ressure-Temperature Chart for Vapors 31. HIRSCH lOha Lee Road, Blackheath, London S. E. 3, England
ARIATIOS of vapor pressure with temperature is given by the Clapeyron equation,
V
where p T L v,,
J
vapor pressure = absolute temp. = latent heat of vaporization = sp. vol. of vapor and liquid, respectively, a t temp. T = mechanical equivalent of heat =
01
logp
where A
JLM 2.3026 RT + log - AH B
(& h)
+
J LSf 2.3026 R T O
____ 2.3026 R
=
log
(1 - - 3
(4-4)
pb
1 760 1.033228 14.696
Numerical Value of c* = L o g q , 0 2,88081 0,01420 1.16720
+
JLM
=
CTo
A new method of plotting pressure against temperature is shown by Figure 1. The y axis is laid off in cycles of logarithmic scale and the z axis in uniform algebraic scale. The isobaric abscissas are horizontal lines. The isotherms form a pencil of lines converging at a point on the y axis for which C. The z axis intersects the y axis at a point atlog p = t,ributed to a value log p = Cb = log p b . The point of conl’ergence lies a t a distance C above the origin of coordinates. Equation 4 is represented by a Yertical line intersecting the 5 axis at the value T = Th which is the normal boiling point. The new method of plotting hss the advantage over methods previously suggested in that interpolation on an algebraic temperature scale is simpler than on a reciprocal temperature scale.
(3)
+
=
=
,)
Unit of pressure Atmospheres Mm. of mercury Kg, per sq. om. Lb. per s q . in.
-
log PO
where C b = log pb has the numerical value given by the following table for different systems of units of pressure:
Equation 2 may be written
or
A
logp=cb+C
=
logp =
=
The lower limit of integration marked by subscript can be taken a t any fixed point such as the normal boiling point or the critical point. Experimental values for the critical point are frequently open to doubt. It seems, therefore, inadvisable to introduce the critical data into the formula and preferable to take the normal boiling point as the lower limit of integration. With suhscript for the normal boiling point, Equation 4 becomes
molecular weight R = the gas constant
where M
CJ
JLX c = 2.3026 RTo
sures using convergent isotherms results in a straight line parallel to the pressure axis where the familiar log p - l / T plot is linear and in a straight line inclined to the pressure axis where the log p - l / T plot is curved. The underlying equation is related to the Clapeyron and van der Waals expressions and to Trouton’s rule, a correction is applied near the critical temperature, and the method is illustrated using data on ammonia, water, carbon dioxide, sulfur dioxide, oxygen, ethane, butane, and the fluoro refrigerants.
Over small temperature ranges the latent heat can be assumed constant. If the vapor conforms to the ideal gas law and the volume of liquid is negligible compared to the volume of gas, the Clausius equation can be integrated. It follows that JLi1f log p - log po = 2.3026 R ~
where
A new method of plotting logarithmic pres-
JLM 2.3026 RTo
A and B are constants over the small temperature range. Equation 3 is represented by a straight line if the loyarithm of pressure is plotted as a function of the reciprocal of absolute temperature. For engineering purposes this semirational straight-line law has proved useful for finding coordinated values by interpolation and even by extrapolation beyond the range over which the properties have been nieasured. The following form of Equation 3 is more convenient for the present investigation logp=-T+ Ac C F + C or log p =
cd
+ c (1
-
p)
Trouton Rule Trouton’s rule presumes Tr
(4) 174
=
MLb ~
Tb
=
a constant
If Trouton’s rule is true and if, over a certain range near the normal boiling point, C is a true constant with a value C = JLbM/2.3026 RTa, Equation 48 can be written
INDUSTRIAL AND ENGINEERING CHEMISTRY
February, 1942
I
I I
\
I\
\
\ \
\
\I\, \ \ I \, \
A \ \
\
\
\ \
\
\
FIGURE 2
FIGURE1
(5)
For the approximate value Tr = 22 and R / J Equation 5 becomes: log p =
cb
(
f 4.8 1
-
3 -
=
1.9869, (5A)
