Putting Clapeyron's Equation into Practical Use with NO Help from Clausius Reflections on the Nature of Scientific Evidence S. Waldenstrgrn, K. Stegavik, and K. Razi Naqvi Institute of Physics, University of Trondheim (NLHT), N-7000 Trondheim, Norway The traditional belief that science can verify its theories encourages the view that a model may be regarded as hasically sound if its predictions are borne out hy experiments. This pragmatic view is accepted so widely that most students of science-and indeed many practicing scientists themselves, until they r e f l e c t t a k e it as a logical truth and the very hasis of the scientific method. I t is, of course, neither. The logical fallacy is seen easily: If A implies B, where A is the set of assumptions on which aparticular model is hased and B is the predicted outcome of an experiment, then a verification of B does not necessarily imply the truth of A ( I ) . What is not readily perceived, and if seen not always fully appreciated, is that the scientific method derives its strength from the contrapositive assertion "if B is false, then A must also be false," i.e., if the experimental observations do not accord with its predictions, the model must he discarded. This is, in fact, the essence of Popper's teaching (2): science does not prove anything at all; rather it disproves a great deal. An example illustrating these pitfalls can he found in the treatment of the liquid-gas equilibrium in terms of the Clausius-Clapeyron (CC) equation presented inmost standard textbooks of physics and physical chemistry: Starting from
the Clapeyron equation (the validity of which we do not dispute here), dP -=-
AH
(1)
d~ TAV' where P is the saturation pressure a t temperature T, AV = V, - VIthe difference in the specific volumes of the gas and liquid phases, and AH = HE - Hi the heat of vaporization, one makes the assumptions (Ad A V = Vs(VE >> VI), (As) PV, = RT
(Aa) AH independent of temperature and obtains by integration of eqn. (1)the CC-relation Ml h P = ---+constant
(2a)
R T
The differential form
AH
1
(d
d ( h P ) = --d (2b) R obtained from eqn. (1)by invoking only assumptions A1 and Az, is frequently given as an intermediate step.
..
Clapeyron.
30
Clausius.
Journal of Chemical Education
.
..
Daniels and Alherty (3) arrive at e m . (2a) and go on to state: "This suggests that aplot of log P veisus 1/T should he linear, and this is borne out by data on both vauorization and suhlimation, as shown. . . .~0vera wide temperature range there are significant deviations from linearity because some of the approximations made in the above derivation do not apply." In the first part they deal with the proposition A implies B and the experimental verification of the prediction B (linearity); the last part of their statement corresponds to not A implies not B, which is logically equivalent to B implies A-an unwarranted deduction, as emphasized earlier. Other authors follow a similar line of reasoning in their comments on the CC-relation; more often than not, the reader is led to conclude, directly or imulicitlv, that linearitv can he achieved if and onlv if the s&ndard assumptions hold. w i t h this in mind, confusion arises when d o t s of In P aeainst 1IT based on good e x ~ e r i mental data are examined: 71) the experimentalboints lie, to a first approximation, on a straight line coverina the entire
T e m p e r a t u r e t i°C 37d 15 300
I
200
1
100
0
'
earity can barely he seen. Abbott and van Ness (4) remark, after giving eqn. (2h): "More specifically, eqn. (2b) indicates that the value of AH is given by the slope of In P versus 1/T. It is a fact, however, that such plots made of experimental data
A~~
~
----------
approximatevalidity'only a t low pressures, at pressures well below the critical pressure for any pure substance considered." Brown (5) shares their views: "It is usually pointed out that the assum~tionslimit the use of this eauation to low Dressures. It th1.n 11cwmr imyus:ihle ta, e x l h n the rxprrinlent31 iact t l ~ tla t I I ~ ,dI i n I' rtrsus 1 7' is frc.~nir~lrlv very" ~>c..arlv linrar up to the critical point, even though'all three of the assumptions made are evidently no longer valid." The confusion and the problems evidently stem from the expectation that lin-
-
T e m p e r a t u r e t i°C 200
r
150
100
I
I 1.0
1.2
1.4
1.8
1.6
2.2
2.0
2.4
l/Tr Figure 1.The natural logarithm of reduced, saturated vapor pressure (P,) for wale< as a function of the reciprocal of T,. the reduced temperature. The data points (triangles) fall, to a first approximation, on the straight line In P, = h ( l - T;'). with h = 7.55.
