RELATIONSHIPS BETWEEN MELTIIL’G POINTS, IL‘ORMAL BOILIKG POINTS AIL’D CRITICAL TEMPERATURES* BY R O B E R T TAFT AND JESSE STARECK
Xiany years ago Guldbergl and Guye2 simultaneously called attention to the fact, that there existed a simple approximate relationship between the critical temperature of a substance and its normal boiling point. This useful relationship, purely an empirical one, is quoted in nearly all text books of physical chemistry under the name of the Guldberg-Guye rule. Briefly it states that the absolute normal boiling point is two-thirds of the absolute critical temperature. Guldberg reached this conclusion after examining the data for j 5 carbon compounds, 7 inorganic compounds and two elements, ~ these 64 substances being 0.6 51. Guye based the average value3 of T B / Tfor his deduction after comparing the data for 75 substances, eleven of which were inorganic; of these eleven, four were elements. The average value of TB:’Tc obtained by Guye was 0 . 6 4 j . Later R. Lorenz4 examined the ratio TB/Tc more extensively and found the value of 0.64 as the average ratio for 137 substances. Van Laarj points out that the limiting values for the ratio T ~ / T B can be obtained from the vapor pressure equation, resulting in the expression T c - log Pk TB
+ I
f,
where Pk is the critical pressure and f, a constant, having an average value of 2.9. Since Pk lies between IOO atm. and 20 atm. for most substances, the value of Tc/Tn would therefore lie between 1.69 and 1.45. Van Laar examined fifty inorganic compounds and fifty-eight organic ones and found 1.60 as the average value of Tc/TB, corresponding to 0.63 for T B / T ~ . S. Young6 has discussed the Guldberg-Guye rule and its limitations. Fielding7 has proposed a relationship between absolute critical temperature and absolute boiling point as follows: T c = 1.714 TB 3.3’ which he applies only to elements and inorganic compounds. This relationship, however, has been adversely criticised by Moles.8 More recently Prud’hommeg has pointed out that for certain groups of ele* Presented before the Kansas Academy of Science, Hays, Kansas, April 18, 1930.
+
2. hysik. Chem. 5 374 (1890). SUE.,(3) 4, 262 (I&).
TB,T c , a n d TF are used throughout this paper for the absolute normal boiling point,
absolute critical temperature, and absolute melting point respectively. When centigrade temperatures are used, these are indicated by a small t with the suitable subscript. 2. anorg. Chem., 94, 248 (1916). 2. anory. Chem., 104, 92 (1918). “Stoichiometry,” 2nd. ed., p. 169 (1918). ’ Chem. News, 117, 379 (1918). J. Chim. phys., 17, 41j (1919). J. Chim. phys., 18, 270 (1920).
2308
ROBERT TAFT A S D JESSE STARECK
ments the ratios T c / T c - TB and Tc/Tc - T B Tc/TB . were characteristic “constants”; especially in the case of the second ratio was the constancy marked. For example, in the case of the helium family the deviations from a value of 4.06 are less than 114of one per cent for any one member. For fifteen organic substances of diverse character the maximum deviation from the mean, 4.20, was less than two per cent. It was but natural since there apparently existed a rough proportionality between boiling point and critical temperature that inquiries should be made as to whether similar proportionality did not exist between melting point and critical temperature. F. W. Clarke’O was the first to consider the ratio of TF/Tc and pointed out that its value lay between 0.33 and o . q j for some thirty substances. W. Herz” compared these two constants for several elements and for the chlorides and hydrides of the elements according to their position in the periodic classification but was unable to show any regularity even in the same group. As Herz states, however, the data for such substances is so meager that it is difficult’ to decide if any such proportionality exists. Lorenz arrived at an average ratio of TFI’Tc by taking the product of ~ he computed from a large number average values of TF/TBand T B / T which of substances. The average value of TB!Tc as determined by Lorenz was 0.64 as stated previously. That for TF/TB he