The point of convergence is thus found a t a constant distance of C = 4.8 logarithmic units above the atmospheric pressure line which is the x axis, and the pressure-temperature correlation for any vapor near the boiling point is represented by vertical lines intersecting the x axis a t the normal boiling point. However, Trouton's rule is approximate only. Value C therefore varies for different vapors. Thus Equation 5A can make no claim t o rationality, and data found from vertical lines in a universal chart designed for an invariable C value are not sufficiently precise. They may be suitable a s a first approximation for the solution of engineering problems involving a vapor where only the normal boiling point is known if the considered range lies near this temperature. Data with a higher degree of approximation can be found from different charts each designed for a special C value which is characteristic for the individual vapor and allows for the variation of Trouton's constant. When vapor pressures and temperatures have been measured for two different points as (pl, T1)and ( p 2 , T2) the individual value C can be calculated from Equation 4: log Pz
- log
175
most vapors do not conform to the ideal gas law which had been presumed for integrating the Clapeyron equation, and only for temperatures considerably lower than the critical temperature can the liquid volume be neglected in comparison with the vapor volume. It is obvious that any degree of consistency can be obtained by the addition of correction terms to Equation 4. The resulting complicated formula, though fully representing experimental data, is evidently unsuited to engineering calculations. The present investigation is intended to determine whether a simple formula using only one correction term and representing a straight-line law would satisfy engineering and possibly scientific requirements.
Clapeyron Equation The Clapeyron equation can be integrated if an equation of state of the hypothetic form,
is consistent, where dTb is a corrective temperature which is constant and characteristic for the individual vapor, and T is a constant with the dimension of temperature. Its value follows from Equation 7 by introducing the data due to the normal boiling point:
Substituting the value of becomes
Pl
which can be rearranged to But even such data may not be sufficiently exact since both C and the latent heat of vaporization can be considered as constant over only a small temperature range. Besides,
T
from Equation 74, Equation 7
INDUSTRIAL AND ENGINEERING CHEMISTRY
176
The form of this equation is similar to the gas law which can be written
where ( ~ u ~ /=T ()R~/ M ) and subscript 1 denotes any fixed point. I n both formulas the values in the three parentheses have similar dimensions of an individual gas constant, absolute temperature, and density, respectively. Equation 7C is consistent for the normal boiling point for which it gives p = pb. For the critical point it becomes
in the third parenthesis is I~
found from the Clapeyron equation,
Vol. 34, No. 2
or with the normal boiling point as the lower limit of integration,
($)b
=
slope of p
-
T curve at normal boiling point
Equations 9 and 9A are similar to Equations 4 and 4A. The only difference consists in the corrective temperature ATb. The constant c' is similar to c = JLbM/2.3026 ETb. Apart from the corrective temperature AT,, the gas constant ( R / M ) = ( p v , / T ) ~is substituted by the value PI, T ( - ' l b ) ( T , - ATb)" both of which have the same dimension. Equation 10 can be rearranged as
whereby ( d p / d T ) , is the slope of the p - T curve a t the critical point; hence and can be compared with
It may be pointed out that the values in the three parentheses are not true values of gas constant, temperature, and density. They are modified true values corrected in such a way that the ideal gas law is satisfied. Equation 7C may be compared with the Tran der Waals equation :
