I
BERNARD E. SALTZMAN Public Health Service, U. S. Department of Health, Education and Welfare, 1014 Broadway, Cincinnati 2, Ohio
Generalized Thermodynamic Properties of Diatomic and Triatomic Gases Simplified generalized nomograms and mathematical expressions permit complete and convenient portrayal of thermodynamic properties with minimal data. Methods illustrated for water and air may be applied to more complex materials
THE
thermodynamic properties of gases are basic data required for understanding combustion processes and other chemical reactions as well as for thermomechanical calculations. Unfortunately, such data are available for only a few substances, and over limited ranges. The irregular nature of these properties makes thermodynamic calculations laborious and requires considerable technical skill. I n this work, such data are presented in new simplified and generalized forms. Heat Capacity, Enthalpy, and Entropy
The ideal (low pressure) molar heat capacities of diatomic and triatomic gases are represented in Figure 1 a8 a function of absolute temperature, in the form of a log-log plot. These data were obtained for the most part from standard references, based largely upon recent spectrographic calculations. Air, CO, COZ, Hz,HtO, Nz,OZ COS, CSt, H C I , H C N , HzS, NO,
so2 Clz, HBr,
(9) (3)
HI
which coincides with T when reference curve is displaced to coincide with that of given gas CPw*= molar heat capacity at constant (low) pressure of reference gas a t temperature T, The values of a and b are given in Table I, with the temperature ranges in which the relationships hold within 1% accuracy (HF, 2%). Two overlapping ranges are given for carbon dioxide. When the tempeiature boundary between these two ranges is crossed, calculations must be stepwise. By using these relationships the thermodynamic properties may be expressed in terms of those of the corresponding reference substances. However, such expressions are valid only over the indicated temperature ranges. Enthalpy may be expressed as
H = s T I C p * d T - (H*
- H)T
(3)
where H = molar enthalpy, relative to 0 pressure and base temperature, To
(H"- H)T= correction term for nonideal deviations at high pressure For the present, low pressures are dealt with, where the correction term for nonideal deviations is negligible. An asterisk is used throughout this report to denote ideal (low pressure) properties. Combining Equations 1, 2, and 3, the change between two temperatures is :
Hz*
- Hi* =
sTl T 2
C*,
dT =
The integral on the right side is the ideal enthalpy of air or water between the appropriate temperatures, available from tables (9) ; however, tabular values for air must be converted from a pound to a molar basis. This enthalpy difference between temperatures Tz/a and T J a is multiplied by b to give the desired value for the ideal enthalpy difference of the gas between Tz and T I . Entropy may be expressed as:
(16 )
Brz, OS,FZ HF
(17)
HZSe, N z O
(11,W
(1)
The air curve can be made to fit those of all the diatomic gases with good accuracy over wide temperatme ranges if it is displaced appropriately, horizontally and vertically. Similarly, the curve for water can be made to fit those of the triatomic gases. If the horizontal displacement is regarded as the logarithm of a constant, a, and the vertical displacement as the logarithm of another constant, (b/a), the basic transformation equations relating the properties of any substance to those of the corresponding reference substance (air or water) are: T = aT, Cp* = ( b / a ) Cpw*
(1) (2)
where a, b
T
= empirical constants for each gas
absolute temperature of gas in ' R (" F. 459.67) Cp* = molar heat capacity at constant (low) pressure of gas T , = absolute temperature of reference gas (air or water vapor) =
+
0
Figure 1. Molar heat capacities at low pressures of diatomic and triatomic gases are presented as functions of absolute temperature (log-log plot) 0 Critical temperatures ____._ Extensions to air and water reference curves to Rt data for Br2, Clz, Fz, and COz, HzS, Nz0 SOz, and Os, respectively
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1593
where the correction term in Equation 5 is negligible,
(S*-
S)T,P
Sz*
(5)
- SI*=
(&*
- + I * ) - R In P z / P I (7)
where
but
S = molar entropy relative to base temperature Toand base pressure Po (generally 1 atm.) P = pressure R = gas constant in appropriate units, 1.987 B.t.u./lb.-mole R or calories/g. mole K (S*- S ) T , p= correction term for nonideal deviations a t high pressure P
The Keenan and Kaye tables (9) also present air and water vapor values * dT, for Jk , which may be denoted
For convenience, the integral term is denoted by 4':
TV
as
where +* is the ideal molar entropy temperature function of a gas. For the low pressure (ideal) conditions
'f
&*.
