Latent Heats of Vaporization H. P. MEISSNER If the critical pressure and critical temperature of a pure liquid are known, its molal latent heat of vaporization at any temperature can be computed from vapor pressure data alone, by the follow-ing equation :
This expression is derived from the Clapeyron equation, and integrates as follows upon assuming that vapor pressure data plot as straight lines on the coordinates (In p o ) against (l/To) :
A graphical solution of this equation is presented, from which values of ( A H / T , ) can be obtained directly, given the reduced vapor pressure and the reduced temperature for the point in question. The maximum error is of the order of * 9 per cent, for both polar and nonpolar liquids, with an average error of less than * 5 per cent.
HE relation between the pressure and temperature of a two-phase system in equilibrium is expressed by the well-known Clapeyron equation ( 2 ) which is usually written as follows:
T
When applied to a pure liquid in equilibrium with its vapor, po is the vapor pressure a t temperature TO,AH is the molal latent heat of vaporization, vg is the molal volume of the saturated vapor, and ut is the molal volume of the saturated liquid. This equation is exact but is difficult to use in computing AH for two reasons. First, data on u, and V L have often been lacking in the past. Secondly, the vapor pressure curve as plotted on the coordinates pa against To shows marked curvature, making the slope d p o / d l h rather awkward t o obtain. The object of this paper is t o present Equation 1 in an integrated form which avoids these difficulties but which in large measure retains the accuracy of the original expression. Clausius (3) simplified Equation 1 by assuming that the molal volume of the liquid is negligible compared with that of the vapor, and that the perfect gas laws apply t o the vapor phase. It then follows that Equation 1 may be rewritten as follows:
Massachusetts Institute of Technology, Cambridge, Mass.
Or rearranging,
Assuming AH t o be constant, this equation can be integrated to give: (3'4)
Unfortunately the above assumptions on which Equations 3 and 3A are based are justifiable only a t comparatively low temperatures and pressures, far removed from the critical. At higher temperatures and pressures, where the molal volume of the liquid phase becomes increasingly great compared to that of the vapor phase, and the deviations of the vapor from the perfect gas laws become serious, these equations fail completely. Many other methods have been proposed for computing latent heats of vaporization. They are all empirical in nature. The most familiar is Trouton's rule (11) which states that the molal entropy change of vaporization for a liquid a t its normal boiling point is 21 entropy units. It has long been recognized that this rule is inexact. Among others, Nernst (10) and Bingham (1) proposed equations modifying Trouton's rule, which are of improved accuracy. However, these equations are not applicable to all substances and have the disadvantage of applying only a t the normal boiling point. Hildebrand (5) proposed a relation which is not limited to heats of vaporization a t one atmosphere. He claimed that the molal entropy change of a liquid being vaporized is a unique function of the molal vapor concentration. Lewis and Weber ( 7 ) and McAdams and Morrell (8) published charts based upon the Hildebrand relation, showing molal entropy changes plotted as ordinates against (1000 P / T ) , a factor which is proportional t o the vapor concentration when the perfect gas laws apply. Inspection of the charts shows that, while chemically similar substances group close together t o form a single line, the spread between lines of dissimilar substances is considerable. Hence data on chemically similar substances must be available before these methods may be applied t o any given liquid. Moreover, it has been found that all forms of the Hildebrand correlation break down a t high temperatures and pressures. Heats of vaporization over a wide temperature range can be better computed by the Watson correlation (lb). This involves use of the chart presented in Figure I, where relative values of ( A H / T ) as ordinates are 'plotted against values of T,, the reduced temperature, as abscissas. The following equation relates the heats of vaporization a t the two temperatures, TI and T2, ( A H / T J = (W/YZ) ( A H / T a )
where y1 and y2 are the ordinates corresponding to T,, and T,, on Figure 1. This chart applies to both polar and non1440
November, 1941
INDUSTRIAL AND ENGINEERING CHEMISTRY
polar substances with a claimed accuracy of 5 per cent except in the immediate vicinity of the critical. However, it cannot be applied until the heat of vaporization is known for one temperature. If an experimental value is not available, Watson recommends using the heat of vaporization at the normal boiling point as estimated by Kistiakowsky’s equation (6) : ( AH/T.) = 8.75 4.571 log T .