While van der Waals allowed for the cohesive force between molecules by introducing an additive correction for the pressure, Equation 7C introduces in the second parenthesis the value AT6 as a subtractive correction for the temperature. The corrective effect is similar for the two methods. Similarity also exists for the correction of the volume in the last member. No further theoretical argument is a t present claimed for Equation 7C which is presented here only as a convenient basis for a simple chart which is sufficiently in accord with experimental data to be suitable for engineering purposes. For extreme conditions near absolute zero temperature, Equation 7C is consistent only for At6 = 0; otherwise zero pressure would be obtained not a t zero temperature but a t a temperature Atb higher than zero. Whenever, therefore, a value of ATb is found other than zero, this value cannot be considered as a true value for temperatures near absolute zero. By combination of Equation 7 with the Clapeyron equation it follows that dP = p
r
(T
as found before. The first member of Equation lOA, Lb.v/ (Tb - AT,) may represent a modified value of Trouton's constant whereby the corrective temperature ATb should allow for variation in the conventional Trouton's value and should possibly make the corrected value a true constant. The third member of Equation 10A represents a modified value of J / R which is different from the universal value, 1/1-9869, and characteristic for the individual vapor. Figure 2 illustrates the method of plotting the correlation given by Equation 9A in a chart with coordinates similar to those of Figure 1. Equation 9A is represented by a straight line which, however, is no longer parallel to the y axis. The slope is given by the straight line being parallel to the isotherm for the corrective temperature ATb. If cocrdinated values of pressure and temperature have been measured for three points (one of which may be the normal boiling point), empirical values of C' and AT, can be found from Equation 9A. With the abbreviations 21
=
TI
--Tbcb and xz = Tz - Tb 1% pZ - Cb
log PI
it follows that
dT
-
ATb)' ATb = T I - XiC'
and by integration
=
Tz
- XZC'
(11-4)
Data on Vapors Introducing the constants C' =
7
2.3026 (To - AT,)
and
co = log PO,Equation 8 may be written,
! o g p = co
+ C' (1
-
'To 1 f2) + C' ( TyT -)- ATTo = co
(9)
The following discussion is concerned with different vapors the properties of which have been measured, and has for its object to determine whether data read from the proposed chart are in good agreement with previous findings. Special vapors are studied by way of example only. The results can therefore be neither complete nor definite. The method of discussion varies in order to illustrate the possible degree of
The corrective temperature AT, was calculated from the above data by presuming hypothetical Equation 7 to be consistent over the range of two following temperatures of the
deviation from the observed data, and different systems of units have been used in order to show t h a t the method can be applied to any consistent system. The usual procedure for tracing such charts should be as follows:
table. With the abbreviation y = p T ("
ATb =
TABLE I. VALUES Temp. range, ATb F. Ehuation 7 Equation 11 a Average.
\I\
\I
\
FIQURE3
=
25.5743247
-3295 1254 T
6.4012471 log, T -
0.0004148279 T
C' =
+ 0.0000014759945 T 2
+
08
-107.80" 60 40
--
4
0
40 80 120 271.4b Triple oint. CritioaPpoint
Volume, Cu. Ft./Lb. Liquid, Vapor Pressure Lb./Sq. l%. ut vu 0.88 0.02182 44.73 0.02278 5.55 24.86 10.41 0.02322 9.116 0.02419 30.42 0.02533 3.971 73.32 1.955 0.02668 163.0 1.047 286.4 0.02836 0.0086 1667 0.0686
....
F.
OF
--40 60 to
(13)
49.0 50.82
AT&FROM EQUATIONS 7 AND 11
-40
to 0
53.2 55.60
(
~
-
0 t o 40
t o 80
40
80 to 120
60 to 120
53.0 53.85
45.5 52.16
34.6 45.70
46.7a 53.74a
~(log73.32~- 1.16720) f = 4.574 ~
log pt? = 1.16720
where p = pressure, lb./sq. in. T = t 459.58 = absolute temp. R. L = heat vaporization, B. t. u./lb. J = 778.26 = mechanical equivalent of heat, ft.-lb./ B. t. u. v, = sp. vol. of vapor, cu. ft./lb. vl = sp. vol. of liquid, cu. ft./lb.