Thus for other substances:
The desired molar entropy function thus may be obtained by multiplying by b / a , the difference between the tabular
20,000
15,000 -5+
10,000
+ Br2
'.Og
4000
4
2000,
W Q
F Y
B 1000
+ CO, +
AIR
N2
3
5i-!
~1~ Y
5
where To is the critical temperature. This function is analogous to the logarithm of reduced temperature (TIT,), being 0 a t the critical temperature, positive above, and negative below this value. Use of Nomograms in Figures 2 and 3. ILLUSTRATIVE EXAMPLE1. Evaluate the entropy and enthalpy changes for carbon dioxide going from 200' F. and I-atm. pressure to 2000" F. and 5atm. pressure. Solution. From Table I, for this temperature interval (range 2), a = 0.44, 6 = 0.45. Hence, Tn/a = (2000 459.67)/0.44 = 5590 Tl/a = (200 459.67)/0.44 = 1499 From Figure 3, the H*/b values corresponding to these T,la values are 61,000 and 12,500. Thus, Hz* - Hi* = 0.45 X (61,000 - 12,500) = 21,830B.t.u./lb. mole By drawing a straight line through the point for COZ (range 2 ) from these T/a values to the molar entropy temperature function scale, the corresponding values are : (+?* - $ 1 ' ) = 17.1 - 1.6 = 15.5 entropy units/mole These values may be compared with tabular values ( 9 ) Hz* - H I * = 27249 - 5163 = 22,084 B.t.u./lb. mole + z * - + I * = 68.479 - 52.934 = 15.543 entropy units/mole From Equation 7, Sz* - SI*= 15.5 - 1.987 In 5 = 12.3 entropy units/mole
+ +
4 (312
30
values for the reference substance at temperatures Tz/a and Tlla. The foregoing relationships are presented in nomographic form in Figure 2 for diatomic gases and Figure 3 for triatomic gases. A straight line drawn through the point for a given substance relates the molar entropy temperature function to values of Tla and Fib. The base temperature for the molar entropy temperature function has been taken as the critical temperature to simplify corrections for high pressure. The reduced ideal molar entropy temperature function is defined as:
700
3f400
If absolute values of ideal entropy are required, the absolute molar ideal entropy temperature function may be obtained :
2
i
00% = A *
200
1594
INDUSTRIAL AND ENGINEERING CHEMISTRY
(11)
$,,* = ideal molar entropy temperature
function relative to ideal gaseous state at absolute zero temperature = constant representing absolute ideal molar entropy temperature function at critical temperature,
100
Figure 2. Nomogram for thermodynamic properties of diatomic gases, terms of a and b (Table I)
+ &*
where
Tc.
in
The absolute entropies at 23' c. listed in Table I (72) are also values of
DI- A N D TRIATOMIC OASES &* at 25' C., as both P and PO are taken a t 1 atm. The value for +?* may be obtained from the appropriate nomogram for the same temperature (536.69' R). These values permit evaluating c$~*in Equation 11, which may then be used through the entire temperature range to convert the reduced functions from the nomograms to absolute entropy temperature functions. The base states for the H*/b scales are taken as the ideal gas at absolute zero temperature for air and water. For other substances it is the same condition, on the assumption that the reference heat capacity curve fits the actual curve down to absolute zero. True absolute enthalpies may be obtained, if required, from one known value by an equation analogous to Equation 11. However, the base state cancels out for changes within the specified applicable temperature ranges, and is ordinarily of no concern. .