+
1441
T,,may be obtained from Figure 4, which was prepared dican also rectly from Figures 2 and 3. The term d(lnp,)/d( be readily evaluated, since vapor pressure data for a pure substance forms a practically straight line over wide ranges of
k)
1.0
0.9
This equation is surprisingly accurate but applies only to nonpolar liquids a t their normal boiling point.
0.8
Derivation of Proposed Equation
0.7
Equation 1 can be converted into a form more easily dealt with by utilizing t h e following generalized equations of state for vapors and for saturated liquids (4,Q). (Consistent units must be used when applying the equations presented in this paper.) POV, figRTo (4A) POVL fiLRTo (4B) where both p~ and pg are unique functions of T,,, the reduced temperature, and pTo,the reduced vapor pressure. Figure 2 shows the relation of pLsto p , and T,; Figure 3 gives the relation of p~ to p,, and T,. ‘Applying Equations 4A and 4B to vapor and liquid in equilibrium, it follows that RT (va - VL) = (fi, - P L ) O(5) Po Substituting the value of (vg - UL) given in Equation 5 into Equation 1 and rearranging,
Ob 0.5
0.4 a30
0.1
0.2
0.4
0.3
0.5
0.6
0.7
a8
0.9
10
PR
FIGURE2.
fi
CHARTFOR VAPORS
temperature when plotted on the coordinates (In PO) against TO), and therefore can be represented by the well known equation: In po =
-;o+B
(7)
where A and B are constants for any given substance but differ from substance to substance. The term d(ln p ~ ) / d represents the slope of this vapor pressure line, as shown by differentiating Equation 7:
(&)
3b
32
It is plain, therefore, that AH can be readily computed by use
28
of Equation 6, provided vapor pressure data for the substance in question are available, and provided the critical pressure and critical temperature are known. Further light is thrown on Equation 6 by subtracting Equation 8 from it:
24 h
2P
w 20
rG d
Ib
I2
0.4
a5
0.6
as
0.7
a9
1.0
FIGURE1. WATSONCHART, SHOWING OF VARELATIVEMOLALENTROPIES PORIZATION us. REDUCEDTEMPERATURE
Equation 6 is the equivalent of Equation 1, and is exact to the extent that the values of po and p L read from Figures 2 and 3 are exact. Equation 6 is, moreover, far simpler to use in computing AH than Equation 1, since its terms can be easily evaluated from vapor pressure data alone. For example, the value of the term (pg pL) a t any given p,, and
-
Evidently the term AH/R(po - p ~ in) Equation 6 remains constant for any given substance when its vapor pressure data obey Equation 7. Hence (pp - p ~ must ) vary in such a way as to offset variations in AH, which equals zero at the critical and increases with decreasing temperature. Examination of Figure 3 shows that (pa - p ~ is) likewise zero at the critical, increases with decreasing temperature, and approaches unity as a limit at low temperatures. [At values of T,,below 0.6, the term (pg - pL) is never less than 0.96. I n this region (pg - pL) may therefore be considered equal to unity, without introducing a serious error into the calculations. When (pg - pL) reaches unity, Equation 6 becomes identical with Equation 3.1 It is now evident that Equation 6 can be integrated if the assumption is made that the vapor pressure data for pure substances conform rigidly to Equation 7. Substituting the critical pressure and temperature into Equation 7 : (10)
INDUSTRIAL A N D ENGINEERING CHEMISTRY
1442
pro
FIGURE 3.
