thepower T~~2 appears in the series (22), which does not allow the left-hand side of eq. (22) to be fitted well. The fit parameters a,, calculated are therefore not presented here. The vapour pressure of 4He is used to define the 4He scale (up to 5.2 K) [6]. For temperatures above the triple-point temperature of H 2 (13.81 K) the International Committee on Weights and Measures have recommended that the internationally 1958
boundary values
y~and
x~may be deduced. This
means that Ye and x~ = in T~can be calculated by minimizing the value of the integral 2.8K
C 1K
I~ P (Ye, x~)I dT pp~,
x~)
(see fig. 3), leading to the numerical values just stated. The selected critical values can be tested to be flumerically correct if the corresponding relations at the critical point are sufficiently satisfied. These 2 ln p/ (din T) ~, rela= 0, tionsread follow from eq. (5), i.e. d and T2
T~ :p dT2 dT dlnpfdlnp \ dlnp = din T~,,dlnT 1). dinT1, -~--~:
for T= T~.
agreed practical temperature scale, IPTS-68, be used [7] in conjunction with the provisional 0.5 to 30 K temperature scale, EPT-76 [2]. A new IPTS-90 is under preparation. Candidates for defining a ternperature scale from 5.2 to 13.81 K are the gas thermometer [8] and the platinum and germanium resistance thermometers. At temperatures below the useful the ‘He vapour thermometer, i.e.range in theofmillikelvin range,pressure temperature scales are proposed which are based on measurements of the specific heat of liquid 3He at melting density [9] and of the magnetization of metallic samples [10].
Acknowledgement (20)
With p,, = 0.1150 J/cm3, T~ = 3.319 K and d in p/ d in T 1~—1 + exp(0.86539) (see fig. 1) one has3 dp/dT1~ = (p~/T~) d in p/d ln T1~=0.1169 J/K cm and d2p/dT~~ = (dp/dT 1~)[dlnp/dln T1~—1 ]/T~= 2 cm3.
The author is grateful to the referees for their valuable criticisms, which helped to clarify the text and were responsible for the insertion of examples 1 to 3.
0.0837 J/K 189
Volume 149, number 4
PHYSICS LETTERS A
References [l]T.R. Roberts, R.H. Sherman, S.G. Sydoriak and F.G. 3He scale of temperatures, in: Brickwedde, The 1962 Progress in low temperature physics, Vol. IV, ed. C.J. Gorier (North-Holland, Amsterdam, 1964) pp. 480—5 14. [2] M. Durieux and R.L. Rusby, Metrologia 19 (1983) 67. [3] H.B. Callen, Thermodynamics (Wiley, New York, 1960) ch. 10.3. [4] A. Eisner, Phys. Lett. A 130 (1988) 225. [5] R.P. Behringer, T. Doiron and H. Meyer, J. Low Temp. Phys. 24(1976)315.
190
24 September 1990
[6] H. Van Dijk and M. Durieux, The temperature scale in the liquid helium region, in: Progress in low temperature physics, Vol. lIed. C.J. Gorier (North-Holland, Amsterdam, 1957) pp.431—462. [7] M. Durieux, The international practical temperature scale of 1968, in: Progress in low temperature physics, Vol. VI, ed. C.J. Gorier (North-Holland, Amsterdam, 1970) pp. 405—425. [8] D.N. Astrov, L.B. Beliansky, Y.A. Dedikov, S.P. Polunin and A.A. Zakharov, Metrologia 26 (1989) 151. [91D.S. Greywall, 18th mt. Conf. on Low temperature physics, LT-18 (1987) pp. 2082—2083. [l0]G.Eska,Lowlemp.Phys.73(1988)207.