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Thermodynamical properties of ethylene under pressures up to 3000 atmospheres and temperatures between 0° and 150°C, tabulated as functions of density
Physica
XII,
no. 2-3
Juni
1946
THERMODYNAMICAL PROPERTIES OF ETHYLENE UNDER PRESSURES UP TO 3000 ATMOSPHERES AND TEMPERATURES BETWEEN 0” AND 150” C, TABULATED AS FUNCTIONS OF DENSITY by A. MICHELS, Van
84th der Waals
M. GELDERMANS Publication Laboratorium,
and S. R. DE GROOT
of the Van der Waals Gemeente-Universiteit.
Fund Amsterdam
Summary The energy of C,H,, the kinetic energy, the spxific heat at constant volume, the entropy, the free energy, the enthalpy and the thermodynamic potential have been calculated as functions of temperature and density for the ranges O”-150’ C and l-3000 atmospheres, from isotherm data.
9 1. Introdzcction. The experimental results of the isotherm measurements on CzH4 have been published in a previous paper I) where the values of the product of pressure and volume (PV) are given. From these data and from specific heats cP at P = 0, and P = 1 atm the thermodynamical properties were to be calculated. $2. The values o/ PV. To facilitate this procedure the values of PV at round values of temperature and density were calculated with the power series and the deviation curve given I). The results are shown in table I expressed in Amagat units: For ethylene one Amagat-unit of energy is equivalent to
RTo -= A0
1.98725 x 273.15
-A 1.007582
= 538.73 calories per mole.
$3. The zerofioint of the thermodynamkql functions. In literature there is some confusion regarding the zeropoint for thermodynamical functions. Amongst the possibilities the two most reasonable seem to be: --- 105 -
106
A.
MICHELS,
M.
GELDERMANS
AND
TABLE PV P 0 I 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475
I
-
0” 1.00758 1 .ooooo 0.82789 0.66948
0.15381 0.2087 1 0.32908 0.53905 0.86586 I .34008 1.99360 2.8500 3.9697
T
in Amagat25”
I .09980 1.09288 0.93667 0.79487 0.67412 0.57503 0.49648 0.43584 0.39088 0.36 I C2 0.34856 0.35972 0.40560 0.50172 0.668 10 0.928 15 1.30861 I .83886 2.5503 3.4749 4.6446
I
units
S. R.
DE
GROOT
I
for round
valws
of 1 and
50”
75”
100”
1.19202 I. 18572 I .04403 0.91619 0.80846 0.72062 0.65 124 0.59935 0.56468 0.54848 0.55472 0.59060 0.66723 0.79987
1.28424 I .27850 1.14980 I .03467 0.93874 0.86173 0.80292 0.76212 0.74014 0.73953 0.76529 0.82546 0.93135 I.09813
1.37646 1.37122 1.25430 1.15108 I .06654 1.00037
1.00770 1.31364 I .74389 2.3274 3.0942 4.0752 5.3027
1.34467 1.49360 2.1706 2.8039 3.6226 4.6574 5.9387
1
D
L
125’ I .46868 I .46392 1.35817 1.26611 1.19250 1.13737 1.10115 I .08478 I .09066 1.12319 1.18869 1.29605 1.4574 I 1.68797 2.0061 2.4338 2.9962 3.7202 4.6336 5.7642 7.1413
C.95269 0.92390 0.91561 0.93130 0.97683 I .06C84 1.19485 1.39408 I .67725 2.0664 2.5869 3.2658 4.1324 5.2161 6.5474
1500 1.56089 1.55654 1.46119 1.38007 1.31731 1.27326 1.24860 1.24510 1.26566 1.31499 1.40028 1.53084 1.71857 1.97903 2.33 I2 2.7969 3.3999 4.1665 5.1248 6.3006 7.7219
System a. Zeropoint at temperature t = 0” C, pressure P = 0, hence also density p = 0 *). From a theoretical point of view this is the most obvious definition for a zeropoint. Some difficulty however arises for the entropy S, and thus for the free energy F = U -- TS and the thermodynamic potential G = U - TS + PV, as now for all actual pressures these functions are infinite. This follows from the fact that in the expression
W’-,p)= S(T,O)+J’(=/ap)~ 4 0
the integral is infinite at the lower limit of integration for any temperature (one of the terms of the integral in which it may be divided giving-R In p (cf. 9 9), the other terms of the expression being finite). It is possible however to circumvent this difficulty by tabulating the value of S with the omission of lim (R In p), p=o *) for
The symbol temperature
t is used for degrees in general.
Celsius;
T stands
for
the
absolute
temperature
or
THERMODYNAMICAL
PROPERTIES
OF
the value of the integral-term at the lower limit This may be expressed by the notation S(T, p) = S(T,O) + (&3p),
of integration.
dp.
System p. Zeropoint at temperature t = 0” international atmosphere. This zeropoint is easier in handling, though grounds of principles. The density at this point, called the Amagat-unit of density (p = 1). The value of any thermodynamical function system A(“) can be transformed into the latter of an amount
Here the latter gives immediately necessary this will ponding value of
107
ETHYLENE
C, pressure P = 1 less satisfactory on taken as a unit, is A in the former A(p) by subtraction
zeropoint will be taken, but Ata) = A@) + A the transformation into the other system. Where be mentioned under the tables with the corresA.
§ 4. The Paths of integration.
ways of integration to obtain the point (t, p). One is indicated
There are in principle two elementary a thermodynamical function A at by the line a in-figure 1,
t \ (I .O)
r------((t ,I) I I I I
-F---
----(t .q’
/k
Fig. 1. Paths
of integration
in the temperature-density
plane.
108
A.
giving
MICHELS,
M.
GELDERMANS
AND
S. R. DE
GROOT
the result:
the second by the line b: A= P
0
= dT +j&) ’
1
I
(2)
=tdP.
In these formulae (8A/i3p)T can be calculated from the compressibility isotherms, whereas (aA/aT), is to be taken from independent is values of the specific heat c,. In practice the path of integration therefore determined by the reliability of these c, data at p = 0 or p = 1. In the above formulae the zeropoint t = 0” C, p = 1 is taken, though, of course, analogous formulae may be written down using any zeropoin t . $5. Definition of internal functions. The increase of any thermodynamical function A for an ideal gas by isothermal compression can be found using the Boy 1 e-G a y-Lu s s a c equation of state: The contributiqn by, the potential field of the molecules may thus be expressed as
(3) Here this value A&'-,
will be referred
to as the internal
function:
PI-
5 6. The energy. The energy was calculated tegration indicated in 5 4 by a. Thus dp + /~G?,-. d
dT
along the path of in-
(4)
+ /($-)t=tdp 0”
where
au (3aP
T
= -(T(g)
-P)/F.’ P
As a2(PV)/aT2 is small, (aPV)/aT), perimental values of (A(PV)/AT),.
= -{T(F)
-+.
(5)
P
could be taken
from the ex-
THERMODYNAMICAL
PROPERTIES
OF
1.09
ETHYLENE
c, = cp - R at p = 0. The values of cp were taken from E uc k e n and P a r t s “). All integrations were carried out graphically. For an ideal gas (XY/+)idcal, T = 0. Hence the definition (3) of the internal energy gives vi = Thus : U(t, p) = -
-p au !( ap ) t=t dp* 0
Ui(Q 1) +;
(6)
(cJp=o dT + Ui (t, p).
(7)
Table II shows the results for Vi and U, given in cal/mol. 3 7. The Kinetic 1 o m b interaction
energy. For a group of particles with C o uthe virial theorem leads to the relation 3PV = K + U. (8) w’here K is the sum of the kinetic energies of all particles concerned, and U is the total energy. Now following path b of fig. 1 K(t,
p)
=/@)p=l~T
i-.
r($)t=tdp
0
-
d
/(f,,-,
dp
t9)
d
where the term
is just the internal
kinetic
iFi?L,.,.
energy Ki(t, p) according
to 5 5, for
(11)
T = (%9;deal. T = O*
As at isothermal compression the kinetic energy of the molecules themselves (e.g. translational and rotational) remains constant by the equipartition law, this Ki measures the increase of the kinetic energy of the internal motion of the molecules (e.g. kinetic energy of the nuclear vibrations and of the electrons). Taking K(O”, 1) = 0 the kinetic energy may be written: K(t,
p) = 3/(p) 0
_ dT -/‘( P--l
!&)p=ldT+K;(i,
p)---Ki(t,
0
The values of Ki and K are represented
in table III.