/r
CHARTFOR SATURATED LIQUIDS
Subtracting Equation 10 from Equation 7 and combining the result with Equation 9: AH
(11)
Vol. 33, No. 11
Equation 11is an integrated form of Equation 1. Inspection of this equation shows ( AH/Te) to be a unique function of p,, and T.,. This relation is shown graphically in Figure 5, which was constructed from Equation 11 and Figure 4. Figure 5 makes it possible to determine the molal latent heat of vaporization directly for any substance, provided the reduced vapor pressure and the corresponding reduced temperature for the point in question are known. (If the vapor pressure a t the temperature in question is not known, it can be estimated from Equation 8 after evaluating the constants in this equation. These constants can be calculated if the temperature and pressure are known for two points on the vapor pressure curve. The points used should lie as near as possible to the temperature in question, though if no other data are available, the normal boiling point and the critical point may often be used without serious error.) To check the accuracy of Figure 5, values of ( AH/T,) for various substances computed from experimental data (taken at random largely from the International Critical Tables) are listed in Table I and compared with values of (AHIT,) read from Figure 5. Inspection shows that the maximum error is of the order of *9 per cent over the entire range, for both polar and nonpolar liquids, while the average error is less than 5 per cent. This degree of accuracy is to be expected from the two assumptions made in deriving Equation 11 from Equation 1. The first assumption-namely, that vapor pressure data are linear when Dlotted on the coordinates In PO against l/To-has loni been recognized as being nearly correct over a range of reduced temperature from 1.0 down to about 0.45. Below 0.45, however, this linear relation no longer holds for many liquids, and so Equation 11 no longer
TABLE I. COMPARISON OF ( A H I T , ) DATA TO, C.
60
0.647 0.0213 0.8 0.1876 0.988 0.90 0.841 0.293 0.628 0.0209 0.60 0.0138 0.783 0,173
14.25 10.45 3.8 9.51 13.1 12.9 10.4
116.8
1.0
0.696 0.0207
17.5
Acetone Ammonia Benzene
56.1 52 280 200 80.2
n-Butane
-17.8
Carbon.tetrachloride Chlorobenaene m-Cresol Cyolohexane Dichlorodifluoromethane Diethylamine Ethane
Ethyl alcohol Ethyl ether
Tro
Atm. 1.0 20.9 43.2 7.25 1.0 0.5 6.3
O
n-Butyl alcohol Carbon dioxide
9,
-( AH/To)*Calcd. Obtained from from exptl. Figure 5 data4
pi,
13.7 10.6 4.2 9.3 12.8 12.4 10.3 18.7
30 0
71.1 34.4
0.996 0.976 0.898 0.471
1.65 7.95
1.4 8.0
280 200 80
43.1 14.4 1.1
0.995 0.959 0.851 0.320 0.635 0.0245
2.9 9.09 12.8
2.9 9.4 11.8
130.6 202 81.6
1.0 1.0 1.0
0.639 0.0224 0.674 0.0222 0.64 0.0247
13.8 15.5 13.i
13.0 15.6 12.7
49
11.7 0.635 1.0 23.56 7.67 59.9 19.44 1.069 1.0
0.838 0.295 0.606 0.016 0,0276 0.66 0.895 0.452 0.764 0.157 0,994 0.95 0.878 0.308 0.704 0.0168
9.3 12.6 13.4 7.4 9.6 3.42 12.17 18.1 13.3
8.8 12.6 13.4 7.6 10.0 3.2 12.8 19.0 13.4
- 40
55.0
0 40 240
-
180 80
34.6
0.66
0.0282
7
To,
Helium Hydrogen chloride Methane Methyl alcohol Methyl aniline Methyl chloride Methyl formate Nitrogen Nitrogen tetroxide Nitrous oxide n-Octane Oxygen Pyridine Tin tetrachloride Toluene Water
C. -268.6 85 --159
AH/Te)-
Tr, pro 0.867 0.442
Calc6. from exptl. data" 4.6
1.0 1.22
0.58 0.0123 0.598 0.0268
12.0 11.3
11.7 10.3
PO
At&. 1.0
Obtained from Figure 5 4.8
1.0
0.656
0.0127
16.4
16.3
194
1.0
0.665 0.0195
14.6
15.1
- 29 38
0.8
12.00 10.5
12.0
8.1
0.585 0.0122 0.748 0.123
31.3 -195.75
1.0 1.0
0.625 0.0189 0.617 0.0298
13.8 10.7
13.4 11.2
21.3 30 - 20 125 -182.9 115.0
1.0
62.6 18.1 1.0 1.0 1.0
0.684 0.0101 0.977 0.873 0.816 0,252 0.0406 0.7 0.584 0.0201 0,627 0.0167
19.1 3.55 9.37 14.2 10.6 13.8
19.7 3.5
64.7
13.4 1.0 0.65 0.027 13.4 1.0 0.645 0.024 3.95 0.987 0.904 197.0 7.78 0.927 0.556 99.2 14.6 1.96 0.607 0.009 Units are €3. t. u,/(lb. mole) ( O R.)or gram cal./(gram mo!e) ( O a Data obtained ohiefly from Volume V of the Internatlonal Tables.