found to be 0.68. Therefore: TF/Tc = TF,’TBX T B / T = ~ 0.68 X 0.64 = 0.44 Lorenz and Herzl* subsequently revised their value of TF/TBto 0.62 as discussed later. This would bringtheir averageratioof TF/’Tc to0.62 Xo.64 = 0.40. Observations on the ratio of the absolute boiling point and absolute melting point were apparently first made by H. Fritz13 who pointed out that TB/TFfor elements was equal to 1.2 for the halogens, 3 ’2 X 1.2 or 1.8 for a second group of elements and 3,’z x 1.8 or 2.7 for still a third group, although the grouping was not that dependent upon their position in the periodic table. hIott’l has also calculated TB,’TF for a large number of elements. The number of times that 1.2 occurs in Mott’s test is quite striking. In the list of 66 elements in Mott’s table, 3 3 ratios are expressible as multiples of 1.2 as suggested by Fritz. The most extensive examination of the average value of this ratio has been made by Lorenz and by Lorenz and Herz in the papers t o which reference has already been made. The average value of TF/TB = 0.62 was based on those of 33 elements, 1 2 2 inorganic compounds, and 2 j; organic substances. The minimum and maximum cases in this list range from 0 . 3 2 2 (Bi) to 1.04 (UFG). Considering organic substances alone, the values range from 0.36 (propane) to 1.01 (acetylene). ’“Am. Chem. J., 18, 618 (1896). 11 12
l3 l4
2. anorg. Chem., 94, I (1916). 2. anorg. Chem., 122, 51 (1922). Monatsheft, 13, 802 (1892). Trans. Am. Electrochem. SOC.,34, 287 (1918).
MELTING A N D BOILING POINTS ASD
CRITICAL TEMPERATURES
2309
Limiting values for the ratio TF/TBhave also been suggested by Longinescu,lj who deduced the relationship for organic substance as follows : As a result of previous work,16he (Longinescu) had shown that the normal boiling point of a liquid on the absolute scale was proportional to the product of the density of the liquid at zero Centigrade and the square root of the number of atoms per molecule. The value of the proportionality constant was computed to be on the average equal, for non-associated liquids,to 100. This relationship may more briefly be stated as ( I ) , TB = I O O dl 2/ n where TB represents the absolute boiling temperature (at 76 cm.), dl the density of the liquid at, o°C, and n the number of a t o m s i e r molecule. Similarly, he deduced the relat,ionship ( z ) , T F = 50 d, 2 / n where TF represents the absolute melting point, d, the density of the solid, and n the number of atoms per molecule for a non-associated substance. Dividing equation ( 2 ) by ( I ) gives TF,/TB= 1/2 . d,/dl. Longinescu further pointed out that the ratio of d,/d, for most organic substances lies between I and 1.2. Longinescu therefore states that the ratio TF/TBshould lie between 0.5 and 0.6. Simultaneous relationships bet’ween all three constants, melting points, boiling points and critical temperatures, which are more or less obvious from the Guldberg-Guye rule and from the observations of Fritz, were not suggested until 1920. In that year Prud’homme17 stated an empirical relationship to which he gave the name of “The Rule of Three Temperatures.” This in its simplest terms states: “The absolute critical temperature is equal to the sum of the absolute melting point and the absolute boiling point,” or TF+TB=Tc. Prud’homme was led to this conclusion by applying analogous considerations to the melting point and critical temperatures as he had applied to the boiling point and critical temperature discussed earlier in this paper, Le., by examination of the ratios Tc Tc Tc - . - Tc = IC1 and . - = Kz Tc - TB TB Tc - TF TF Dividing the first by the second gives: ~
The value of R lies, in general, between 1.1 and the above equation reduces to Tc = TB TF
I.
I n the case where R =
+
I,
+
In subsequent papers’* Prud’homme computed the value of the ratio TF -- TB Tc and found that it likewise approximated a constant not greatly different than Bull. Chim. pura applicata, SOC.Romano Stiinte, No. 46, j I - 2 (1923). J. Chim. phys., 1, 289 (1903). J. Chim. phys., 18, 307 (1920). ** J. Chim. phys. 18, 359 (1920); 19, 188 (1921); 21, 243 (1924).