Tern . t ,
Tn - TI d&l- 1
Figure 3 is the chart based on the data T , a= 431.55, C' = 4.574, A T , = 53.74. If this straight line is extended beyond the range from -60' to 40" F. as far as to the triple point and to the critical point, the following pressures are found from the chart. For the triple point:
The latent heat of vaporization had been calculated from the Clapeyron equation,
An extract of the original data follows:
TI-
-
Properties of ammonia have been elaborately measured by the National Bureau of Standards (I), and the following equation was formulated :
'
there follows
Between -60" and 40" F. the value of AT, (from Equation 7) is practically constant, with an average value of 52.0" F. As far as AT, could be considered as constant, for ammonia the hypothetical Equation 7 would be consistent. The proposed chart would then prove suitable as an easy means for representing the pressure-temperature relation with sufficient accuracy over a considerable range. Values of AT, calculated from Equation 11 are given in the third line of Table I. The average value over the range vJ/L from -60' to 40" F. is 53.74" F. The values pT (vu entering into Equation 13 are identical with the values p / ( d p / d T ) since L had been calculated from the Clapeyron equation. Equations 13 and 11 are therefore both based on pressure-temperature data only, and equal values of AT, should fdlow when calculated from one or the other equation. The discrepancies between the values of ATbfrom Equations 7 and 11 are due to slight inconsistencies in the data and are smaller than the usual allowances for engineering calculations. The value AT, = 53.74 found as average between -60' to 40" F. from Equation 11 has been considered as sufficiently consistent for the following calculations. C' calculated from Equation 12 for 40' F. is:
Ammonia
TrRR \
- "),
The result is given in Table I (second line).
The value of AT, fOllOWS automatically without further calculation.
I
L
from Equation 7,
1. Calculate C' from Equation 12A. 2. Design the system of coordinates for that value of C'. 3. Mark experimental values in that system. 4. F i n d t h e characteristic line by.connecting those points.
log p
177
INDUSTRIAL AND ENGINEERING CHEMISTRY
February, 1942
= 0.94181
- 431.55 + 4.574 (351.72 351.72 - 53.74)
-1
pt, = 0.8746 pound per square inch
compared to 0.88 pound per square inch as given in the Bureau of Standards table. The difference is insignificant as far as engineering problems are concerned though the range of extrapolation below -60" F. is conLatent Heat L, siderable, B. T.U d b .
eid .'8 597.0 568.9 530.2 498.7 455.0 0
For a temperature of 120' F: log PI20 =
1.16720
- 431.55 = 2.45483 + 4.572 (579.58 579.58 - 53.74 )
= 284.99 pounds per square inch
~
~
INDUSTRIAL AND ENGINEERING CHEMISTRY
178
Vol. 34, No. 2
or (C - C') X
FIGURE -1
which is only 0.49 per cent lower than the true value of 286.4 pounds per square inch though extrapolation went 80' F. beyond the upper limit of range for which constants had been calculated. For the critical point: log pe
1.16720
730.98 - 431.55 + 4.574 (_730.98 - 53.74)
=
3.18952
p c = 1547.1 pounds per square inch
This is considerably lower than the true value of 1657 pounds per square inch. Simple extrapolation up to the critical point apparently is not reliable. Fortunately it is possible to allow for this discrepancy simply by using a corrected value of AT, for and near the critical point. Cox (9)found that for hydrocarbon and various other vapors the value C in the general Equation 4 4 log pat,.