1). (12)
225 250 275 300 325 350 375 400 425 450 475
0 I 25 50 75 100 125 150 175 200
P
-
I 0 -193.0-391.1
8.1 -201.1 -399.2
-
!-
I 7 - 8.1
I
0
ui
00
0 7.8 192.5 378.1 555.0721.5876.3 -1018.0 -1149.6 -1274.6 -1397.6 -1521.6 -1649.9 -1783.6 -1922.0 -2063.4 -2205.7 -2345.2-2134.7 -2477.3 -2598.0 -2702.1
-
25’
--2266.8 -2387.5 -2491.6
210.5 202.7 l&O167.6344.5511.0665.8807.5 939.1 -1064.1 --I 187.1 -131 I.1 -1439.4 -1573.1 -I71 1.5 -1852.9 -1995.2
+ + + -
I
0 7.5 183.0 357.6 523.4680.3830.4 969.4 -I 102.1 -1231.2 -1359.4 --1488.8 -1621.2 -1756.9 -1895.5 -2035.5 -2174.4 -2309.4 -2436.4 -2551.1 -2648.4
-
+ + + +
426.5 419.0243.5 68.996.9 253.8 403.9 542.9 675.6 804.7 932.9 -1062.3 -1194.7 -1330.4 -1469.0 -1609.0 -1747.9 -1882.9 -2009.9 -2124.6 -2221.9
50”
!
-
-
u(a)
=
662.7
656.2 649.1 483.4 318.1 160.510.8 133.1 269.0 402.4 532.7
u(B)
-
793.6 927.1 -1062.9 -1200.1 -1337.2 -1472.0 -1601.6 -1721.9 -1829.0 -1917.4-
-
i+ + ++ +
in cal/mol 75”
7.1 172.8 338.1 495.7 645.4 789.3926.0 1058.6 1188.9 1318.9 1450.01583.3 1719.1 1856.3 1993.4 2128.2 2257.8 2378.1 2485.2 2573.6
0
Energy
T
-
-
a.1
I
+ 6.6 + 163.8 + 321.6 + 472.9 + 617.9 + 757.0 + 891.9 + 1023.9 1154.3 1284.6 1416.0 1548.9 1683.4 1818.5 1952.6 2083.4 2207.8 2322.0 2421.9 2502.5-
0
100”
899.8 893.2 736.0578.2426.9281.9 142.87.9 124.1 254.5 384.8 516.2649.1 783.6 918.7 1052.8 1183.6 1308.0 1422.2 1522.1 1602.7
!
-
-
-
-
6.3 156.7 309.1 456.0 597.6 734.4 867.8 999.1 1129.3 1259.6 1390.7 1523.1 1656.3 1789.5 1921.2 2048.8 2169.4 2279.4 2374.5 2449.5-
0
+ + + + + + + + + + -
125”
1157.4 1151.1 1000.7 a48.3701.4559.8 423.0289.6 158.3 28.1 102.2 233.3 365.7 498.9 632.1 763.8 891.4 1012.0 1122.0 1217.1 1292.1
I
-
-
-
6.1 151.3 299.8 443.6 583.0 718.2 850.1 980.4 1110.3 1240.3 1371.1 1502.7 1634.4 1765.5 1895.0 2020.1 2137.7 2244.1 2334.5 2404.0-
0
+ + $ + + + + + + + + + -
150”
1429.5 1423.4 1278.2 1129.7 985.9 846.5 711.3 579.4 449.1 319.2 189.2 58.4 73.2 204.9 336.0 465.5 590.6 708.2 814.6 905.0 974.5
I
I
25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 45c 475
0
P
I
L
T Ii
-
0 k 4.1 0 4.1 I - 09.: 3- - 85.; -147.: !- -143.1
Iii
0” /
-
-
+ 325.5 + 527.9 f 817.0 -I- 1224.3 ,- 1786.0 t2543.2 t3539.7 t4821.6 ;-6436.7 ta431.2
!
I
2.7 56.2-
j
50”
Ii
i
I
104.6 27.9 75.0 213.3
204.3 212.6 197.7 159.7
+ + + + + + -1. + + -I-
+ + + +
+
116.1 118.8 172.3-
+ 400.7 + 656.9 +1007.0 ‘t 1481.5 +21 16.0 +2950.2 +4028.2 +5394.5 +-7094.8 +9176.0
88.2 96.5 .31.643.6 Il.5 88.2 191.1 + 329.4 i-
0
Iii
+ 516.8 i- 773.0 ;1123.1 + 1597.6 + 2232.1 -1-3066.3 +4144.3 ;-5510.6 +7210.9 f9292.1
163.9 182.2175.8-148.0 104.3 t 45.4 + 31.4 + 134.2 +
49.2 52.6120.4~-
Ii
+ 276.3 + 478.7 + 767.8 ; 1175.1 -+ 1736.8 -1-2494.0 +3490.5 t4772.4 +6387.5 +a382.0
55.1 3.8 80.6 + 183.4 +
114.7 133.0 126.698.8
-. t -t +
3.471.2-
-
0
Ii,
25”
Kinetic
-
I
708.5 1013.0 1418.3 1954.0 2655.0 3560.6 4713.9 6157.4 7936.8 10096.0
114.6 17.6 I I 1.7 283.4
262.1 259.5 234.3 185.4
196.8 199.0 241.3
I< I .9 33.6 42.7 -2&O10.1 72.1 -
-
100”
291.3
130.8 12.2 143.5 347.4
334.0 319.3 281.2 219.2
293.2 324.9
+ 614.6 + 964.1 + 1420.6 + 2013.3 + 2776.4 + 3748.3 + 4970.1 i- 6484.8 + 8336.2 +10568.4
160.5 279. I 434.8 + 638.7 +
0
+ 905.9 + 1255.4 + 1711.9 + 2304.6 + 3067.7 + 4039.6 + 5261.4 + 6776.1 + 8627.5 +10859.7
+ i+ +
+ +
I
in ml/m01
111
+ 511.7 + 816.2 + 1221.5 + 1757.2 +2458.2 +3363.8 +4517.1 +5960.6 +7740.0 $9899.2
+ +
2.2 44.5 65.3 62.7 37.5 I 1.4 82.2 179.2 3oe.5 480.2
0
Ki
75”
Energy
TAlS1.E
1.4 21.9 18.3 9.7 62.2
3481.1 4517.5 5808.3 7394.5
i+ + t
+ 9316.9 +11617.5
1111.7 1504.9 2010.7 2658.2
140.4 247.3 388.1 570.9 807.1
0
+ + f +
+ + + + +
-I+
5408.4 6994.6
3081.2 41 17.6
711.8 1105.0 1610.8 2258.3
980.7 1322.5 1757.6 2310.2 3010.4
457.8 799.6 1234.7 1787.3 2487.5
522.9 523.9 532.7 515.3 473.0 404.8 309.4 183.2 19.7 190.;
+ 3369.7 i- 4469.3 + 5a26.0 -I- 7481.2 + 9471.9 +1183a.5
+ + i+ +
+
-
-
150”
1.0 9.8 7.649.9 118.1 213.5339.7503.2712.9
0
+ 3892.6 -‘r 4992.2 + 634a.9 i- 8004.1 + WA +12361.4
+ + + +
+ + + 259.5 + 152.6 + 11.8 + 171.0 + 407.2 +
399.9 401.3 421.8 418.2 390.2 337.7
i- 8917.0 +11217.6
+ +
+ +
t + f +
-+ +
125”
112
A. MICHELS,
M. GELDERMANS
AND
S. R. DE GRCIOT
$8. Th.e specific heat at constant volume. pression (7), for c, is written :
rv= (gj
Using
for U the ex-
= (c”)p=o + (fg . P
‘P
The second term (aU,/aT),, giving the increase of c, by compression, is called A c, ; the results are shown in the first columns of table IV. The second columns give the total value of the specific heat c,. TABLE Specific
-
IV
heat C,, in ml/degree
mol
I
0”
P 0 1 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 -475
c, 7,8;A&0
AC - v 0 w 1 7,a: 0,3; 2 a,11 O,& 5 8,6f
-
- -