‘5
*’
2310
ROBERT TAFT A S D J E S S E STARECK
R. Further, its value computed at other pressures than I atmosphere was sensibly constant for a given group at the same pressure but diminished in value as the pressure decreased. Prud’homme considered in all some 8 j substances and found that with
+
but thirty exceptions, T F TB was equal to or greater than unity, the Tc average value being 1.05, TVe have thought it worthwhile to compute and examine the values of the terms TF/TB, TB/Tc, T F / T ~and , TF TB from the data now available in the International Critical Tables. These data are presumably more reliable in general than that used by any one of the investigators previously mentioned and contain more extensive data on the critical temperatures than that used by these authorities. In general, me have considered only those substances for which values of TF, TB,and Tc are all available, except in the case of the ratio TF/’TB. The Ratio T F / T B ~
+
I t is apparent that the limits of this ratio will lie between zero and I . While in a few cases the melting point is higher than the boiling point, the number of times that this occurs, makes it quite ex~eptional.’~The cases of carbon dioxide and acetylene form the most familiar exceptions. The value of the above ratio for carbon dioxide is 1.1, that of acetylene is 1 . 0 1 ; in the case of hexachlorethane the ratio is exactly 1.0. Simply on the basis of the above limits, the average value of this ratio might be expected to lie in the neighborhood of 0.j. In the case of 1 3 9 ’ ~of the I ;;substances listed in I C T Yol. 3, pp. 248-249, the average value of this ratio proved to be 0.607. If the carbon compounds alone ( I I I in number) in this list are averaged, the value 0. j j 8 is obtained. The average value found by Lorenz and Here was 0 . 6 2 . The individual values fluctuate so considerably that it is our judgment that these average values are without significance and without usefulness. The question remains, however, as to whether or not any relationship exists between these two constants for more restricted groups of substances. K e have already called attention to the work of Fritz who showed that the above ratio was not a periodic property. There still remains the possibility that there does exist a relationship between these two constants in the case of a group of closely related compound, Le., in an homologous series. The data for such an examination is more extensive for the aliphatic hydrocarbons and alcohols than it is for other groups. Data for these two groups are shown in Figs. I , 2, and 3. I n Fig. I , the two curves comprising group 2 were obtained by plotting the absolute melting points (TF) as ordinate against the absolute boiling point TBas abscissa for the aliphatic hydrocarbons. The upper curve is that of the l 9 Whether or not it is logical to consider these cases, depends chiefly upon the definition of the normal boiling point. If this be defined as follows: “The normal boiling point is that temperature at which the vapor pressure of the liquid is equal to 76 cm. of mercury,” they should not be considered as they are not liquids when their vapor pressures are equivalent to 76 cm. Possibly it would be best to call the temperature a t which the vapor pressures of these solids are equal to 76 cm. of mercury, the norvial sublimation temperature. 20 Data on mp’s and bp’s for only 139 of these were available.
M E L T I S G AND BOILING POINTS AND CRITICAL T E X P E R A T U R E S
23 I I
even-membered compounds, the lower for the members containing odd numbers of carbon atoms. If the first points (Le. the members containing two and one carbon atoms, respectively) in each curve are neglected, the remaining points lie on smooth curves of slight curvature which tend to coincide as the temperature becomes greater. This is clear evidence that a relationship does exist between melting and boiling points for the odd (or even) members of an homologous series. That the relationship very likely is parabolic in character is evident from the curve.