=
c (1 -
2
Equation 1 5 represents a hyperbolic relation between C and T,. It is evident that constants C' and AT, can be found such that over a moderate range the hyperbola represented by Equation 18 coincides with the parabola represented by Equation 14. According t o Equation 14, C for the critical point is equal to the value for T , = 0.85 and passes through a minimum for TR = 0.925. The discrepancy between the results of Equations 14 and 15 becomes, t h e r e f o r e , app a r e n t mainly over the range 1 > T n > 0.85. For T , = 0.85, E q u a t i o n 1.5 gives
C' 1 - - A Tb
0.85 T ,
FIGURE 5
and for T , = 1,
follows the parabolic relation between log C and T,: log C = log C,
+ E (1 - T R ) (0.85 - T R )
(14)
where T , = T / T , is the reduced temperature, and C , and E are constants characteristic for the individual vapor. The modified Equation 98,
would be identical with Equation 4A if
There results a modified value, AT,' = ilT,:0.85, for calculating critical data from Equation 9 9 while the value AT, may be considered consistent up to a temperature T = 0.85 To. When using the modified corrective temperature AT,' = 53.74/0.85 = 63.23' F. for ammonia, the critical pressure is found : log PC = 1.16720
+ 4.574 (730.98 730.98 - 431.55 63.23 ) = 3.21826
p, = 1653.0 pounds per square inch
INDUSTRIAL AND ENGINEERING CHEMISTRY
February, 1942
179,
4.2747 -I\
KuprianoH 18)
\
\
\
\
\ \
\ \
FIGURE 7
FIGURE 6
Compared to the true value of 1657 pounds per square inch the difference of 4 pounds is only 0.24 per cent and insignificant for engineering calculations. I n Figure 3 the main straight line is consistent up to the limit temperature 0.85 T , = 621.33' R. or 161.75' C. For practical purposes it may suffice to connect this point with the critical point by a straight line. The critical pressure is found graphically without calculation as the point a t which the isotherm for the critical temperature intersects the straight line running from the boiling point parallel to the isotherm for AT,' = AT,/O.85. With reference to Figure 4 the straight-line relation for temperatures between 0.85 Toand T , can be represented by the formula,
n-Heptane and 2,2,4_Trimethylpentane An equation similar in form to 9A was used by Smith (10)) who found experimental values consistent with the formulas:: for n-heptane: log p = 6.905113
1269.821 - 217.110 +t
for 2,2,44rimethylpentane: log p = 6.820137 -
where p t
= =
pressure, mm.,H temperature,
1262.707 221.307 -t t
8.
These equations can be written: n-heptane:
log p = 2.88081
+ 4.02430
1 -
"h5:
i6E05
2,2,4-trimethylpentane: where, with the abbreviations z = log p , log p0.85 Tc - log p b :
- log p ,
and z' =
(17) aTo'
=
- 2') - 2'
Tb (2/0.85 z
To' corresponds to the temperature a t which the extended straight line intersects the z axis. ATo' is the temperature due to the isotherm which is parallel to the extended straight line. A treatment similar to that for ammonia was given to pressure-temperature data for n-butane, sulfur dioxide, methyl chloride, water, and ethane and is recorded in Table 11. AT, and C' were computed from entries marked in Table I1 by using Equations 11 and 12A, respectively. Calculated pressures, other than the critical, were found from Equation 9A, while the modified corrective temperature, AT,' = ATb/0.85, was used in calculating the critical pressures. '
l o g p = 2.88081
+ 3.93933
372.39 - 51.85
where 371.59' and 372.39' K. are the normal boiling points, 4.02430 and 3.93933 the values of C', and 56.05 and 51.85 the values of AT,, respectively. Values stated by Smith are evidently identical with those resulting from the proposed chart.
Carbon Dioxide The properties of carbon dioxide make the applicability of the proposed chart doubtful. The critical temperature of 31' C. is low. The limit value of 0.85 T , = 258.52' K. comes near the triple point T,, = 216.56' K. or t,, = -56.6' C. The available range for finding values of C' and AT, is, therefore, as small as 258.52 - 216.56 = 41.96' K. I n addition, the saturation temperature a t atmospheric pressure of 194.6" K. is not the true value of the normal boiling point since carbon dioxide is solid a t atmospheric pressure., If'the liquid could be supercooled down to atmospheric pressure, the t r u e
Vol. 34, No. 2
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
180
DATAFOR VAPORS TABLE 11. PRESSURE-TEMPERATURE Vapor
Eeferenae, Figure
Pressure Obsvd. Calcd. Pounds/sq. in.
Temperature F.
14.7
. I .
17.2
17.7
Deztion
...
-2.8
2
/9000 8 6
ATb = 45.4' R.; C' = 4.138 Sulfur dioxide
----
hiehl (6)
0
c.