Equations 5 and 10 reduced entropy is defined as:
(Table I) of those to the water curve at the same reduced temperatures. Corrections for the effect of high pressure on enthalpy (Equation 3) and entropy (Equation 5 ) , based on the law of corresponding states, may be obtained from standard references ( 5 ) . However, the use of qjr* from the nomogram permits a simplified procedure on a different basis, using Figure 5 for estimating entropy and Figure 6 for enthalpy. The entropy indicated in Figure 5 is based on the hypothetical ideal gas state at the critical pressure and temperature as a reference Thus, from
S, =
+?*
- R h P , - (S* -
S)T,F
(12)
where
S. = reduced molar entrop (definition) P7 = reduced pressure, P A c where Pc is critical pressure Lines of constant reduced entropy are plotted in Figure 5 on axes of ideal molar entropy temperature function and logarithm of reduced pressure. When the correction term (S* S ) T , p is negligible (ideal region), all
-I -5
Construction of Nomograms. The & * scales are linear. The ordinates of the T/a scales are proportional to &,* from the tabular value (9) for each Tw (= T'/a). The enthalpy scale was made to correspond to the T, scale by making the ordinate for each Hw*(= H*/b) proportional to the associated &* value. T h e point for each substance was located horizontally according to b / a and vertically so that &* equaled 0 a t Tc/a. If the critical temperature was outside the accurate temperature range, the vertical location was made to give a true value of dr* at a selected temperature within the range.
-t 03
N20 -t
Corrections for Nonideal Behavior at High Pressures
Figure 4 presents a plot of compressibility from actuaI data for steam (70), and from data for oxygen (75), and a generalized plot (27). The choice of the axes log reduced temperature and log reduced pressure with lines of constant compressibility differs from that in the usual plots of compressibility and pressure (or log pressure) with lines of constant reduced temperature. This plot gives a better representation of the temperature variable and reveals the striking fact that the lines of constant compressibility are linear through most of their length, down to the vapor pressure curve. The close numericaI agreement between the solid and dashed plots indicates the basis of the law of corresponding states. The vapor pressure curves for any of the 23 diatomic and triatomic substances listed have the same shape as that given by the dashed line for water on this plot. The horizontal distances from reduced pressure = 1 to the curve of a substance are a constant fraction c
CS*++
cos
Figure 3. Nomogram for thermodynamic properties of triatomic gases, in terms of a and b (Table I) VOL, 50, NO. 10
OCTOBER 1958
1595
20-
P :
P -
a
06I
001
I
I
I , 1 / 1 1 1
002
005
I
REDO;CED
I
I
I
I I l l l l
VRESSURE PI& 05
I
I
I
I
10
critical - Oxygen (combined with data point
0007001
OW
,
I
1
Y P , ~I
OREDUCED M 001 01P R E S ~ J R E
I I I I I I I
I
' 1
I
' I
l
l
Figure 5. Reduced molar entropy as a function of &* and log reduced pressure shows behavior in nonideal region (S, < 61
-...__ Water (IO).
in generalized plots)
substances should give the identical linear plot with slope R (Equation 12). The water and air curves agree for all values of reduced entropy exceeding 6. From reduced entropy 6 to 0 the slope remains unchanged, although the lines are more closely spaced. At 0 to -6 the lines remain almost linear, although the slopes decrease. There is a striking parallel between Figure 5 and Figure 4, which represents compressibility. I n spite of the widely different physical properties of air and water, the solid and dashed lines agree fairly well through most of the region. This suggests that each may be used in a general manner for diatomic and triatomic gases, respectively. If large entropy corrections are thus indicated, the alternative methods based on the law of corresponding states may also be used for comparison. In Figure 6 lines of constant enthalpy on the same ordinates of reduced molar entropy temperature function and logarithm of reduced pressure are shown. A parallelism between the two sets of lines may be noted, although the numerical enthalpy values have no special significance. The reduced ideal molar entropy temperature function may be regarded as merely a temperature function from which, using figures 2 and 3, the corresponding values of T and H* may be obtained. The figure may be used in a generalized manner for all 23 gases. To determine nonideal enthalpy the point for a given &* and P, is located, and is then traced to the left, following a line of constant enthalpy to the ideal low pressure region, giving new ordinate The Ho* value corresponding to (+r*)o is equal to the nonideal H at the high pressure condition. The temperature change for a Joule-Thompson expansion is equal to T - To,the temperature values from the nomogram corresponding to and (41*)0. Thus, Figure 6 may be used conveniently in conjunction with
1596
1 1 l l I l
0003
Figure 4. Lines of constant compressibility are linear through most of their length (log-log plot)
_._.__ Water, C.P.