360
0
K e have not seriously attempted to evaluate the constants of these curves. In the case of the even-numbered carbon atoms a relationship of the form,
TB = K
+ ATF + BT;
has been used. The values of K, -4,and B computed were - I , 2.25, and -0.00187 respectively. This, in general, gave agreement between observed and calculated values of TB to within jTc. The inclusion of higher terms (CTF3, etc.) would likely give TB from TF with considerable precision. We have drawn for reference in Figure I the straight line representing the ratio TF/’TR= 1/2. This is the line which passes thru the origin and is labeled “ 1 i 2 ” on the right hand vertical axis. A comparison of the two curves with this straight line shows their curvature more distinctly. The value of the ratio T F i ’ Tis~also plotted against the number of carbon atoms for the aliphatic hydrocarbons in Fig. 2 and for the primary alcohols in Fig. 3. In Fig. z the form of the function between this ratio and the number of carbons atoms is crudely that of an oscillatory wave of decreasing amplitude. I t is to be noted that both in the case of the hydrocarbons and of the alcohols that the ratio approaches a value of 0.5 with increasing molecular weight. I t is interesting to note that this is the value obtained by combining
ROBERT ?'AFT A S D JESSE STARCCK
2312
the Guldberg-Guye equation with that of the Prud'homme rule. That is from TB/Tc = 2/3 and TB TF = TCthe ratio TF/TB= 1,'2 is obtained by conibining these two equations and eliminating TCfrom them. .As we will point out later, the Guldberg-Guye relationship and Prud'homme's rule are more nearly universal than are any of the other relationships herein mentioned. ICT normal boiling point data for n-C23Ha8and n-Cs4Hjoare tabulated as 3 2 4 O C and 320.7'C. Using these values the ratio of melting to boiling point are in the neighborhood of 0. j4. We have been unable to locate the original data, but we are inclined to doubt the accuracy of the above figures. The following data bears upon this point.
+
No. of C atoms
BPT I j mm.
BP'C
31;~
18
181.j
20
2 0 j-
22
224.j
24
243+
76 em.
324+
s o . of
C atoms
BPT mm.
B76 c m . T 303+ 330L
Ij
77
170
I9
I93
21
21j-
23
234
320.7~
The starred values are those of the ICT, the remainder are taken from values tabulated by Cosale.?' I t will be noted that the boiling points at I j mm. show a considerable increase with increasing molecular weight, whereas the value at 76 cm. for C23H4Sis recorded as less than that for CI9H4o;that for Cn4Hsais also apparently low as there is only a seven degree increase from C&g to Cz4HSo. Farther, Casole (IC) has pointed out that the ratio TB:, TB,, shows but a gradual decrease from Cl5H32 to Cl9Ha, with no tendency of a sudden change in this neighborhood. The value of T B ~calculated ~ ~ ~ from . the relation TF/TB = 0.j is 368°C for C23H48 and 381°C for CzAHjo. As these are in the neighborhood of the cracking temperatures, it is doubtful if the values of T B ~have ~ any ~ significance ~ . as constants for these higher members. The data for other homologous series are not sufficiently extensive to warrant further discussion. The Ratio TF/Tc The average value of 146 substances o u t of the I 7 j (ICT, v.3, p. 248) for this ratio is 0.386. For 113 organic substances the average value is 0.36;. Individual variations are again considerable, ranging from .13 (Hg) to ,723 (SiF,), so that it is very doubtful if any value can be attached to the average values. Further, we are in doubt as to the possible limits of this ratio. The limits are less than one and more than zero. If the Guldberg-Guye rule is combined with that of Prud'homme, eliminating TBbetween them, the ratio TF/Tc = 1 / 3 is obtained. The average values above are considerably in excess of this value, however. Examination of the constants TF and TC as functions of each other in the case of aliphatic hydrocarbons does not yield extensive information due t o the lack of data. In Fig. I , the two curves of group I show TF (ordinate)
__
~ ~ G B I45 z .I,, 577 (1915).