51.18b 40 24.95 19.7 11.17 9.96Q 5.4 0.2 15 22.65
4
M m . mercury 80.2 162.4 372 483.2 721 760 924.8 1172.7 2070 2698
... ...
...
... ... ... ~ . . ... ... ...
KQ./SQ.cnt. 50 8.350 8.478 157.ZC 77.7 79.0 ATb = 39.4' K.; C' = 4.325 O c. Kg./sq. om. Riedel (91, Fig. 9 50b 0.118 ... 20a 3.370 50 8.583 8.'566 157.2O 77.7 81.0 ATb = 37.2O K.; C' 4.380 ' F. Pounds/sq. in. Tanner et al. (11),Fig.Q 80 1.963 1.956 - 405 6.878 20 11.71d ii:ige - 10.660 14.70 0 18.9Od 18:84S 80 86.26 85.66 17Ob 283.9 289.6c 969.2 914:i 3Tb = 56.71' R.; C' = 4.0777
-
-
Methyl chloride
-
c. Water
c. Plank and Kambeitz (71, Fig. 9
--100b 90 - 88.6" 80 - 70
- 15b 60
-
-.,.
-
I
1.5 1.7
2
I
/
,
/
,,,
1
1000
B 6
4
... ...
-0.2
4.2
\ \ 2
\
\I
i
Q
0
L
I I
'i
I\
0.2
...
-0.2
.. . 4
-0.3 -0.7
...
-5.6
*. 4:&7 5.122 5.258 5.398 5.541 5.689 5.837 5.990 6.145
... ... ... ...
225: 6
... ...
...
0:04 0.04
0.02 0.02
0.02 0.05 0.02 0.02 0.02
... ... ... ... ...
0.02
FIGURE 8
Ka./sq. cm. 0.5354 0.9596 1.0333 1.606 2.549 3.861 16.63 50.3
32.2C -110 120 - 130 - 140 -150 C' = 4.002 from -lOO' C' = 3.636 from 90' C' = 4.123 from 80' C' = 4.00Gfrom 70'
-
... ... ... ... ... ... ... ...
K g . / s q . cm.
Osborne and Myers (e), Fig. 5
0.006228 01 0.12578 50 1.033228 100a 4.8535 150b 4.985 151 5.120 152 5.257 153 5.397 154 5.540 155 5.686 156 5.836 157 5.989 158 6.144 159 6.5023 16Ob 15.857 200 40.560 250 87.611 300 168.63 350 225 65 374.15c ATb = 48.1' K.; C' 5.040
Ethane
... ...
... ... ...
... ... ... ... 48.6
... ... ... ...
... ...
-3.4
..... 0.2703 0.1249 ..... 0.0511 ..... 0.0180 ..... 0.0052 ..... ... and -15' data&. - and -50' datab. -- and -70" datab and -GOo datab:
ATb = 13.93' K.; ATb = 28.98' K.; ATb 7.20° K.: ATb 12.73O K.: Boiling point. b D a t a used t o calculate ATb and C'. C Critical point. d ATb from these pressures ie the irrational value -218.0' R. A modi5ed ATb' = ATb/0.74 instead of ATb/0.85 would bring t h e calculated critical pressure in agreement with the 0x erimental value. e A # i from these pressures is 51.3' R. Z Triple point.
boiling point would be found at a temperature lower than 194.6' K. These difficulties are partly overcome by UBing the triple point, instead of the boiling point, for calcuIating values of C' and AT,. I n Figure 6 an auxiliary z axis is used with an origin of coordinates a t the value log pt7 instead of log pb where pi,is the pressure at the triple point. The point of convergence lies a t a distance C' - (log ptr - log pb) above the origin of coordinates. Carbon dioxide data were revised ;by Plank and Kuprianoff (8); the following are extracted from their data:
INDUSTRIAL AND ENGINEERING CHEMISTRY
February, 1942
Temperature ' C. Pressure, kg.)sq. cm. a Triple point.