6-
I 1 1 / 1 1 1
5
2
the nomogram and indicates the small regions where more rigorous methods may be necessary. Vapor Pressure and Heat of Vaporization
I t has been reported (4)that linear plots of vapor pressure and heat of vaporization may be obtained over wide temperature ranges, when they are plotted against the same properties of a reference substance at equal reduced temperature. This corresponds to: In P, = c In P,, (13) where
-
Air (79)
empirical constant for each substance P, = reduced vapor pressure of liquid substance P,, = reduced vapor pressure of reference substance (water) at same reduced temperature c
=
L = dL,
(14)
where d = empirical constant for each substance L = heat of vaporization of liquid substance L, = heat of vaporization of reference substance (water) at same reduced temperature
Table I. Thermodynamic Constants for Diatomic and Triatomic Gases Substance AiP Brz c19
co coz
(1) ( 2)
cos csz
F2 Hz
Gaseous Heat Capacity -jbs. ~~l~~ Constants Entropy a t Crit. Temp. 536.69 R Temp., a b range, ' R (25'C.) 'R
1.00 1.00 0.126 0.1234 0.22 0,215 1.00 1.00
100-6500 536-2290' 540-3600' 200-5400
46.33 58.63 53.31 47.31
0.24 0.44 0.75 0.95 0.40 1.65
200-750 580-5400' 485-1710 490-1620 18O-530Ob 5 10-4 100
51.05
547.67
55.32 56.86 48.56 31.22
680.69 983.09 259.5 59.87
0.20 0.45 0.95 1.36 0.40 1.65
...
Crit. Press., Atm.
238.41 37.2 1035.65 121. 750.89 76.1 239 33 34 53 I
...
Liquid Freezing Vapor! point, pressure T/T, constant c e . .
...
0.464 0.413 0.513
0.872 0.796 0.785
72,85
0.709
0.914 0.803c
72.9
0.355 0.297 0.371 0.423
a
... 61 .O 55.
12.80
...
...
0.817d 0.749 0.4421 0.5748 0.803 0.822 1.020 0.75gh 0.767 1.000 0.811 0.800 0.758 0,862 1.241 0.750; 0.880 0.947
540-2700 47.48 653.69 84.44 0.513 44.66 584.6 81.55 0.489 460-4270 821.99 50.0 0.569 510-2340 48.20 906.05 94.5 0.376 540-3060 41.49 HF 0.524 763.49 82.0 540-3600 49.33 HI 1165.03 218.17 0.422 300-5400 45.13 H20 0.502 672.41 88.9 450-3800* 49.10 H2S 0.510 738.29 91 .o 540-2700 52.8 HZSe 0.501 226.91 33.5 200-6400 45.77 N2 0.589 540-2340b 52.55 557.39 71.7 Rz0 322.49 64.6 0.628 460-5000 50.34 NO 0.353 49.01 277.85 49.7 200-3000 0 2 0.085 513.47 54.6 540-1900b 56.85 OS 0.465 774.65 77.7 59.24 490-5400' so2 Volume composition assumed t o be N Z 78.03%, 0 2 20.99%, -40.98%. Molecular weight HBr HCI HCN
1.10 1.35 0.70 1.40 1-50 1.00 0.67 0.67 1-09 0.30 1.00 0.70 0.26 0,295
1.10
1.36 0.76 1.40 1.58 1.00 0.65 0.67 1.09 0.286 1.02 0.70 0.229 0.28
Pressure range. 28.970. * Upper temperatures using empirically extended reference curves. 40 mm. Hg-72.9 atm. 0.01-1.6 atm. (range of d a t a ) , using 10 mm. Hg-61.0 atm. 1-12.8 atm. 10 pseudo P, = 44.5 atm. f 60-760 mm. Hg, using pseudo P , = 6.7 atm. 200 mm. Hg-49.7 atm. mm. Hg-1 atm., using pseudo P, = 50 atm.