0,Ol 0,36 0,84 1,31 I,75 l,a9 2,00 I,94 1,71 1,42 1,15 0,92 0,79 0,72 0,77 0,86 0,97 I,14 1,34 I,58
0 lo,59 1 Ai5’; 0 ’ ll,;e a,37 / ACTC, 0 a,91 1 k&0 9,46 1 A::, 0 IO,02 / AC’,‘“;, a,38 0,02 a,93 0,02 9,4a 0,Ol IO,03 0,Ol lo,60 0,Ol 1 I,19 a,73 0,41 9,32 0,40 9,86 0,33 IO,35 0,26 lo,85 0,16 1 I,34 9,21 0,al 9,72 0,73 IO,19 0,5a IO,60 0,44 11,03 0,29 II,47 9,6a 1,19 lo,10 1,02 lo,48 0,79 lo,81 0,sa 11,17 0,40 11,5a IO,12 I,53 IO,44 I,25 IO,71 0,95 IO,97 0,69 11,2a 0,4a ll,66 lo,26 1,75 IO,66 1,49 IO,95 1,09 I I,11 0,77 ll,36 0,53 II,71 IO,37 I,86 IO,77 I,56 11,02 I,14 II,16 O,a3 11,42 0,60 11,7a IO,31 l,a4 IO,75 I,60 II,06 I,17 11,19 O,a6 11,45 0,65 I I,83 lo,08 I,74 IO,65 1,5a 11,04 I,17 II,19 o,a7 11,46 0,66 ll,a4 I,53 IO,99 9,79 I,60 IO,51 1,la 1 I,20 O,a7 11,46 0,6a ll,a6 9,52 I,46 IO,37 I,53 IO,99 1,la II,20 c,a9 Il,4a 0,70 11,aa 9,29 I,37 IO,28 I,55 II,01 1,la 1 I,20 0,92 I I,51 0,71 1 I,89 1,Sa I I,04 I,24 II,26 0,97 I I,56 0,77 Ii,95 9,16 I,33 IO,24 9,09 I,36 IO,27 I,64 11,lO I,33 II,35 I,04 1 I,63 O,a2 12,OO 9,14 I,46 IO,37 I,75 I I,21 I,43 1 I,45 1,ll II,70 O,a7 12,OS 9,23 1,63 IO,54 I,92 11,3a I,58 I I,60 I,22 1 I,81 0,94 12,12 9,34 I,84 lo,75 2,15 I I,61 1,76 II,78 I,35 11,94 1,04 12,22 9,51 2,09 II,00 2,4l 1l,a7 I,96 II,98 I,51 12,lO 1,2012,3a 9,71 2,35 II,26 2,72 12,la 2,19 12,21 I,69 12,2a I,39 12,57 3,02 12,4a 2,49 12,51 1,90 12,49 I,61 l2,73 9,95 2,67 II,58
$9. The entropy. The entropy was calculated the path indicated by b in $4 and fig. 1. The expression for S is:
integrating
along
Using
THERMODYNAMICAL
PROPERTIES
the definition
(3) of internal (g)idr.l,t=
OF
113
ETHYLENE
functions
-and the relat’ion
-+
this reads : SK PI =*/ ‘*
dT -
R In p + &(t, p) -
S&, 1)
(16)
where .w, P) -j{(i).-
(g)i,a,
,)dp =/{-if?
0
(&),t-
+j dp.
(17)
0
For numerical calculation the values of c, at p = 1 are needed. Those for cp at P = 1 are known 2). Substitution of the series expansion of K a m e r 1 i n g h 0 nn e s, PV = A +- B . v-’ + C . P + . . . . in the thermodynamical relation cp-c,
= -T(g(g-):
(18)
gives :
2 ++; (
p2+
-%!?!?-A$
g
. .
I
1
. . (19)
This series expansion converges rapidly in the neighbourhood of p = 1, the second term being of the order: 0,05 cal/mol. degree and the third 0.001 cal/mol. degree. As the error in cp is about lo/, the third and following terms of the series expansion may obviously be neglected. The second term is easily found from data of the second virial coefficient B(T) published before I). Table V gives the values for SJt, p) and S. When t’ = 0” C, p = 0 is taken as the zeropoint the difficulty arises that all total entropies become infinite (with a negative sign). To get the expression for total entropy with zeropoint 1 = 0” C, p=OtoS,theform:’
! as
0”
I( 1
Physica
ap XII
I=0
dp = -[[R
In p]: f $(O,l)
= -
00 f- S,(O,l)
(20) 8
0 I
450 475
50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425
25
P
-
-
T -
p S
-
0 +w -0,Ol. 4 0 - 0,36’ 9- - 6,75: - 0,75: 2- -8,511
S {CT,
-
.-
0 += 0,014 + c,7130,3426,0120,680 7,728 1,OOy 8,862 1,322 9,747 1,614-10,4821,882-11,1122,132-ll,6692,37712,1792,631 - 12,667 2,905 - 13,151 3,210 - 13,645 3,557 - 14,165 3,946 -- 14,713 4,377 - 15,291 4,854 - 15,905 5,375 - 16,554 5,936 - 17,236 6,536 - 17,951 7,173 - 18,694 -
--
-
3,848 4,276 4,743 5,251 5,795 6,376 6,990
-. -
c G,013 i 0,3120,615 0,907 1,1871,4551,713-lO,2451,9672,2252,495 2,787 3,106 3,459 -
s(a)
13,917 14,492 15,097 15,732 16,398 17,092 17,813
10,80611,3Z 11,833 12,335 12,843 13,369
+ca 1,4125,284 6,964 8,0628,9149,625-
Entropv
= SW)
-
-
V
-
3,736 4,155 4,611 5,102 5,627 6,183 6,771 0,014
-
0 0,011 + 0,282 0,557 0,824 1,084 1,3371,5891,8432,104-10,5232,380 2,676 2,998 3,351 -
in ml/degree
TABLE
-
13,120 13,686 14,279 14,899 15,544 16,214 16,909
11,033 11,539 12,050 12,576
+=J 2,0994,569 6,221 7,294 8,1268,8229,4369,997-
00
mol
2,291 2,588 2,909 3,258 3,636 4,046 4,490 4,966 5,474 6,013 6,577
--
0,763 1,009-1,2531,4991,7522,013-
-
-
0,258 0,512
-
0 O,OlO+ -
10,266 10,774 11,284 11,805 12,343 12,900 13,480 14,086 14,714 15,367 16,038
-
6,556 -. 7,374 8,061 8,669-9,2299,755-
3,868 5,500-
t-W 2,777-
-
1,191 1,4341,6841,9452,221 2,518 2,838 3,184 3,558 3,963 4,399 4,867 5,363 5,889 6,439
-
-
+
0,235 0,475 0,715 0,953-
0 0,009
7,3307,9448,4929,0189,528 10,035 10,544 11,063 11,597 12,148 12,721 13,318 13,934 14,574 15,230
3,176 4,793 5,839 6,649-
+CC 3,447-
-
-
1,1461,3861,6341,8952,171 2,466 2,785 3,127 3,495 3,894 4,323 4,782 5,270 5,784 6.321
0 0,008 0,217 0,448 0,6800,912-.
-
-
-
+ -
6,622 7,224 7,780 8,305 8,814 9,320 9,828 10,343 10,870 11,416 11,982 12,570 13,177 13,806 14,449
+m 4,111 2,495 4,103 5,141 5,945
THERMODYNAMICAL
PROPERTIES
115
OF ETHYLENE
.
has to be added where -
00 is to be considered as the limiting iii;
(R In p).