-50 6.97
-40 10.25
-30 14.55
-20 20.06
-56.6a 5.28
The values of C' and AT, calculated from these data are given in the following table, together with the calculated values of the hypothetical normal boiling point and of the critical pressure: Teomp. range,
C.
ATb
a
c.
C' -I (log p t r log Pb)
C'
Tb. tb
c.K.
p ; , kg./sq. om.
-
-5Oto -40 34.34
-40tO -30 27.37
-30 t o -20 20.53
3.4507 4.1591 185.52 -87.64 76.65
3.5714 4.2798 185.24 -87.92 76.75
3.6846 4.3930 184.95 -88.21 76.54
181
-50tO -20 27.98 3.5663 4.2747 185.31 -87.85 77.06
The value of Tb,the hypothetic normal boiling point, was calculated from the equation,
and is about 9' lower than the temperature of -78.6" C.at which the pressure over solid carbon dioxide is equal t o the normal atmospheric Pressure. The average value of p , of the critical pressure was calculated from the equation, log
4.2747 (304.1 - 185.31) = 1.8868 0*01420+ 304.1 - (27.99/0.85)
where T , = 304.1 O K. is the critical temperature. The overall value of p , = 77.06 kg./sq. cm. is 2.8 per cent higher than the true value of 74.96 kg. recommended by Pla&. This discrepancy is not surprising, considering the allowance to be made for the basic data of carbon dioxide. The true value
INDUSTRIAL AND ENGINEERING CHEMISTRY
182
mould have been found nhen using a value of the hypothetic normal boiling point of T , = 186.07' K. instead of 185.31 K. or a value AT,' = 27.95 0.8976 instead of 27.95i0.55. It may be pointed out that the calculated \-alues of p , Trary only slightly over the different ranges although the values C' and AT, vary considerably. This may he due partly to the fact that the tabulated data are not the original values measured by experiment but are calculated from empirical equations n hich closely agree n i t h the expeiimental results. When, therefore, a different Equation 9 is used for calculating the ~ a l u e sC' and AT, from the tabulated data, the resulting values may not be found constant over the whole range. However, the variation of one value should be compensated by the variation of the other value in such a v a y that interpolated and extrapolated data can be found by using constant values calculated for a m a l l range or as averages for LL wider range. The pressure a t the limit temperature of 0.85 T, = 258.52' K. = - 14.64" C. follows froin O
log
p2jg.j
p,,,
(258.52 - 18t5.31) + 4.2737258.52 = 1.3717 - 27.99
=
0.01420
=
23.53 kg./sq. cm.
The modified values of ATO' and To' entering into Equation 16 for the straight line between 0.85 T , and T , are found from Equations 17 and 18 to be 39.50' and 189.07' K., respectively. Instead of 0.85, the value of 0.8976, found to be more consistent for calculating the critical pressure, was used. Pressures calculated from Equation 16 follow, together with Plank's values: C. Pressure, kg./sq. Equation 16 Plank Deriation, yo Temp.
31
30
20
73.17 73.34 0.18
58.94
46.44
58.46: 0.82
45.95 1.07
10
0
- 10
35.86 35.54 0.90
37.05 26.99
CIIL
7.5.02
74.96 0.08
0.22
Thus, an error up to about 1 per cent occurs when a straight chart for carbon line is assumed between 0.55 T , and T,. dioxide is given in Figure 7 .
