MAY, 1935
IKDUSTRIAL AND ENGINEERING CHEMISTRY
Goodwin, N., and Park, C. R., IND. ENG.CHEM.,20, 621 (1928). Ibid., 20, 706 (1928). Johnson, C. R., Ibid., 20,904 (1928). Ibid., 21, 1288 (1929). Kayser, H., Ann. Phys., 121 12, 526 (1881). King, A., J . Chem. Soc., 1933, 842. Kolthoff. I. M., J . A m . Chem. Soc., 54, 4473 (1932). Kruyt, H . R.. and de Kadt, G. S.,Kolloid-Z., 47, 4 4 (1929). Ibid., KoZloidchem. Beihefte, 32, 249 (1931). Lamb. A. B., and Elder, L. W., J . Am. C h m . SOC.,53, 157 (1931). (36) Langmuir, I., J . Am. Chem. Soc., 37, 1154 (1915). (37) LeRlanc, M., Kroger, hl., and Klos, G., Kolloidchem. Beihefte, 20, 356 (1925). tennard-Jones, J. E., Trans. Faraday Soc., 2 8 , 3 3 3 (1932). Lowry, H. H., and Hulett, G. A,, J . Am. Chem. SOC.,42, 1408 (1920). Meyer. H., and Raudnits, H., Ber., 63, 2010 (1930). Meyer, H.. and Steiner, K., Monatsh., 3 5 , 3 9 1 (1914). Ibid., 35, 475 I 1914). Michaelis, L., and Rona, P., Biochem. Z., 102,268 (1920). Miller, E. J., J . Am. Chem. Soc., 46, 1150 (1924). Millcr. E. J.. J . Phys. Chern., 30, 1162 (1926). NBray-Seabo, S. v., 2. Elektrochem., 33, 15 (1927).
571
(47) Nellensteyn, F. J., and Thoenes, D., Chem. WeekhZad, 29, 582 (1932). (48) Rhead, T. F. E., and Wheeler. R. V., J . Chem. SOC.,101 846 (1912). (49) Ibid., 103, 461 (1913). (50) Rideal, E. K., and Wright, W. M., J. Chem. Soc., 127, 1347 (1925). (51) Shilov, N., and Chmutov, K., Z. physik. Chem.. A148, 233 (1930). (52) Shilov, N., Shatunovska, H., and Chmutov, K., Ibid., A149, 211 (1930). (53) Smith, R. A,, PTOC. Roy. SOC.,12, 424 (1863). (54) Spear, E. B., and Moore, R. L., IND. ENQ. CHIM., 18, 418 (1926). (55) Spear, E. B., and Moore, R. L., Rubber Aye (London), 9, 123 (1928). (56) Twiss, D. F., and Murphy, E. A., J . SOC.Chen. I?& 45, 121T (1926). (57) Wiegand, W. B., Can. Chem. Met., 13, 269 (1929). (58) Wiegand, W. B., and Snyder, J. W., IND. ENO.CHEJI.,23, 646 (1931). 29, (59) Wiegand, W. B., and Snyder, J. W., Rubber Age (N. Y.), 311 (1931). RECEIVED November 16, 1934.
Activity Coefficients of Gases
8
e
Application to Calculation of Effect of Pressure on Homogeneous Chemical Equilibria and to Calculation
@A
PREVIOUS paper by one of the authors (19) dealt with the definition of the activity coefficient of a pure gas (fugacity divided by pressure) and the calculation of values of this function for a large number of gases over wide ranges of pressure and temperature. Practically all of the published pressure-volume-temperature data on gases were utilized in these calculations so that the results represent as complete a picture of the subject as our present data permit. The data so calculated were then correlated and presented in graphical form by making use of the law of corresponding states. This made it possible to present a large mass of data in a compact and easily available form, and it also made possible the prediction of activity coefficients for any gas nhose P-Y-T data are not known, provided only that its critical pressure and temperature can be obtained. The agreement betm een values read from the graphs and values calculated from observed data is remarkably good for almost all subbtances over wide ranges. It is the purpose of the present paper to show how these graphs may be utilized to calculate in a very simple manner certain quantities that arise in connection with important chemical engineering problems. It is recognized that the calculations are not exact but they are sufficiently accurate for many purposes. The first application is to the calculation of the effect of pressure on the equilibrium in a gaseous reaction. The use of high pressures makes it possible to carry out on an industrial scale several important reactions which would give an entirely negligible yield a t ordinary pressures. I n the further development of existing high-pressure processes and in the search for new ones, i t would be most convenient to be able to calculate the maximum possible or equilibrium yield under pressure either from lon~-pressureequilibrium measurements or from free energy data based on the third law of
of Integral Joule-Thomson Effects ROGER H. NEWTON AND BARNETT F. DODGE Yale University, New Haven, Conn.