Temperature t (15350
300
250
ioC
1 200
180
lb0
50
.
Fig-re 2 Tne aata d;spn)w n F glre 1 n a c Dean rop orloll here as n Pwrsus recnprocal ="so ble Iemperal.rr Tnc h ~ ran:a r x* s nos "em o cel no low pimr Ii 1100-200°C~ 5200-300°C~. \1 1300-374.15'C! aria n cum re5 on 3 F'ragnl ne, n P - A - 10 Ti. corrcrponulng lu m e lumpcrat.ro nlerrsls !0-I0O0C~. has been fined to the data. The four temperature intervals and the corresponding values of B have been indicated.
Volume 59
Number 1 January 1982
31
earity will hold only over a minute range, i.e., only i f the assumptions hold. The traditional approach of linking linearity to the standard assumptions is like putting the cart before the horse: facts are made to suhserve forecasts based on assumptions of limited validity. The main purpose of this paper is to show that, by abandoning the assumptions altogether, i.e., by combining experimental facts with the exact form of Clapeyron's equation, eqn. (11, one can arrive at far reaching conclusions and deduce an accurate vapor-pressure relation. In the next sec. tion, we accomplish our aim by taking water as a concrete examwle: in the last section, we consider the problem of estimating k from vapor-pressure data, and go on to discuss the implications of invoking the standard assumptions. In taking the last step, we follow Brown (5):"By postponing the three approximations to the last step, the student can obtain a clearer idea of how these approximations enter into the final result." Union of Experiment and Theory: A Simple but Accurate Vaoor-Pressure Relation Throughout this section we take water as the test-case, though anv other wure substance could serve inst as well to most commonly used in student laboratories, secondly, data are easily available. Our data are from the comprehensive compilation "Properties of Water and Steam in SI-Units" ( 6 ) ; for a few selected values of temperature, the relevant data from this work are reproduced in the table, together with the (derived) values of the dimensionless quantities $ and Z, (to be discussed later). For a restricted interval ( T , T AT) of arbitrary location, experimental (P,T)-data can he represented rather accurately by the relation
+
InP=A--
B
(3)
T where A and B are constants. Indeed, as a first approximation, a straight line can he fitted to the data over the entire liquid range, i.e., from the triple point, (Pt,Tt),to the critical point, (P,,T,); introducing the reduced variahlesp, =PIP, and T, = TIT,, eqn. (3) can he written
it is seen that if $ is known a s a function of T,, a relation of the form in P = f(T,) follows by integration. Given a semi-reduced (or reduced) vapor-pressure equation of this standard form, $ can he calculated and AHIPAV determined; combined with volumetric data, this may he utilized to find AH. I t becomes evident that the simple straight-line fit in eqn. (4) corresponds to
# = h (first approximation)
(74
and that, if eqn. (3) is fitted to (P,T)-data over an interval AT, the variation of$ with temperature can be estimated from the relation
Thus. il. mav be viewed as a warameter measuring the deviation from the straight line inAeqn.(41, and can in principle he determined from (P,T)-data, for a given value of T,, by numerical differentiation of eqn. (3) (AT very small). Referring to Figure 2 and the above comments to that figure, it follows that (for water): (a) $is a slowly varying function of T,, (b) the ($,T,)-curve is U-shaped, with a minimum around T, = 0.8.