Oxygen Figure 8, a chart for oxygen, illustrates the applicability of the proposed method. Consistent data for oxygen were published by Henning ( 3 ) . They were used for calculating average values C' = 3.5245 and AT, = 5.963" C. from pressures of 25.17 and 629.13 mm. of mercury at 66.69' and 88.42' K., respectively. The pressure at the triple point, T,, = 54.33' K., calculated from these values is I .I1 nim. compared to a true value of 1.2 mm. The critical presure at I', = 154.36' K. is found to be46.4atmospheres compared to the true value of p , = 49.7 atmospheres. The result is satisfactory as far as the triple point is concerned and not inconsistent for the critical point, considering the critical piessure lying far beyond the range for which the values of C" and AT , had been calculated. The discussion of the proposed chart has been limited to characteristic examples which may encourage investigators to compare measured data m-ith values found from the chart and to discuss its practicability more definitely than this first study can do. For similar reasons the modified Trouton's constant has not been considered further. Future investigations may find it to be a true constant which then could be used for calculating the latent heat of vaporization or a consistent value of AT, for vapors for which the latent heat of vaporization is known. The examples indicate that if the values of C' and AT, vary, the increase or decrease of C' is compensated by a de-
Vol. 34, No. 2
crease and increase of A T,, respectively. Consistent results are thus obtained when using constant values of C' and ATb over a range n-hich differs from the range over which those d u e s had been found. For beat results C' and ATb should be determined from data as near the range of application as possible. Though no precise results can be expected when using an arbitrary value of C' considerably lower or higher than the true value, such variation may sometimes be suitable for comparing different vapors. One system of coordinates using one arbitrary value of C' may thus be utilized for representing pressure-temperature relations for a series of vapors of similar chemical constitution.
Universal Chart Such a chart is given by Figure 9. It goes a step farther by dealing with vapors of different chemical constitution. The arbitrary value of C' = 4.2 has been assumed. KO consistency may be expected for ammonia which has a true value of C'considerably higher than 4.2. Following a straight line down to the triple point, the change of slope occurs a t a temperature of about 30" C. which is considerably lower than 0.85 To. Although certain limitations are given for using such a universal chart, it is suitable for solving general problems where exact data are missing such as: 1. Flnding the approximate normal boiling point of iin unknown vaDor. 2. Staiing approximate pressure-temperature relations for a vapor the pressure of which is known only at one temperature, 3. Identifying a vapor by its approximate normal boiling point if its pressure has been meas-14'61 ured at onlv one temDerature. 23.40 A verticd line passing through the reference point ... represents the approximate relation for those three . .. cases in which data are known only for one point. If, in case 2, data are given for two points, a line with a fixed slope results and greater precision is attained. It will often be possible to presume the slope in case 3 and to foretell somewhat the probable constitution of the vapor. Interpolating data between twn points. /?. D. Extrapolating data beyond the measured range. For cases 4 and 5 individual charts are preferable and may give some help even for scientific purposes such as precise interpolation over a sniall range or extrapolation beyond the critical point in order to find the partial pressure of vapors dissolved in liquids at temperatures higher t'han the critical temperature.
Literature Cited (1) Bur. of Standards. Circ. 142 (1923). ( 2 ) Cox, E. R., IND. ENG.CHEX, 28, 613 (1936). (3) Henning, F., and Otto, J., PTGC.7th Intern. Congr.. R ~ f r i g . , The Hague-Amsterdam,, 1936,I, No. 3, 174 (1939). (4) Kay, W. B., IND. ENG.CHEM.,32,358 (1940). (5) Mehl, W., Bull. Intern. Inst. R e f r i g . , 15, 3 3 - 4 l h (1934). (6) Osborne, N. S., and Myers, C. H., J . Recezrch Natl. BZLT. Standards, 13, 1-20 (1934) (Research Paper 691). (7) Plank, R., and Kambeits, J., 2. ge3. Ktilte-lnd., 43,203 (1936). (8) Pl.ank, R., and Kuprianoff, J., 2 . ges. Kd(te-Industrie, 1, l ( 1 9 2 9 ) . (9) Riedel, L., Bull. Intern. Inst. R e f T i o . , 20, No. 4 , Annex No. 5 , B1-10 (1939). (10) Smith, E. R., J . Research S a t l . B I W .Standards, 24, 229 (1940).
(11) Tanner, H. G., Bonning, A. F., and Mathewson, W. F.. IND. ENO.C H E M .31, , 575 (1030).
Citrus Pectates-Correction An errnr has just come to our attention in this article in the March, 1941, issue. On Figure 4, page 291, the original latex should contain 35%, not 25%, rubber. Vir. E. BAIER