(21)
9 10. The free energy. The relation F = U for F. The results are given in table VI. TABLE
p 0 I 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 _-~
1
0” --00 0 + 1651,3 + 1933,9
/
25” --cc 9,9 + 1810,5 +2136,5 + 2297,7 +2395,1 + 2459,4 + 2505,6 + 2540,O + 2567, I + 2589,6 + 2609,9 + 2628,9 + 2650,2 + 2675.2 + 2706,l + 2746,9 i-2800,9 + 2872, I + 2964,6 + 3082,o
)
value :
VI
Free energy-. in d/m01 50” / 75” 1 L-3 37,2 + 1951,l +2319,3 + 2508,3 + 2626,? + 2706,4 + 2767,7 +2a16,4 +2a56,2 + 2890,9 t- 2923,a +2955,5. + 2989,a +3028,3 +3074,1 $-3130,7 + 3200,9 +3289,2 + 3398,7 +3534,4
--co al,6 +2074,1 +2484,0 + 2700,O + 2839,9 +2938,3 +3015,3 t-3078,1 +313o,a +3178,4 +3223,5 +326a,l +3315,4 +3367,6 + 3427,6 +3499,2 +3585,5 +3689,a +3815,9 + 3969,4
a,=--4,s a?,=-3,9 a,,=--3,6-a,,=-3,2 in Ffa) = F(p) - a.1 - T(-0,014
TS gives the value
100”
-co 143,l +2179,3 + 2630,6 + 2873,3 +303&S +3150,7 + 3242,7 +3319,7 -f-3385,6 +3446,0 +3504,1 +3561.5 +3621,4 +36a7, I +3760,8 + 3846,4 + 3948,2 + 4068,3 +4212,1 +43ai,a -.__
1
125”
)
-cc - 221,3 + 2265,2 + 2756,6 + 3026,2 +3207,1 +3341,4 +3452,5 +3539,4 $3618,6 +3691,4 + 3762,l +3832,4 +39OS,8 + 3985,2 + 4072,9 +4173,5 +4290,6 i-4425,8 + 4585,5 +4771,7
150” --co - 316,l + 2334,O + 2865,9 + 3 16 I ,3 +3362,1 +3513,4 +3636,2 +3741,2 + 3833,5 +3918,9 + 4002,2 + 4085,s +4171,7 + 4263,6 + 4365,2 + 4479,6 +46lo,a +476l,Z + 4937,0 +5139,6
a,,,=-2,91a,,,=-2,5a,,,=-2,i -- CO) = F(p) + at + m
5 11. The enthalpy. Similarly the enthalpy w=u+lv. Data are tabulated in table VII.
is obtained
from
$ 12. Thermodynamic eotential. As G = U - TS + PV, the thermodynamic potential is derived from the free eneigy by adding PV. Table VIII shows the results. § 13. Discussion. The accuracy of the isotherm data is & 1 in 10.000. Hence the accuracy obtained in Ui and on the average in
116
A. MICHELS,
M.
GELDERMANS
1
Enthalpy 50” )
AND
TABLE p
1
0
0” + + + -
I 25 50 75 100 125 150 175 200 225 250 275 330 325 350 375 400 425 450 475
/
25”
550,9 538,7 253,0 30,4
+ 803,O + 791,5 +- 522,6 + 260,6 + 18,7 -- 201,2 - 398,3 - 572,7 - 728,5 - 869,6 - 999,3 --I 117,3 - 1220,9 - I302,8 --1351,6 --1352,9 -1290,2 --I 144,o - 892,9 - 515,4 + IO,6
+ + + + + + + +
IO68,7 1057,8 806,O 562,5 338,6 134,4 53,1 220,o 371,4 509,2 634,l 744,l 835,2 899,5 926,1 901,3 808,4 629,l 343,0 70,9 634,8
S. R. DE
GROOT
VII in cnl/mol 75” / + + + + + + + + + + +
1348,l 1337,9 1102,8 875,5 666,2 475,0 299,5 140,8 3,7 134,3 250,4 349,l 425,3 471,3 475,7 424,a 302,7 91,o .=9,7 680,l 1281,9
1000 + + + + + + + + + ‘+ + + + + + + + +
1
1641,3 1631,9 141 I,7 1198,3 1001,5 820,8 656,0 505,6 369,2 247,2 141,5 55,3 5,4 32,6 15,l
60,5 210,O 451,4 804,l 1288,O 1924,6
125” + + + + + + + + f + + + i+ + + + + + + $-
)
1948,6 1939,8 l732,4 1530,4 1343,8 1172,5 lOl6,2 874,O 745,9 633,2 538,2 464,9 419,5 410,5 448,7 547,4 722,7 992,2 1374,3 I888,3 2555, I
1504 + 2270,4 +2262,0 -I- 2065,4 + 1873,2 + 1695,6 + 1532,4 + 1384,O + 1250,2 -t ll31,O +1027,6 + 943,6‘ + 883,l + 852,7 + 861.3 + 919,9 + 1041,3 + 1241,O + 1536,4 + 1946,3 -+ 2489,3 +31a5,5
W(Q) = WC/3 -8.1 TABLE p 0 1 25 50 75 100 125 150 175 200 225 250 275 330 325 350 375 400 425 450 475
(
0’ --cc3 + 538,7 +2097,3 + 2294,6
a,=-4,3
1
Thermodynamic 25” 1 50”
VIII
potential 1 75”
in cal/mol 1 1000
1
125’
--00 -cm -m -co --cc $ 578,9 + 601,6 + 607,2 + 595,6 + 567,4 +2315,1 +2513,6 + 2693,5 + 2855,0 + 2996,9 + 2564,7 +2812,9 +3041,4 + 3250,7 j-3438,7 + 2660,9 + 2943,8 + 3205,7 +3447,9 + 3668,6 + 30i9,% + 2704,9 +3014,9 +3304,1 +3572,4 f 2726,9 +3057,2 + 3370,9 +3663,9 i-3934,6 + 2740,4 +3090,6 +3425,9 + 3740,4 f 4036,9 + 2750,6 +3120,6 +3476,8 +3813,0 +4127,0 +276l,6 +3151,7 +3529,2 + 3887,3 -t 4223,7 -l-2777,4 -l-3189,7 + 3590,7 +3972,3 +4331,8 + 2803,7 +3242,0 +3668,2 + iO75,6 + 4460,3 +2847,4 -+3315,0 i-3769,9 + 4205,2 +4617,6 +2920,5 + 3420,7 +3907,0 + 4372,4 4 48 15,2 +3035, I +3571,2 + 4092,O + 4590,7 + 5066,O + 3206,l +3781,8 +4340,0 +4074,1 +- 5384, I +3451,9 -+- 4070,2 + 4668,5 + 5240,O +-5787,6 +3791,6 +4454,7 + 5096,l +5707,6 + 6294,8 + 4246,0 + 4956, I +5641,4 + 6294,6 + 6922,l + 4836,7 + 5594,2 + 6325,0 + 7022,2 + 7690,9 + 5584,2 +6391,1 t-7168,7 + 7909, I +8618,9 ---~. t&=-3,9 a&,=-3,6 a,,=-3,2 al,,=-2,9a,,,=-2,5a,,,=--2,: in G(a) = G(b) + at + co
)
150” --co + 522,5 +3121,2 +3609,4 $- 387 I ,o + 4048,O +4186,1 + 4307,o +4423,1 +4541,9 + 4673,3 + 4826,9 +5011,4 + 5237,9 +5519,5 + 5872,0 3-631 I,2 + 6855,4 + 7522,l +a331,3 + 9299,6
THERMODTNAMICAL
PROPERTIES
OF
ETHYLENE
117
Ki may be estimated to be of the order of 1 in 1000. The accuracy in U and W varies with growing density between 1 and 4 cal/mol.
at low temperatures and from 7 to 9 cal/mol. at high temperatures. The error estimated in AC, is f 50/5, in c,, 1Oh. The order of accuracy in .Si is 5°/oo and varies in S, F and G from I-10 per mille with increasing temperatures. Received
March
3 rd.
1946
REFERENCES 1)
2)
.4. MiLhcls and AI. Geldcrmans, Physica !), 967, 1942. MTith the values from this paper (PY),,oo, p=l = 0,999896. A better approximation to the correct value (PV)+oo, p=l = 1 is 0,999996 and is obtained by changing somewhat the calculated normal volumes and the series coefficients. The normal volumes mentioned diminish 1 in 10000. In tables IIa, 116 and III the absolute value of A increases 1 in 10000, of B 2 in 10000, of C 3 in lOOOC, of Z 4 in 10.000, of D 5 in IOCOO, of Y 6 in 10000 and of E 7 in 10000. In table I the values of 3~ (named p here) decrcase 1 in 10000, those of PV increase 1 in 10000. In table IV only th: values of dA decrease 1 in 10000. For 0” C the values for the low pressure series (table 11~) are now: il = 1.007582. R = - 7.60211 1Om3, C = 16.474. 10-e and D = 126.8. 10-12. A. E o c 1~ e n and A. P a r t s, Z. phys. Chem. B. 20, 184, 1933.