That these two properties are shared by most compounds, as pointed out in Ref. (8) (pp. 201-2051, makes $ aparameter of general interest. [Further, specific examples of ($,T,)curves are given by Waldenstr6m and Sylevik (9);it is noteworthy that the curves pertaining to quantum liquids deviate from the normal shape, thus reflecting the special properties of these substances.] When values of AH and volumetric data are available, in addition to (P,T)-data, $ can he calculated from eqn. (6b); the Temperature t (OCI
where h (=BIT,) is a dimensionless constant. A plot illust~atingthe applicability of eqn. (4) to so-called "simple fluids" is given by Scott (7); Figure 1displays the corresponding plot for water. In Figure 2, we have plotted in P versus l l T ; the figure brines out two important features: (a) the simple " vapor-pressure relation in eqn. (3) holds over surprisingly laree intervals AT, (b) the magnitude of the slope, B, decreases first as the temperature rises, goes through a miniT,. mum, and increases again as T Clapeyron's equation, eqn. (I), can he written in the form
-
or, by introducing the reduced temperature T, and noting that d(ln P ) = d(lnP,),
xo
with AH PAV
#-T,-
The dimensionless quantity $-first considered by Reid, Prausnitz, and Sherwood (8)-is very convenient for discussing the characteristics of vapor-pressure curves. Writing eqn. (6a) as 32
Journal of Chemical Education
R e d u c e d t e m p e r a t u r e T, Figure 3. The parameter #, given in the table, as a function of reduced temperature. The intervals l-IV used in Figure 2 have also been indicated here; the average value of # in each interval is consistent with the corresponding value of B in Figure 2 [cf, eqn. (7b)l. The figure also shows the origin (yo.xo)andthe rotation angle (#)of the (displaced and rotated) coordinate system in which the #-curve may be approximated as a parabola of the form y = r x 2 .
Properties of Saturated Water and Steam t
T
P
'4
"g
('c)
(K)
(bar)
lm3/kg)
(m-'lkg)
0.00 0.01 50 100
273.15 273.16 323.15 373.15
150 200
423.15 473.15
6.108. 6.112. lo@ 0.12335
1.0002. l.000Z.10-3
1.0133
1.0437 I O F
4.760
1.0908. I O W 1.1565.
206.3
206.2
1.0121~10~~
.
15.549
12.05 1.673
392.4 127.2
values so determined must agree, by virtue of Clapeyron's equation, with those derived from (P,T)-data alone. ~brivater, &-valuesobtained from e m . (6b) are given in the table and are Shown graphically in ~ i ~ ; r 3.e I t is seen that the approximation $ = h = 7.55 (cf. Fig. I ) , which corresponds to the gross average value of $J,is rather crude. Vapor-pressure relations based on a simple, analyt~c representation of $. T o reproduce the U-shape, we assume, as a second approximation, that the ($,T,)-curve can he represented by a parabola of the form y = ax2(u > 0) in a displaced and rotated b,x)-coordinate system, as indicated in Figure 3. For the special case 4 = 0, for which y = ax2corresponds to $(T,) = y o a(T, - x#, one obtains directly by integration of eqn. (fic),
+
I ~ P=, L T ' z - ~ + ( z ) d z= (yo+ o x : - r ) - (YO
A semi-reduced equation of the form
+ mi) T,
- 2az0ln T,
+ oT,
AH (kJ1kg)
.lo@
.