INDUSTRIAL AND ENGINEERING CHEMISTRY
DI- AND TRIATOMIC OASES Values of c (Table I) for all 23 substances were determined graphically using water (70) as a reference substance, The vapor pressure data were taken from Stull (ZO), except for carbon monoxide and nitric oxide, for which the data in Lange (74) gave a better fit, fluorine (G), and ozone (8). Values of critical temperature and pressure listed in Table I were selected from standard sources (7, 2, 4, 73, 74, 77). Good straightline plots were obtained; the erratic nature of the small deviations suggests errors in data rather than systematic errors. Agreement was within 6% of vapor pressure, corresponding to a temperature error of less than 2” F. up to reduced temperature 0.5 and less than 5” F. u p to reduced temperature 0.7. This agreement is of the same order as that between values of vapor pressure from different sources for many of the substances. The method does not apply to the vapor pressure of the solid phase; hence, the reduced temperature must be above the freezing points listed in Table I. Values of d in Equation 14 were graphically obtained, using water as the reference substance, from Perry’s data (77): COz
Nz
0.426 0.146
0 2
0.171
SO2
0:621
IO
IFigure 6. Enthalpy as a function of and log reduced pressure shows behavior in nonideal region
+,*
.----. -
Water ( 7 0 ) Air ( 7 8)
I
&--
C T , I , , I I
0003
OL1070M
I
,
OR2
I
I
I 1 1 1 1 1
004
REDU%
point from standard references (74, 77) [hydrogen fluoride (7), ozone (8), and fluorine ( G ) ] . Although the ideal slope of this plot should be 1165 (critical temperature of water) from Equation 16,
I
I
I
I l I l 1 1 ,
TRESGRE P”;P, ’
I
,
I
O7
I
‘
,
the best empirical fit was actually 1197. This gives the empirical relationship: d = c T,/1197
(Iba)
Deviations from this line were less than
‘‘O3
5,000
O
dL
Agreement was within 4% except for a few erratic values. Because of the limited data for heat of vaporization as a function of tem perature, values of d may be derived from c by both ideal and empirical methods. If the Clausius-Clapeyron equation is applied to a liquid at a low pressure :
Applying this to water at the same reduced temperatures :
Dividing 15 by 15a and noting that the quotient of the left side is equal to c from Equation 13, we have:
4
From Equation 14 the ratio of heats of vaporization equals d. Rearranging: c
To = (TwM
19,000
(16)
As the Clausius-Clapeyron equation is inexact at higher pressures, an empirical plot was made of the, product of c and the critical temperature us. d. The above-mentioned values of d were used, with others obtained from single values of heat of vaporization, generally at the normal boiling
Figure 7. Nomogram for generalized relationships of vapor pressure P, temperature 7, and heat of vaporization f, in terms of empirical constants c (Table I) and d d is empirically related to c as shown in Equation 160
VOL. 50, NO. 10
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1507
5% except in a few instances (H2, HCN, HF, HI, and NO). Part of this deviation may be due to experimental errors, as heat of vaporization data for many substances differed widely. These relationships are given in nomographic form in Figure 7. A straight line through the proper value of G gives corresponding values of reduced vapor pressure and reduced temperature. The reduced temperature scale was actually constructed with ordinates corresponding to the logarithm of the vapor pressure of water (70) for each temperature. For each r e d y e d temperature the corresponding value of L/d [heat of vaporization of water ( l o ) ] is given on the opposite side of the scale. Use of Figure 7. ILLUSTRATIVE Ex2. What are the vapor pressure and heat of vaporization of sulfur dioxide at 150 O F.? Solution. From Table I, T, = 774.65‘R., P, = 77.7 atm., c ’ = 6.947. Hence, T / T , = (150 459.67)/774.65 = 0.787. On Figure 7, a straight line is drawn from T / T , through c to give P/Pc = 0.155. Then, P = 0.155 X 77.7 atm. = 12.04 atm., or 177 p.s.i.a. AMPLE
+
The value of L/d corresponding to T I T , on the nomogram is 13,800. Two values of d may be used : 1. Heat of vaporization data
d = 0.621
L = 0.621 X 13,800
=
8570 B.t.u./lb. mole 2. Empirical method from vapor pressure constant (Equation 16a) d = (0.947 X 774.65)/1197 = 0.613 L = 0.613 X 13,800 = 8460 B.t.u./lb. mole These values may be compared with those in thermodynamic tables for sulfur dioxide at this temperature (77, p. 275).
P
= 182 p.s.i.a. L = (217.2 - 83.0) X 64.07 8598 B.t.u./lb. mole
Application to Other Substances. Figure 7 should also be applicable to a wide variety of other polar and nonpolar liquids. The heat of vaporization relationships given by Equation 14 were reported 10 agree with the data generally within 5YO for substances containing up to 15 atoms per molecule; values of d have been listed (4). If the critical temperature and pressure of a substance and a single value of vapor pressure or heat ofvaporization are known, Figure 7 can be used to estimate the entire remaining range of both properties. Critical properties may also be conveniently estimated by rapid trial and error procedures using Figure 7. The critical temperature may be estimated from two widely different values of heat of vaporization.