MELTING AND BOILISG
POINTS AND CRITICAL TEMPERATURES
23 13
plotted against T, (abcissa) for the members possessing an even number of carbon atoms (upper) and for the odd members (lower). Omitting the first member, the remaining members of both groups fall on a smooth curve, indicating some functional relationship between the two terms, but not sufficient data are available to predict the form of the relationship. We have shown for oa
or
5
06
3
d,
0 05;
-6
t
e
04 03
3
f
z
02 0
I
2
3
4 5
6
1
8 9 IO
I1
12 I3
14 I5 16 17 I 8 I9
20 21
n u m b e r of carbon atoms FIG 2
0.8 07 0.6
5
3
3
F
TS
0.3 0
2e 0.4
L
03 5
TG
02. 0
I
2
3 4 5
6 7
8 8
IO
/I
I2
I3
1 4 IS IO 17
18 19 2 0 21
number o f carbon atom5 FIG. 3
comparison here, (Fig. I ) , the ratio T F / T ~ = 1/13 by drawing the straight line passing thru the origin and marked 1/3 on the right hand vertical axis. It will be seen that the curve for the even membered compounds is nearly coincident with this for the range given. In Figs. 2 and 3 the ratio T F , / Tis~ plotted against the number of carbon atoms for aiiphatic hydrocarbons and for primary alcohols. The data are not
2314
ROBERT TAFT AXD J E S S E STBRECK
sufficiently extensive to determine the limiting value of the ratio but it does appear that the fluctuation in the value of the ratio will diminish with increasing molecular weight. In connection with this ratio an observation of kourbatowzzis interesting. Kourbatow states that those temperatures which are the same fractions of the temperature of fusion are corresponding temperatures for substances in the solid state. He cites as evidence the facts that the electrical conductances (atomic) are approximately the same, and that the law of Dulong and Petit is exact at corresponding temperatures thus defined. It is obvious if two substances were compared at equal fractionsof their freezing points, they would also be at equal fractions of their critical temperatures (the usual definition of corresponding temperature) provided that the ratio of melting point to critical temperature were the same for both substances. The Ratio T B I T ~ This ratio has caused more comment than has any of the other quantities herein mentioned. The fluctuations of this ratio from the average value, is considerably less than for either of the preceding ratios, and as will become apparent, are of a different character than either TF T c or TF,'TB. The average value of 166 substances (ICT Y. 3 p. 248) is 0.653; for 134 carbon compounds it is 0.667. As we have mentioned, the variations from the general average are not as large as in the case of the first two ratios, yet they are of sufficient magnitude to show that the ratio of TB'Tc is not a constant, and that the Guldberg-Guye rule is but a rough, tho useful, approximation. That a real, altho not simple, relationship does exist between TBand Tc for a given homologous series is apparent from Fig. I , curve 3 . I n this case Tg is plotted as ordinate against Tc as abcissa for the aliphatic hydrocarbons. I t will be noted that the points, with one exception fall upon a smooth curve of slight curvature. Further, it will be noted that separation into odd and even numbered carbon atoms is not necessary to bring out this relationship. The form of this curve is not that of any of the common ones so that a solution for the equation of this curve has not been attempted. The fifth point lies considerably off from the curve. We believe that this indicates an error in the determination of the critical temperature of this substance (pentane), being approximately I O degrees too low. There is the possibility of the boiling point being too high, but due to the greater ease of determination of the boiling point than the critical temperature, reinforced by the fact that the corresponding point in the curve TB,TF falls smoothly on its curve, leads us to suspect the accuracy of the critical temperature measurement. That the relationship between TB and TC is of somewhat different character than that existing between TF and TC or TF and TBis shown also by Figs. z and 3 . I t will be seen that the ratio TB;TCin the case of both aliphatic hydrocarbons and primary alcohols (omitting the zero members, H z and 22
J. Chim. phys.,
6, 339 (1908(.
M E L T I S G b S D BOILING P O I S T S A S D CRITICAL TEMPERATURES
water) gradually rises with increasing numbers of carbon undergoing the up and down fluctuations of the other ratios.