thermodynamics, without the large errors attendant upon the use of the ideal gas laws. The method here presented is not new but the graph given in the previous paper (19) simplifies the calculations and facilitates the application t o a wider variety of substances. The second application is to the calculation of integral Joule-Thomson effects or the change in temperature which results from the expansion of a gas from a constant higher pressure to a constant lower pressure without the performance of any external work. Stated in another way, it permits the calculation of difference in heat content or enthalpy between a gas a t two different pressures but the same temperature. This effect is utilized directly in the low-temperature separation of gases and may have other important applications as yet undeveloped. The calculation of the effect of pressure on heat content is necessary if one is to have a complete picture of the thermal properties of any gas or vapor. Such data would be useful whenever gases and vapors under high pressure are to be heated or cooled.
General Theory Although the paper referred to (19) m7as concerned only with the fugacity and the activity coefficient of pure gases, i t is a fact that the most important applications of these functions are to solutions. From the definition of free
INDUSTRIAL AND ENGINEERING CHEMISTRY
578
energy and the concept of the ideal gas and of partial pressure, it follows that,
where F
=
free energy of a gaseous soln.
n, = number of moles of any component P, = partial pressure of component i f ( T ) = a pure temp. function
dF
(5i> p . T . n . is called the "partial molal free energy" or the "chemical potential" and may be represented by the symbol 1%. By analogy with this equation, the fugacity of a component of a solution has been defined as follows: =
RT1n.E + f ( T )
5
(2)
where is the fugacity of the ith component of the solution or, as it may be termed, the partial fugacity; is a rather complex function of the pressure, volume, and composition of the mixture which cannot, in general, be evaluated because of our very limited knowledge of the behavior of solutions. T h e general form of the function is obtained by the application of the laws of thermodynamics to a multicomponent system. This leads to the following equation for 1.1, details of the derivation of which may be obtained from the monographs by Kuenen (11) and by van der Waals and Kohnstamm (26):
5
.E
VOL. 27, NO. 5 =
x,j,
(8)
This rule, known as the "fugacity rule" of Lewis and Randall, has been widely used in the calculation of partial fugacities. The foregoing discussion presents nothing new but was considered dePirable because many who have used this rule have not realized that it rested on the assumption of additive volumes, an assumption that is not yet very well tested. Lewis and Randall (14) stated that this assumption was involved in the rule but gave no proof. Gillespie (4) was the first to offer a definite proof which differs considerably in detail from the one sketched above. The available data a t the present time are too meager to permit any generalizations about the validity of the law. Gibson and Sosnick (3) have calculated the fugacity of ethylene and argon a t 24.95' C. in binary mixtures of these two gases using the data of Masson and Dolley (17). The deviations from the fugacity rule are small as long as the mole fraction of the component concerned is near 1.00, but, for the limiting case where it approaches zero, the deviation for argon was nearly 100 per cent and for ethylene was about 50 per cent. Merz and Whittaker (18) have made similar calculations for nitrogen and hydrogen mixtures a t 0" C., and they find a maximum deviation of only 20 per cent with pressures as high as 1000 atmospheres. KO investigations have been made on the effect of temperature on these deviations.
Effect of Pressure on Gas Reaction Equilibria For the generalized reaction, a4
where
?r
=
= RT In Z,
or
the equilibrium constant K, is represented by the equation :
&JVdP
+ RT + A T ) $J~
6s
= x,e
(5)
Additive Volume Law In the absence of the necessary data for rigorous calculation of the fugacity of a component in a solution by Equation 5, various assumptions have been made to permit an approximate calculation. One of the simplest assumptions that can be made is that the volume of a solution is the sum of the volumes of the individual pure components when the latter are measured at the temperature and total pressure of the solution. This '(additive volume law" is expressed mathematically by the equation,
+
+
(9)
(4)
where 4% stands for the expression in brackets. From Equations 2 and 4 i t follows that
j
+ bB = cC + dD
+
V = Visi VZZZ V S S S . .. (1 - x i - xz . . . . xn-~)Vn (6) where V = total volume of 1 mole of the 3oln. a t P and T Vl, V2,Va. . . . V , = molal volumes of individual components a t P and T xl, xZ,xa . . , . (1 - zi - x p . . . . x,+~) = mole fractions of components
where fa, f s , etc., are the partial fugacities (or the activities, since fugacity and activity are generally made identical for a gas) of the various components in the gaseous solution represented by the equilibrium mixture. They are related to the variables of state by relations of the type of Equation 5 . The constant K, is independent of pressure and composition and depends only on temperature. Another equilibrium constant, K,, which depends on all three variables, is defined as follows:
where
P = total ressure on the gaseous mixture
x A ,xB, etc., =
mole !actions
A t a low pressure, Po,where the gases may be assumed ideal, the two constants are identical, or K, = K,,,. At high pressures, however, this is by no means true. Substitution of Equation 8 in Equation 9 and multiplying and dividing each term of the right-hand side by P gives:
With the aid of this equation the expression for 4Gsimplifies to (7)
but & I T 7 @
at P and 2'.