A reasonable critical temperature is
1 598
assumed, and the two temperature values are converted to estimated reduced temperatures. The ratio of the corresponding L/d values (from the nomogram) is then compared with the ratio of the two known heat of vaporization values. The assumed critical temperature may then be adjusted until the two ratios agree. If more than two values of heat of vaporization are available, a graphical plot may be made of known L values (ordinate) us. nomographic L/d values (abscissa). Various critical temperatures are assumed until the data fall on a straight line passing through the origin. The slope of this line is d. Because c may be estimated from the value of d, the critical pressure may be found from a single known value of vapor pressure divided by the corresponding reduced vapor pressure obtained from the nomogram. The best procedure for estimating critical pressure from vapor pressure data and a known critical temperature is as follows: Any reasonable value of critical pressure is assumed (conveniently an integral power of 10, in the same units as the data), and the data are converted to estimated reduced pressure and reduced temperature. Lines are drawn on the nomogram connecting these corresponding values. All intersect close to a central point. A vertical line drawn from this central point to the c scale gives the best value of c. A line is then selected which goes very closely through the central point, and a second line from the same reduced temperature is drawn through the value of c on the c scale to give the true reduced vapor pressure. The best estimate of the critical pressure is the actual vapor pressure divided by this reduced value. The errors of these procedures may be illustrated by their application to the data (74) for various substances. The errors in absolute critical temperatures estimated from heats of vaporization were: ammonia -2%, benzene -6Yo, ethane +3YO:and propane +2Yo. The errors for critical pressures estimated from vapor pressure values were: ammonia -3%; benzene +$&,and propane -6%. Constants c from the latter calculations were converted to constants d by Equation 16a The errors in d were found to be: ammonia -2%, benzene -2%, and propane -6%. Other trial and errbr procedures using Figure 7 may be used for characterizing substances where few data are available, such as from t\vo values of vapor pressure and a single value of heat of vaporization. Conclusions The ideal molar heat capacity data and related properties for 23 diatomic and triatomic gases may be represented within 1% over wide temperature regions by two empirical constants for
INDUSTRIAL AND ENGINEERING CHEMISTRY
each substance in terms of the corresponding properties of air and Tirater. Simplified corrections for nonideal behavior at high pressures are given, asing the concepts of reduced molar entropy temperature function and reduced molar entropy. The critical pressure and temperature and one constant for each substance give both the vapor pressure and heat of vaporization over the entire liquid temperature range. These representations fit the data with little more error than the variations between different sets of data. They are convenient to use in nomographic form, and permit complete portrayal of thermodynamic properties, in spite of scanty data for many substances. Acknowledgment
The writer is grateful to Robert Lemlich, University of Cincinnati, for many valuable discussions and for review, and to Nathan Gilbert. literature Cited (1) “American Institute of Physics Handbook,” McGraw-Hill, New York, 1957. (2) Gratch, S., Trans. Am. Sac, Mech. Engrs. 70, 631-40 (1948). (3) Hougen, 0. A., Watson, K. M., “Chemical Process Principles,” pp. 21214, Wiley, New York, 1947. (4) Ibid.,pp. 234-8. (5) Ibid.,pp. 493-9. (6) Hu, J. H., White, D., Johnston, H. L., J . Am. Chem. Sac. 75, 5642-5 (1953). (7) “International Critical Tables,” vol. V, p. 136, McGraw-Hill, New York, 1929. (8) Jenkins, A. C., Birdsall, C. M., J . Chem. Phys. 20, 1158-61 (1952). (9) Keenan, J. H., Kaye, J., “Gas Tables,” Wiley, New York, 1950. (IO) Keenan, J. H., Keyes, F. G., “Thermodynamic Properties of Steam,” Wiley, New York, 1946. (11) Kelley, K. K., U. S. Bzu. Mines, Buli. 476 (1949). (12) Zbid., 477 (1950). (13) Kirk, R. E., Othmer, D. F.,