XII,
no. 2-3
Juni
1946
THERMODYNAMICAL PROPERTIES OF ETHYLENE UNDER PRESSURES UP TO 3000 ATMOSPHERES AND TEMPERATURES BETWEEN 0” AND 150” C, TABULATED AS FUNCTIONS OF DENSITY by A. MICHELS, Van
84th der Waals
M. GELDERMANS Publication Laboratorium,
and S. R. DE GROOT
of the Van der Waals Gemeente-Universiteit.
Fund Amsterdam
Summary The energy of C,H,, the kinetic energy, the spxific heat at constant volume, the entropy, the free energy, the enthalpy and the thermodynamic potential have been calculated as functions of temperature and density for the ranges O”-150’ C and l-3000 atmospheres, from isotherm data.
9 1. Introdzcction. The experimental results of the isotherm measurements on CzH4 have been published in a previous paper I) where the values of the product of pressure and volume (PV) are given. From these data and from specific heats cP at P = 0, and P = 1 atm the thermodynamical properties were to be calculated. $2. The values o/ PV. To facilitate this procedure the values of PV at round values of temperature and density were calculated with the power series and the deviation curve given I). The results are shown in table I expressed in Amagat units: For ethylene one Amagat-unit of energy is equivalent to
RTo -= A0
1.98725 x 273.15
-A 1.007582
= 538.73 calories per mole.
$3. The zerofioint of the thermodynamkql functions. In literature there is some confusion regarding the zeropoint for thermodynamical functions. Amongst the possibilities the two most reasonable seem to be: --- 105 -
106
A.
MICHELS,
M.
GELDERMANS
AND
TABLE PV P 0 I 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475
I
-
0” 1.00758 1 .ooooo 0.82789 0.66948
0.15381 0.2087 1 0.32908 0.53905 0.86586 I .34008 1.99360 2.8500 3.9697
T
in Amagat25”
I .09980 1.09288 0.93667 0.79487 0.67412 0.57503 0.49648 0.43584 0.39088 0.36 I C2 0.34856 0.35972 0.40560 0.50172 0.668 10 0.928 15 1.30861 I .83886 2.5503 3.4749 4.6446
I
units
S. R.
DE
GROOT
I
for round
valws
of 1 and
50”
75”
100”
1.19202 I. 18572 I .04403 0.91619 0.80846 0.72062 0.65 124 0.59935 0.56468 0.54848 0.55472 0.59060 0.66723 0.79987
1.28424 I .27850 1.14980 I .03467 0.93874 0.86173 0.80292 0.76212 0.74014 0.73953 0.76529 0.82546 0.93135 I.09813
1.37646 1.37122 1.25430 1.15108 I .06654 1.00037
1.00770 1.31364 I .74389 2.3274 3.0942 4.0752 5.3027
1.34467 1.49360 2.1706 2.8039 3.6226 4.6574 5.9387
1
D
L
125’ I .46868 I .46392 1.35817 1.26611 1.19250 1.13737 1.10115 I .08478 I .09066 1.12319 1.18869 1.29605 1.4574 I 1.68797 2.0061 2.4338 2.9962 3.7202 4.6336 5.7642 7.1413
C.95269 0.92390 0.91561 0.93130 0.97683 I .06C84 1.19485 1.39408 I .67725 2.0664 2.5869 3.2658 4.1324 5.2161 6.5474
1500 1.56089 1.55654 1.46119 1.38007 1.31731 1.27326 1.24860 1.24510 1.26566 1.31499 1.40028 1.53084 1.71857 1.97903 2.33 I2 2.7969 3.3999 4.1665 5.1248 6.3006 7.7219
System a. Zeropoint at temperature t = 0” C, pressure P = 0, hence also density p = 0 *). From a theoretical point of view this is the most obvious definition for a zeropoint. Some difficulty however arises for the entropy S, and thus for the free energy F = U -- TS and the thermodynamic potential G = U - TS + PV, as now for all actual pressures these functions are infinite. This follows from the fact that in the expression
W’-,p)= S(T,O)+J’(=/ap)~ 4 0
the integral is infinite at the lower limit of integration for any temperature (one of the terms of the integral in which it may be divided giving-R In p (cf. 9 9), the other terms of the expression being finite). It is possible however to circumvent this difficulty by tabulating the value of S with the omission of lim (R In p), p=o *) for
The symbol temperature
t is used for degrees in general.
Celsius;
T stands
for
the
absolute
temperature
or
THERMODYNAMICAL
PROPERTIES
OF
the value of the integral-term at the lower limit This may be expressed by the notation S(T, p) = S(T,O) + (&3p),
of integration.
dp.
System p. Zeropoint at temperature t = 0” international atmosphere. This zeropoint is easier in handling, though grounds of principles. The density at this point, called the Amagat-unit of density (p = 1). The value of any thermodynamical function system A(“) can be transformed into the latter of an amount
Here the latter gives immediately necessary this will ponding value of
107
ETHYLENE
C, pressure P = 1 less satisfactory on taken as a unit, is A in the former A(p) by subtraction
zeropoint will be taken, but Ata) = A@) + A the transformation into the other system. Where be mentioned under the tables with the corresA.
§ 4. The Paths of integration.
ways of integration to obtain the point (t, p). One is indicated
There are in principle two elementary a thermodynamical function A at by the line a in-figure 1,
t \ (I .O)
r------((t ,I) I I I I
-F---
----(t .q’
/k
Fig. 1. Paths
of integration
in the temperature-density
plane.
108
A.
giving
MICHELS,
M.
GELDERMANS
AND
S. R. DE
GROOT
the result:
the second by the line b: A= P
0
= dT +j&) ’
1
I
(2)
=tdP.
In these formulae (8A/i3p)T can be calculated from the compressibility isotherms, whereas (aA/aT), is to be taken from independent is values of the specific heat c,. In practice the path of integration therefore determined by the reliability of these c, data at p = 0 or p = 1. In the above formulae the zeropoint t = 0” C, p = 1 is taken, though, of course, analogous formulae may be written down using any zeropoin t . $5. Definition of internal functions. The increase of any thermodynamical function A for an ideal gas by isothermal compression can be found using the Boy 1 e-G a y-Lu s s a c equation of state: The contributiqn by, the potential field of the molecules may thus be expressed as
(3) Here this value A&'-,
will be referred
to as the internal
function:
PI-
5 6. The energy. The energy was calculated tegration indicated in 5 4 by a. Thus dp + /~G?,-. d
dT
along the path of in-
(4)
+ /($-)t=tdp 0”
where
au (3aP
T
= -(T(g)
-P)/F.’ P
As a2(PV)/aT2 is small, (aPV)/aT), perimental values of (A(PV)/AT),.
= -{T(F)
-+.
(5)
P
could be taken
from the ex-
THERMODYNAMICAL
PROPERTIES
OF
1.09
ETHYLENE
c, = cp - R at p = 0. The values of cp were taken from E uc k e n and P a r t s “). All integrations were carried out graphically. For an ideal gas (XY/+)idcal, T = 0. Hence the definition (3) of the internal energy gives vi = Thus : U(t, p) = -
-p au !( ap ) t=t dp* 0
Ui(Q 1) +;
(6)
(cJp=o dT + Ui (t, p).
(7)
Table II shows the results for Vi and U, given in cal/mol. 3 7. The Kinetic 1 o m b interaction
energy. For a group of particles with C o uthe virial theorem leads to the relation 3PV = K + U. (8) w’here K is the sum of the kinetic energies of all particles concerned, and U is the total energy. Now following path b of fig. 1 K(t,
p)
=/@)p=l~T
i-.
r($)t=tdp
0
-
d
/(f,,-,
dp
t9)
d
where the term
is just the internal
kinetic
iFi?L,.,.
energy Ki(t, p) according
to 5 5, for
(11)
T = (%9;deal. T = O*
As at isothermal compression the kinetic energy of the molecules themselves (e.g. translational and rotational) remains constant by the equipartition law, this Ki measures the increase of the kinetic energy of the internal motion of the molecules (e.g. kinetic energy of the nuclear vibrations and of the electrons). Taking K(O”, 1) = 0 the kinetic energy may be written: K(t,
p) = 3/(p) 0
_ dT -/‘( P--l
!&)p=ldT+K;(i,
p)---Ki(t,
0
The values of Ki and K are represented
in table III.