23 I 5
rather than
The Rule of Three Temperatures The line of reasoning which led Prud’homme to this rule in its simplest terms has already been given. It will be noted that Prud’homme employed the ratio TF,’Tc(rather its reciprocal) in arriving a t his destination. X much simpler deduction of this rule can be obtained by taking the sums of the ratios TF/Tc and TB/Tc,thus
T F Tc
=
Iil and
TB - = K2, then Tc
Employing the values of Iil (.386) and Kz ( . 6 j 3 ) given above, gives 1.04 as the value for K3, nearly identical with the average value of Prud’homme’s Of I . 0 5 . 2 4
+
The simple form of Prud’homme’s rule, i.e. TB TF = Tc, holds for quite a large number of common substances. I n Table I are listed the critical temperatures as computed from this rule and the experimental values as taken from ICT for all substances therein listed the sum of whose absolute melting and boiling points is within zycof their absolute critical temperatures. A few substances which exceed this limit ( 2 % ) are also given, the percentage discrepancy in such cases being indicated. ITe have found it convenient to employ Prud’homme’s rule in the form, tF $. tB
+ 273
=
tC
as centigrade temperatures are universally employed for recording constants rather than those of the absolute scale. It will be noted that many of the common solvents fall within this class; the esters are more strongly represented than any other group of organic derivatives, altho it should be kept in mind that other esters not listed show considerable discrepancies. If one has any difficulty in remembering the critical temperature of water, the difficulty should immediately disappear if the rule of three temperatures is applied. Young (1.c.) has already called attention to this fact. -4 relationship between the three temperatures is also deducible on this basis by assuming that the difference of the two ratios could be taken, giving TB - T r =, (Kz - KI)Tc Le., the difference between the absolute boiling points and melting points should be proportional to the critical temperature. Using the values of K Zand K, given above yields, T a - TF = .27Tc An examination of a number of compounds shows t h a t the proportionality constant is nearer 0.40 and further does not show as satisfactory agreement between observed and calculated values as does Prud’homme’s rule. 23
l4
23 16
ROBERT TAFT AXD JESSE STARECK
TABLE I Substance
Chlorine Helium Hydrogen Iodine IVater Hydrazine Phosphine Phosgene Chloroform Methyl chloride Methyl alcohol Methyl formate Ethyl alcohol Dimethyl sulfide Ethylamine Propionitrile Acetone Ethyl formate Methyl acetate Trimethylamine Thiophene Methyl propionate Propyl formate
t(.(ICTl
tF+tB+ 2 i 3
144 - 268
- 268
- 240
-239
136 137)
553
571
37-1
3 73
3 80
388
51
53
182
172
2 63
270
I43
I jl
2 40
2 40
214
2 0j
2 43
235 226
230
76
183 291
I
235
235
235
246
234 161
232
3'7
318
257
265 261
265
(2.2cr)
2.3% (2.25)
I57
I94
Ethyl ether
I94
Isobutyl alcohol Pyridine Isobutyl formate Methyl butyrate Propyl acetate B romobenxene Iodobenzene Ethyl butyrate Isobutyl acetate Propyl propionate Ethyl isovalerate Isobutyl isobutyrate
26 j
272
344
346
278 281
276 280
276
282
39i
448
398 43' ( 2 . 3 5 )
293
301
288
293
305
3 18
315
309
329
341
185
MELTISG AND B O I L I S G POISTS AND CRITICAL TEMPERATURES
23 I 7
Further, attention may be called to the diverse character of the substances listed in Table I ; elements, inorganic and organic compounds are all rcpresented. It is our judgment that this is a more useful approximation than are relationships between any of the pairs of constants previously considered, but that a fundamental theoretical treatment for its cause is still lacking.
Summary X brief review of the literature on relationships between melting and boiling points and critical temperatures is given. 2. The data of the International Critical Tables for the ratios T F ‘Tc, T B / T ~TF/TB , and for the relation T F TB = T c have been examined. 3. TF/Tc and TF/TB are not even approximately constant, but definite relationships between these constants do exist for restricted groups of corn; pounds. In the case of the aliphat’ichydrocarbons the limiting value of the ratio TF/TBis 0.5. 4. TB/TC is more nearly “a constant” than either of the above ratios, but the relation T F Tg = TC is a more useful approximation than is the Guldberg-Guye rule. I.
+
+
Cniuersity o j Kansas, Lawrence.