Putting f / P = y, and using Equation 10:
is the log of the fugacity of component i
It follows a t once that
where K r is defined as (rc)c (?'A).
(rD)d
MAY. 1935
IXDUSTRIAL AND ENGINEERING CHEMISTRY
K, is obtained from lon-pressuie equilibrium data or from the free energy change for the reaction in question by means of the familiar equation : AFo = -RT
In K,
(13)
579
aid of his equation of state. The equation of state for the three-component solution was obtained by a simple combination of the constants of the equations for the individual pure gases. The agreement was, in general, not quite as good a s that obtained by the present authors' method, and the
T h e composition of the equilibrium mixture a t any pressure, P , is readily calculated from Equation 12 when means are available for evaluating K y . This is most easily done by means of the graphs given in the previous paper (19).
Application t o Animonia Equilibrium The data on the ammonia equilibrium a t high pressures are the only data so far published which permit a test of the accuracy of the predictions that can be made by means of Equation 12 combined with the activity coefficient chart previously published. The simplicity of the method may be illustrated by the folloa ing sample calculation: Assume one wishes to calculate the equilibrium constant K , a t 300 atmospheres and 450" C., given the value a t 1 atmosphere and 450" C. The necessary data to permit the use of Figures 2 and 3 of the previous paper (19) are given in Table I.
I
-
I
I
l
,
1 1
XITROGEN, AND TABLE I. CRITICAL CONsT.4STS O F HYDROGEN, AMMONIA,AND REDUCED V.4RIABLES FOR 453" c. ASD 300 Critical Pressure Gas O K. Atm. 33.2 12.8 H2 126.0 33.6 N2 111.6 406 "3 P 0 For H2, T R = T,: and P R = -' Pc 8
Reduced Temp.
Reduced Pressure
17.53 5.73 1.78
14.4 8.94 2.69
+
From the figures, 0.91, Ky =
= 1.09, Y N * = 1.14, and
YSHz (y~?)'/'(yH2)
''
-
YP;H~
=
0.91 = 0.750 (1.14) (1.09)3'2
'''
T h e observed value of K,, as calculated by Gillespie (6) from the experimental measurements of Larson ( I d ) is 0.008770. The equiIibrium constant a t 1 atmosphere has not been measured, but a small extrapolation from the 10 atmosphere values using the empirical equation [log,, K,, = (2679.35/2') - 5.88331 developed by Gillespie (6) will give such values, the particular one for the present case being 0.00664. From these two figures, K , (observed) = 0.757. In this manner the calculated values of K , in Table I1 were obtained. The observed values are based on Kno obtained from Gillespie's equation and on the experimental K , values given by Larson and Dodge (13)and Larson (12) as smoothed by Gillespie. The agreement up to 600 atmospheres is very good, and even at 600 atmospheres it is fair. The marked deviation a t 1000 atmospheres is probably due to the breakdown of the fugacity rule. Although the rule has been shown to hold quite well for nitrogen-hydrogen mixtures even a t 1000 atmospheres, the presence of ammonia to the extent of nearly 70 mole per cent a t this pressure would probably cause the nitrogen and hydrogen to deviate considerably from the rule. The agreement between an observed and calculated mole fraction of ammonia in the equilibrium solution is better than the agreement between the values of K,. For example, at 450" C. and 600 atmospheres the calculated and observed values for this mole fraction are 0.516 and 0.536, respectively. Calculations of the effect of pressure on the ammonia equilibrium have been made by Keyes (IO), by Gillespie (6), and by Gillespie and Beattie ( 7 ) . Keyes' method involved the integration of the general thermodynamic equation with the
400
600
800
/wO
PRESSURE. ATPI
FIGURE 1. VALUES OF Kr FOR T H E METHANOL SYNTHESIS REACTION
ATYOSPWERES" Critical Temp.