1). (12)
225 250 275 300 325 350 375 400 425 450 475
0 I 25 50 75 100 125 150 175 200
P
-
I 0 -193.0-391.1
8.1 -201.1 -399.2
-
!-
I 7 - 8.1
I
0
ui
00
0 7.8 192.5 378.1 555.0721.5876.3 -1018.0 -1149.6 -1274.6 -1397.6 -1521.6 -1649.9 -1783.6 -1922.0 -2063.4 -2205.7 -2345.2-2134.7 -2477.3 -2598.0 -2702.1
-
25’
--2266.8 -2387.5 -2491.6
210.5 202.7 l&O167.6344.5511.0665.8807.5 939.1 -1064.1 --I 187.1 -131 I.1 -1439.4 -1573.1 -I71 1.5 -1852.9 -1995.2
+ + + -
I
0 7.5 183.0 357.6 523.4680.3830.4 969.4 -I 102.1 -1231.2 -1359.4 --1488.8 -1621.2 -1756.9 -1895.5 -2035.5 -2174.4 -2309.4 -2436.4 -2551.1 -2648.4
-
+ + + +
426.5 419.0243.5 68.996.9 253.8 403.9 542.9 675.6 804.7 932.9 -1062.3 -1194.7 -1330.4 -1469.0 -1609.0 -1747.9 -1882.9 -2009.9 -2124.6 -2221.9
50”
!
-
-
u(a)
=
662.7
656.2 649.1 483.4 318.1 160.510.8 133.1 269.0 402.4 532.7
u(B)
-
793.6 927.1 -1062.9 -1200.1 -1337.2 -1472.0 -1601.6 -1721.9 -1829.0 -1917.4-
-
i+ + ++ +
in cal/mol 75”
7.1 172.8 338.1 495.7 645.4 789.3926.0 1058.6 1188.9 1318.9 1450.01583.3 1719.1 1856.3 1993.4 2128.2 2257.8 2378.1 2485.2 2573.6
0
Energy
T
-
-
a.1
I
+ 6.6 + 163.8 + 321.6 + 472.9 + 617.9 + 757.0 + 891.9 + 1023.9 1154.3 1284.6 1416.0 1548.9 1683.4 1818.5 1952.6 2083.4 2207.8 2322.0 2421.9 2502.5-
0
100”
899.8 893.2 736.0578.2426.9281.9 142.87.9 124.1 254.5 384.8 516.2649.1 783.6 918.7 1052.8 1183.6 1308.0 1422.2 1522.1 1602.7
!
-
-
-
-
6.3 156.7 309.1 456.0 597.6 734.4 867.8 999.1 1129.3 1259.6 1390.7 1523.1 1656.3 1789.5 1921.2 2048.8 2169.4 2279.4 2374.5 2449.5-
0
+ + + + + + + + + + -
125”
1157.4 1151.1 1000.7 a48.3701.4559.8 423.0289.6 158.3 28.1 102.2 233.3 365.7 498.9 632.1 763.8 891.4 1012.0 1122.0 1217.1 1292.1
I
-
-
-
6.1 151.3 299.8 443.6 583.0 718.2 850.1 980.4 1110.3 1240.3 1371.1 1502.7 1634.4 1765.5 1895.0 2020.1 2137.7 2244.1 2334.5 2404.0-
0
+ + $ + + + + + + + + + -
150”
1429.5 1423.4 1278.2 1129.7 985.9 846.5 711.3 579.4 449.1 319.2 189.2 58.4 73.2 204.9 336.0 465.5 590.6 708.2 814.6 905.0 974.5
I
I
25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 45c 475
0
P
I
L
T Ii
-
0 k 4.1 0 4.1 I - 09.: 3- - 85.; -147.: !- -143.1
Iii
0” /
-
-
+ 325.5 + 527.9 f 817.0 -I- 1224.3 ,- 1786.0 t2543.2 t3539.7 t4821.6 ;-6436.7 ta431.2
!
I
2.7 56.2-
j
50”
Ii
i
I
104.6 27.9 75.0 213.3
204.3 212.6 197.7 159.7
+ + + + + + -1. + + -I-
+ + + +
+
116.1 118.8 172.3-
+ 400.7 + 656.9 +1007.0 ‘t 1481.5 +21 16.0 +2950.2 +4028.2 +5394.5 +-7094.8 +9176.0
88.2 96.5 .31.643.6 Il.5 88.2 191.1 + 329.4 i-
0
Iii
+ 516.8 i- 773.0 ;1123.1 + 1597.6 + 2232.1 -1-3066.3 +4144.3 ;-5510.6 +7210.9 f9292.1
163.9 182.2175.8-148.0 104.3 t 45.4 + 31.4 + 134.2 +
49.2 52.6120.4~-
Ii
+ 276.3 + 478.7 + 767.8 ; 1175.1 -+ 1736.8 -1-2494.0 +3490.5 t4772.4 +6387.5 +a382.0
55.1 3.8 80.6 + 183.4 +
114.7 133.0 126.698.8
-. t -t +
3.471.2-
-
0
Ii,
25”
Kinetic
-
I
708.5 1013.0 1418.3 1954.0 2655.0 3560.6 4713.9 6157.4 7936.8 10096.0
114.6 17.6 I I 1.7 283.4
262.1 259.5 234.3 185.4
196.8 199.0 241.3
I< I .9 33.6 42.7 -2&O10.1 72.1 -
-
100”
291.3
130.8 12.2 143.5 347.4
334.0 319.3 281.2 219.2
293.2 324.9
+ 614.6 + 964.1 + 1420.6 + 2013.3 + 2776.4 + 3748.3 + 4970.1 i- 6484.8 + 8336.2 +10568.4
160.5 279. I 434.8 + 638.7 +
0
+ 905.9 + 1255.4 + 1711.9 + 2304.6 + 3067.7 + 4039.6 + 5261.4 + 6776.1 + 8627.5 +10859.7
+ i+ +
+ +
I
in ml/m01
111
+ 511.7 + 816.2 + 1221.5 + 1757.2 +2458.2 +3363.8 +4517.1 +5960.6 +7740.0 $9899.2
+ +
2.2 44.5 65.3 62.7 37.5 I 1.4 82.2 179.2 3oe.5 480.2
0
Ki
75”
Energy
TAlS1.E
1.4 21.9 18.3 9.7 62.2
3481.1 4517.5 5808.3 7394.5
i+ + t
+ 9316.9 +11617.5
1111.7 1504.9 2010.7 2658.2
140.4 247.3 388.1 570.9 807.1
0
+ + f +
+ + + + +
-I+
5408.4 6994.6
3081.2 41 17.6
711.8 1105.0 1610.8 2258.3
980.7 1322.5 1757.6 2310.2 3010.4
457.8 799.6 1234.7 1787.3 2487.5
522.9 523.9 532.7 515.3 473.0 404.8 309.4 183.2 19.7 190.;
+ 3369.7 i- 4469.3 + 5a26.0 -I- 7481.2 + 9471.9 +1183a.5
+ + i+ +
+
-
-
150”
1.0 9.8 7.649.9 118.1 213.5339.7503.2712.9
0
+ 3892.6 -‘r 4992.2 + 634a.9 i- 8004.1 + WA +12361.4
+ + + +
+ + + 259.5 + 152.6 + 11.8 + 171.0 + 407.2 +
399.9 401.3 421.8 418.2 390.2 337.7
i- 8917.0 +11217.6
+ +
+ +
t + f +
-+ +
125”
112
A. MICHELS,
M. GELDERMANS
AND
S. R. DE GRCIOT
$8. Th.e specific heat at constant volume. pression (7), for c, is written :
rv= (gj
Using
for U the ex-
= (c”)p=o + (fg . P
‘P
The second term (aU,/aT),, giving the increase of c, by compression, is called A c, ; the results are shown in the first columns of table IV. The second columns give the total value of the specific heat c,. TABLE Specific
-
IV
heat C,, in ml/degree
mol
I
0”
P 0 1 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 -475
c, 7,8;A&0
AC - v 0 w 1 7,a: 0,3; 2 a,11 O,& 5 8,6f
-
- -