*
112
109.6
366.0 327.0 120
10.6
8.9
14.2 10.4 14.2 13.0 13.2 3.9 8.6 14.2
K).
Critioal
November, 1941
1443
INDUSTRIAL AND ENGINEERING CHEMISTRY
not so good for polar liquids, as the original
papers point out (4, 9). The term (,ug - ELL) can, however, be considered a unique function of ppoand even !Profor polar liquids, since errors in pg tend to be offset by compensating errors in p ~ .This is shown to be the case by the high degree of accuracy with which AH can be computed for polar liquids from Equation 11, as Table I illustrates. Nomenclature AH = molal heat of vaporization P = pressure
pa po
P T
To TO
2 PO
PL
v, VL
= critical pressure = va or pressure at temperature
= refuced vapor pressure, polpa
TO
= gas constant = temperature
critical temperature saturation temperature at pressure PO reduced saturation temperature, T(Ta temperature at normal boilingpoint, K. POV,/RTO POVL/RTO molal volume of saturated vapor molal volume of saturated liquid
= =
= = = = = =
Literature Cited (1) Bingham, J . Am. Chem. SOC.,28,723 (1906). 0
0.1
0.3
0.2
0.4
0.5
0.6
0.7
0.8
0.9
(2) Clapeyron, J. 6coZe poZ&ch., 14, 143 (1834). (3) Clausius, Ann. P h w i k . , 81, 168 (1860). (4) Cope, Lewis, and Weber, IND.ENQ.CHEM., 23, 887 (1931). (5) Hildebrand,J . Am. Chem. SOC.,37,970(1915). (6) Kistiakowsky, 2.physik. Chenc., 107,65(1923). (7) Lewis and Weber, J. IND.ENQ.Cnma., 14,485
1.0
Pro
FXGURE 4. CHART OF
(po
- PL) us. REDUCED TEMPERATURE AND REDUCED PRESSURE
(1922).
applies in this region. Equation 6, however, is subject to no such limitations and may be used at any temperature. The second assumption, that PO and ELLare Unique functions of pro and T,,,is entirely justifiable for nonpolar liquids but
(8) MdAdEms and Morrell, ZW., 16,376 (1924). (9) Meissner and Paddison, Ibid., 33, 1189 (1941). (10) Nernst, “Theoretical Chemistry”, tr. of 6th German ed., London, Macmillan Co.,1914. (11) Trouton, p h a . ~ a g . [5] , 18, 64 (1884). (12) Watson, IND. ENQ.CHBM.,23, 361 (1931).
20
16
16
14
12
AH TC
IO
8
6
4
2
0 0
.02
.04
.06
.06
.IO
JD
.30
.20
.40
.50
.BO
,70
.80
.90
1.0
Pro TEE MOLAL LATENTHEATOF VAPORIZATION OF A PUREI LrQum, FIQURE5. CHARTFOR PREDICTINQ GIVENTEE REDUCED VAPOR PRESSURE AND REDUCED TEMPERATURE AT TIIE POINT IN QUESTION The units of ( A H / T a ) &reB. t. u./Qb. mole)(O R.) or (gram oal.)/(grarn mole)(O KJ.