T, = TIT,
iji = T,AHIPAV
2, = W , I R T
2501.6 2501.6 2382.9
0.422
2256.9
0.576
8.377 8.376 8.00 7.68
0.999 0.997 0.984
2113.2
0.654
7.42
1938.6
0.731
7.23
0.422 0.499
0.999
0.956 0.906
If the constants (QliQ2,Q3)and x + are determined by fitting eqn. (10) to (P,T,)-data, the rotation angle 4, the parameter a , and the coordinates (y0,xo) can he calculated as follows [inversion of eqn. (9b)l:
Application of eqn. (10) to water. Figure 3 indicates that the rotation angle 4 is between 5' and 6", and a between 13 and 14; with (yo,xo)= (7.13,0.85), d = 5.6", and a = 13.8, eqn. (9b) gives
(8a)
frequently discussed in the literature and conforming to the analytic expression given by Nernst in 1906 (101,may thus he interpreted as a ($,T,)-parabola without rotation:
By allowing for a positive rotation, 0 < 4 < r (cf. Fig. 3), it follows from the general coordinate relations y = -(T, X O ) sin $ + ill.- yo) cos 4
-
, \ + ~ ~ n i nthe g c,m,canr3 ~hc.1111nwric,11 v.iIues i n rqn. 11'21. rqn. 1931 reprud11~1.s lht. V I I ~ oU f~< ~ I W Ni n llw 13bIe wilnir 3.K;.e w w l fbr f h r \.it1110 i ~ T, t = ,I 939. . . \I here I hr (1, \.i>itim is of the order 2.5%. The larger deviation in the critical region 1) is not unreasonable: in the first lace, the &-curve (T, inFigure 3 becomes exceedingly steep, secondly, $, a ratio of two infinitesimals (cf. eqn. (fib), AH and AV both tend to zero I ) , is hard to determine experimentally. Using the as T, constants in eqn. (121, the integration constant Qo in eqn. (10a) can be evaluated by requiring, e.g., that the point (P,T,) = (P,,l) (PC= 221.2 bar) should fit the expression; this gives Qo 24.87. Without further refinements, the formula
-
-
-
x=(T,-ro)cosd,+(Ji-yo)sin$
that y
=
ax2 transforms to
W,)= QI - Q2T, - Q3 & F T where
* QI=Y~+G cos
Q z = cot 4,
Qa
[I
+ 2 ~ x sin 0 m]
= [csin3m]-112
x + = x o + q , q=--
(94
eas2 4
40 sin 4
(9b)
turns out to reproduce the tabulated vapor pressures for water (6) within 1%throughout the entire liquid range. Estimation of AH; The Standard Assumptions Recalling that AV = V, - Vi and introducing the compressibility factor Z, = PV,/RT, the definition of $ in eqn. (6h) yields the exact relation
(20)
The constraint (x+ - T,) 2 0 requires x+ 2 1;the choice of the sign in front of (x+ - T,)'I2 in eqn. (9a) is dictated by the very U-shape (the curvature). With $ given by eqn. (gal, integration of eqn. (6c) yields the equation (semi-reduced form)
where Qo denotes the integration constant and
With $ determined as explained in eqn. (7h), one arrives at the approximate relation ( 5 )
The standard assumptions A1 and A2 correspond simply to setting (1 - V,V,-') = 1and Z, = 1, respectively. The true variation of these factors along the saturation curve-as functions of temperature-are sketched in Figure 4a; the data used are those given in the table. I t is seen that the factor (1 - V,Vp-') stays close to unity even for high temperatures, but i t falls, on the other hand, very rapidly to zero as one apVolume 59
Number 1 January 1982
33
most conveniently done, of course, by representing the (P,T)-datafor the whole range by an accurate vapor-pressure relation. I t should finally be mentioned that even for very low pressures, where the assumptions A 1 and A2 may hold almost exactly and AH = RT,$ according to eqn. (14), AH can vary considerably with temperature, as it does for water. Conclusion The approach to vapor-pressure relations based on a ($,T,)-parabola can he taken a step further by extending the hvoothesis to include conic sections in general. This leads to .. a class of relations of the same form as that given in eqn. (lOa), but the function F enterine the last term will now d e ~ e n don
I
100
200
100
I
simplest one.-with suitably chosen conic sections, tabulated vapor pressures for various types of liquids (quantum, polar, and normal) have been reproduced with high accuracy (maximum deviation less than, or of the order, 0.5%). For water, this accuracy (max. dev. 0.1%) was attained with an ellipse; for neon (max. dev. 0.25%),with a parabola, i.e., by the use of eqn. (10) (9). That vaDor pressures for water can be re~roducedwithin
L
,7413
Temperature t (OCI
Figure 4. The temperature d e p e n d e n c e of (1 - v~v,'). Zg. and A H f o r water: (a)The d a s h e d curve refers to t h e volumetric factor ( I - V,