0,Ol 0,36 0,84 1,31 I,75 l,a9 2,00 I,94 1,71 1,42 1,15 0,92 0,79 0,72 0,77 0,86 0,97 I,14 1,34 I,58
0 lo,59 1 Ai5’; 0 ’ ll,;e a,37 / ACTC, 0 a,91 1 k&0 9,46 1 A::, 0 IO,02 / AC’,‘“;, a,38 0,02 a,93 0,02 9,4a 0,Ol IO,03 0,Ol lo,60 0,Ol 1 I,19 a,73 0,41 9,32 0,40 9,86 0,33 IO,35 0,26 lo,85 0,16 1 I,34 9,21 0,al 9,72 0,73 IO,19 0,5a IO,60 0,44 11,03 0,29 II,47 9,6a 1,19 lo,10 1,02 lo,48 0,79 lo,81 0,sa 11,17 0,40 11,5a IO,12 I,53 IO,44 I,25 IO,71 0,95 IO,97 0,69 11,2a 0,4a ll,66 lo,26 1,75 IO,66 1,49 IO,95 1,09 I I,11 0,77 ll,36 0,53 II,71 IO,37 I,86 IO,77 I,56 11,02 I,14 II,16 O,a3 11,42 0,60 11,7a IO,31 l,a4 IO,75 I,60 II,06 I,17 11,19 O,a6 11,45 0,65 I I,83 lo,08 I,74 IO,65 1,5a 11,04 I,17 II,19 o,a7 11,46 0,66 ll,a4 I,53 IO,99 9,79 I,60 IO,51 1,la 1 I,20 O,a7 11,46 0,6a ll,a6 9,52 I,46 IO,37 I,53 IO,99 1,la II,20 c,a9 Il,4a 0,70 11,aa 9,29 I,37 IO,28 I,55 II,01 1,la 1 I,20 0,92 I I,51 0,71 1 I,89 1,Sa I I,04 I,24 II,26 0,97 I I,56 0,77 Ii,95 9,16 I,33 IO,24 9,09 I,36 IO,27 I,64 11,lO I,33 II,35 I,04 1 I,63 O,a2 12,OO 9,14 I,46 IO,37 I,75 I I,21 I,43 1 I,45 1,ll II,70 O,a7 12,OS 9,23 1,63 IO,54 I,92 11,3a I,58 I I,60 I,22 1 I,81 0,94 12,12 9,34 I,84 lo,75 2,15 I I,61 1,76 II,78 I,35 11,94 1,04 12,22 9,51 2,09 II,00 2,4l 1l,a7 I,96 II,98 I,51 12,lO 1,2012,3a 9,71 2,35 II,26 2,72 12,la 2,19 12,21 I,69 12,2a I,39 12,57 3,02 12,4a 2,49 12,51 1,90 12,49 I,61 l2,73 9,95 2,67 II,58
$9. The entropy. The entropy was calculated the path indicated by b in $4 and fig. 1. The expression for S is:
integrating
along
Using
THERMODYNAMICAL
PROPERTIES
the definition
(3) of internal (g)idr.l,t=
OF
113
ETHYLENE
functions
-and the relat’ion
-+
this reads : SK PI =*/ ‘*
dT -
R In p + &(t, p) -
S&, 1)
(16)
where .w, P) -j{(i).-
(g)i,a,
,)dp =/{-if?
0
(&),t-
+j dp.
(17)
0
For numerical calculation the values of c, at p = 1 are needed. Those for cp at P = 1 are known 2). Substitution of the series expansion of K a m e r 1 i n g h 0 nn e s, PV = A +- B . v-’ + C . P + . . . . in the thermodynamical relation cp-c,
= -T(g(g-):
(18)
gives :
2 ++; (
p2+
-%!?!?-A$
g
. .
I
1
. . (19)
This series expansion converges rapidly in the neighbourhood of p = 1, the second term being of the order: 0,05 cal/mol. degree and the third 0.001 cal/mol. degree. As the error in cp is about lo/, the third and following terms of the series expansion may obviously be neglected. The second term is easily found from data of the second virial coefficient B(T) published before I). Table V gives the values for SJt, p) and S. When t’ = 0” C, p = 0 is taken as the zeropoint the difficulty arises that all total entropies become infinite (with a negative sign). To get the expression for total entropy with zeropoint 1 = 0” C, p=OtoS,theform:’
! as
0”
I( 1
Physica
ap XII
I=0
dp = -[[R
In p]: f $(O,l)
= -
00 f- S,(O,l)
(20) 8
0 I
450 475
50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425
25
P
-
-
T -
p S
-
0 +w -0,Ol. 4 0 - 0,36’ 9- - 6,75: - 0,75: 2- -8,511
S {CT,
-
.-
0 += 0,014 + c,7130,3426,0120,680 7,728 1,OOy 8,862 1,322 9,747 1,614-10,4821,882-11,1122,132-ll,6692,37712,1792,631 - 12,667 2,905 - 13,151 3,210 - 13,645 3,557 - 14,165 3,946 -- 14,713 4,377 - 15,291 4,854 - 15,905 5,375 - 16,554 5,936 - 17,236 6,536 - 17,951 7,173 - 18,694 -
--
-
3,848 4,276 4,743 5,251 5,795 6,376 6,990
-. -
c G,013 i 0,3120,615 0,907 1,1871,4551,713-lO,2451,9672,2252,495 2,787 3,106 3,459 -
s(a)
13,917 14,492 15,097 15,732 16,398 17,092 17,813
10,80611,3Z 11,833 12,335 12,843 13,369
+ca 1,4125,284 6,964 8,0628,9149,625-
Entropv
= SW)
-
-
V
-
3,736 4,155 4,611 5,102 5,627 6,183 6,771 0,014
-
0 0,011 + 0,282 0,557 0,824 1,084 1,3371,5891,8432,104-10,5232,380 2,676 2,998 3,351 -
in ml/degree
TABLE
-
13,120 13,686 14,279 14,899 15,544 16,214 16,909
11,033 11,539 12,050 12,576
+=J 2,0994,569 6,221 7,294 8,1268,8229,4369,997-
00
mol
2,291 2,588 2,909 3,258 3,636 4,046 4,490 4,966 5,474 6,013 6,577
--
0,763 1,009-1,2531,4991,7522,013-
-
-
0,258 0,512
-
0 O,OlO+ -
10,266 10,774 11,284 11,805 12,343 12,900 13,480 14,086 14,714 15,367 16,038
-
6,556 -. 7,374 8,061 8,669-9,2299,755-
3,868 5,500-
t-W 2,777-
-
1,191 1,4341,6841,9452,221 2,518 2,838 3,184 3,558 3,963 4,399 4,867 5,363 5,889 6,439
-
-
+
0,235 0,475 0,715 0,953-
0 0,009
7,3307,9448,4929,0189,528 10,035 10,544 11,063 11,597 12,148 12,721 13,318 13,934 14,574 15,230
3,176 4,793 5,839 6,649-
+CC 3,447-
-
-
1,1461,3861,6341,8952,171 2,466 2,785 3,127 3,495 3,894 4,323 4,782 5,270 5,784 6.321
0 0,008 0,217 0,448 0,6800,912-.
-
-
-
+ -
6,622 7,224 7,780 8,305 8,814 9,320 9,828 10,343 10,870 11,416 11,982 12,570 13,177 13,806 14,449
+m 4,111 2,495 4,103 5,141 5,945
THERMODYNAMICAL
PROPERTIES
115
OF ETHYLENE
.
has to be added where -
00 is to be considered as the limiting iii;
(R In p).
(21)
9 10. The free energy. The relation F = U for F. The results are given in table VI. TABLE
p 0 I 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 _-~
1
0” --00 0 + 1651,3 + 1933,9
/
25” --cc 9,9 + 1810,5 +2136,5 + 2297,7 +2395,1 + 2459,4 + 2505,6 + 2540,O + 2567, I + 2589,6 + 2609,9 + 2628,9 + 2650,2 + 2675.2 + 2706,l + 2746,9 i-2800,9 + 2872, I + 2964,6 + 3082,o
)
value :
VI
Free energy-. in d/m01 50” / 75” 1 L-3 37,2 + 1951,l +2319,3 + 2508,3 + 2626,? + 2706,4 + 2767,7 +2a16,4 +2a56,2 + 2890,9 t- 2923,a +2955,5. + 2989,a +3028,3 +3074,1 $-3130,7 + 3200,9 +3289,2 + 3398,7 +3534,4
--co al,6 +2074,1 +2484,0 + 2700,O + 2839,9 +2938,3 +3015,3 t-3078,1 +313o,a +3178,4 +3223,5 +326a,l +3315,4 +3367,6 + 3427,6 +3499,2 +3585,5 +3689,a +3815,9 + 3969,4
a,=--4,s a?,=-3,9 a,,=--3,6-a,,=-3,2 in Ffa) = F(p) - a.1 - T(-0,014
TS gives the value
100”
-co 143,l +2179,3 + 2630,6 + 2873,3 +303&S +3150,7 + 3242,7 +3319,7 -f-3385,6 +3446,0 +3504,1 +3561.5 +3621,4 +36a7, I +3760,8 + 3846,4 + 3948,2 + 4068,3 +4212,1 +43ai,a -.__
1
125”
)
-cc - 221,3 + 2265,2 + 2756,6 + 3026,2 +3207,1 +3341,4 +3452,5 +3539,4 $3618,6 +3691,4 + 3762,l +3832,4 +39OS,8 + 3985,2 + 4072,9 +4173,5 +4290,6 i-4425,8 + 4585,5 +4771,7
150” --co - 316,l + 2334,O + 2865,9 + 3 16 I ,3 +3362,1 +3513,4 +3636,2 +3741,2 + 3833,5 +3918,9 + 4002,2 + 4085,s +4171,7 + 4263,6 + 4365,2 + 4479,6 +46lo,a +476l,Z + 4937,0 +5139,6
a,,,=-2,91a,,,=-2,5a,,,=-2,i -- CO) = F(p) + at + m
5 11. The enthalpy. Similarly the enthalpy w=u+lv. Data are tabulated in table VII.
is obtained
from
$ 12. Thermodynamic eotential. As G = U - TS + PV, the thermodynamic potential is derived from the free eneigy by adding PV. Table VIII shows the results. § 13. Discussion. The accuracy of the isotherm data is & 1 in 10.000. Hence the accuracy obtained in Ui and on the average in
116
A. MICHELS,
M.
GELDERMANS
1
Enthalpy 50” )
AND
TABLE p
1
0
0” + + + -
I 25 50 75 100 125 150 175 200 225 250 275 330 325 350 375 400 425 450 475
/
25”
550,9 538,7 253,0 30,4
+ 803,O + 791,5 +- 522,6 + 260,6 + 18,7 -- 201,2 - 398,3 - 572,7 - 728,5 - 869,6 - 999,3 --I 117,3 - 1220,9 - I302,8 --1351,6 --1352,9 -1290,2 --I 144,o - 892,9 - 515,4 + IO,6
+ + + + + + + +
IO68,7 1057,8 806,O 562,5 338,6 134,4 53,1 220,o 371,4 509,2 634,l 744,l 835,2 899,5 926,1 901,3 808,4 629,l 343,0 70,9 634,8
S. R. DE
GROOT
VII in cnl/mol 75” / + + + + + + + + + + +
1348,l 1337,9 1102,8 875,5 666,2 475,0 299,5 140,8 3,7 134,3 250,4 349,l 425,3 471,3 475,7 424,a 302,7 91,o .=9,7 680,l 1281,9
1000 + + + + + + + + + ‘+ + + + + + + + +
1
1641,3 1631,9 141 I,7 1198,3 1001,5 820,8 656,0 505,6 369,2 247,2 141,5 55,3 5,4 32,6 15,l
60,5 210,O 451,4 804,l 1288,O 1924,6
125” + + + + + + + + f + + + i+ + + + + + + $-
)
1948,6 1939,8 l732,4 1530,4 1343,8 1172,5 lOl6,2 874,O 745,9 633,2 538,2 464,9 419,5 410,5 448,7 547,4 722,7 992,2 1374,3 I888,3 2555, I
1504 + 2270,4 +2262,0 -I- 2065,4 + 1873,2 + 1695,6 + 1532,4 + 1384,O + 1250,2 -t ll31,O +1027,6 + 943,6‘ + 883,l + 852,7 + 861.3 + 919,9 + 1041,3 + 1241,O + 1536,4 + 1946,3 -+ 2489,3 +31a5,5
W(Q) = WC/3 -8.1 TABLE p 0 1 25 50 75 100 125 150 175 200 225 250 275 330 325 350 375 400 425 450 475
(
0’ --cc3 + 538,7 +2097,3 + 2294,6
a,=-4,3
1
Thermodynamic 25” 1 50”
VIII
potential 1 75”
in cal/mol 1 1000
1
125’
--00 -cm -m -co --cc $ 578,9 + 601,6 + 607,2 + 595,6 + 567,4 +2315,1 +2513,6 + 2693,5 + 2855,0 + 2996,9 + 2564,7 +2812,9 +3041,4 + 3250,7 j-3438,7 + 2660,9 + 2943,8 + 3205,7 +3447,9 + 3668,6 + 30i9,% + 2704,9 +3014,9 +3304,1 +3572,4 f 2726,9 +3057,2 + 3370,9 +3663,9 i-3934,6 + 2740,4 +3090,6 +3425,9 + 3740,4 f 4036,9 + 2750,6 +3120,6 +3476,8 +3813,0 +4127,0 +276l,6 +3151,7 +3529,2 + 3887,3 -t 4223,7 -l-2777,4 -l-3189,7 + 3590,7 +3972,3 +4331,8 + 2803,7 +3242,0 +3668,2 + iO75,6 + 4460,3 +2847,4 -+3315,0 i-3769,9 + 4205,2 +4617,6 +2920,5 + 3420,7 +3907,0 + 4372,4 4 48 15,2 +3035, I +3571,2 + 4092,O + 4590,7 + 5066,O + 3206,l +3781,8 +4340,0 +4074,1 +- 5384, I +3451,9 -+- 4070,2 + 4668,5 + 5240,O +-5787,6 +3791,6 +4454,7 + 5096,l +5707,6 + 6294,8 + 4246,0 + 4956, I +5641,4 + 6294,6 + 6922,l + 4836,7 + 5594,2 + 6325,0 + 7022,2 + 7690,9 + 5584,2 +6391,1 t-7168,7 + 7909, I +8618,9 ---~. t&=-3,9 a&,=-3,6 a,,=-3,2 al,,=-2,9a,,,=-2,5a,,,=--2,: in G(a) = G(b) + at + co
)
150” --co + 522,5 +3121,2 +3609,4 $- 387 I ,o + 4048,O +4186,1 + 4307,o +4423,1 +4541,9 + 4673,3 + 4826,9 +5011,4 + 5237,9 +5519,5 + 5872,0 3-631 I,2 + 6855,4 + 7522,l +a331,3 + 9299,6
THERMODTNAMICAL
PROPERTIES
OF
ETHYLENE
117
Ki may be estimated to be of the order of 1 in 1000. The accuracy in U and W varies with growing density between 1 and 4 cal/mol.
at low temperatures and from 7 to 9 cal/mol. at high temperatures. The error estimated in AC, is f 50/5, in c,, 1Oh. The order of accuracy in .Si is 5°/oo and varies in S, F and G from I-10 per mille with increasing temperatures. Received
March
3 rd.
1946
REFERENCES 1)
2)
.4. MiLhcls and AI. Geldcrmans, Physica !), 967, 1942. MTith the values from this paper (PY),,oo, p=l = 0,999896. A better approximation to the correct value (PV)+oo, p=l = 1 is 0,999996 and is obtained by changing somewhat the calculated normal volumes and the series coefficients. The normal volumes mentioned diminish 1 in 10000. In tables IIa, 116 and III the absolute value of A increases 1 in 10000, of B 2 in 10000, of C 3 in lOOOC, of Z 4 in 10.000, of D 5 in IOCOO, of Y 6 in 10000 and of E 7 in 10000. In table I the values of 3~ (named p here) decrcase 1 in 10000, those of PV increase 1 in 10000. In table IV only th: values of dA decrease 1 in 10000. For 0” C the values for the low pressure series (table 11~) are now: il = 1.007582. R = - 7.60211 1Om3, C = 16.474. 10-e and D = 126.8. 10-12. A. E o c 1~ e n and A. P a r t s, Z. phys. Chem. B. 20, 184, 1933.