ZOO
0
method of calculation was a great deal more complex and rvould only be applicable to cases where the constants of the Keyes equation are known. Gillespie's method starts with the general equation which is then simplified by the assumption of the additive volume lam. The integration is then performed with the aid of the Keyes equation of state. The agreement between observed and calculated K , is excellent up to and including 100 atmospheres. At 300 atmospheres i t is not as good as was obtained by the present method. KO calculations were given for 600 and 1000 atmospheres. The limitations of the method are much the same as for the method of Keyes. Gillespie and Beattie ( 7 ) have integrated the general massaction equation by using Beattie's approximate equation of state which is explicit in the volume combined with certain assumptions as to the combination of the constants for the individual gases to obtain those for the solution, and by dropping TABLE 11. Pressure Atm. 10
BGREEMENT BETWEES OBSERVED AND CALCUL.4TED V.4LUES O F K , FOR AUMOXIASYNTHESIS
Kr Calcd. Obsrd.
-
Temperature, C . 325 350 375 400 425 450 475 500 0 , 9 8 1 0.983 0.984 0.986 0.987 0.988 0.990 0.992 0.986 0.987 0.988 0.990 0.991 0.992 0 993 0.994
...
30
Calcd. Obsvd.
50
Crtlcd. Obsvd.
..
100
Calcd. Obsvd.
.., ..,
300
Calcd. Obsvd.
,,, ,
600
Calcd. Obsvd.
,
Calcd. Obsvd.
.., . .,
1000
...
.. ..
.,
. ..
0 . 9 6 0 0.963 0.965 0.967 0.969 0.971 0.973 0.963 0.968 0.972 0.974 0 . 9 7 8 0.982 0 . 9 8 5 0 . 9 3 5 0.942 0.946 0.950 0.953 0.956 0.959 0.937 0.946 0.954 0.958 0 . 9 6 5 0.970 0.978
., . .. .
...
.., ... .., ... ..,
0.889 0.895 0,900 0.905 0 . 9 1 0 0.914 0.894 0.907 0.918 0 , 9 2 9 0.941 0.953
..,
. .. .., .., . ..
..,
.. ..
.. . .. .. , ., , ,
.. ., ... ... ... ...
0.750 0.768 0.788 0.757 0 765 0.773 0.573 0.594 0.612 0.512 0.538 0.578 0.443 0.473 0.487 0 , 2 8 5 0.334 0.387
certain terms. They obtain excellent agreement with the Larson and Dodge data over the whole pressure and temperature range; however, they do not strictly. calculate the high-pressure equilibrium constants from the low-pressure ones, but rather leave two adjustable constants in their final
INDUSTRIAL AND ENGINEERING CHEMISTRY
580
equation and select their values to fit the data best. Furthermore, the method is applicable only to gases whose P-V-T behavior is known so t h a t t h e constants of the equation of state may be evaluated.
VOL. 27, NO. 5
ods by which the Joule-Thomson effect may be calculated for a limited set of substances. I n the first three papers the calculation is limited to hydrocarbons containing more than three carbon atoms, and somewhat tedious calculations are involved. The method of Watson and Nelson is simpler as it involves merely reading a chart to obtain the difference between the heat content at high pressure and that a t one atmosphere, but i t mas applied to hydrocarbons only, The method to be described is applicable to any substance whose critical pressure and temperature are known, and the amount of calculation is reduced to a minimum. The relation between the fugacity and heat content has been shown t o be
Organic Equilibria The method should find its greatest usefulness in the fieId of organic reactions where i t is very difficult to measure directly the equilibrium constant either a t low or high pressures, and where the P-V-T data for the individual components are generally lacking. If lowpressure equilibrium data are available, then the method enables one to calculate with reasonable accuracy the effect of pressure. If no equilibrium data of any kind are available, then the lowpressure equilibrium constant] K p omust be calculated from the necessary thermal data. A commercially important case of this kind is the methanol synthesis reaction:
H* - H RT' where H = heat content of a mole of gas at the temp. and pressure in question H* = heat content at same temp. but at some low pressure where j = P, which for most substances is 1 atm. P
where C,
K , for this reaction is YC" and it has been calculated YCO'YH22' over a range of temperature and pressure; the results are presented in Figure 1. Inasmuch as no systematic study
=
(Ti - Tf) =
of the effect of pressure on this reaction has been made, no direct test of the accuracy of this method of calculating the effects of pressure is possible. The K,, for this reaction has been measured by the present writers (20) who also collected all the data for this reaction, corrected them for the effect of pressure in this manner, and presented them in the form of a single graph. Considering the spread of the points ob-
H* - H = C,(Ti - Tj) (15) molal sp. heat of the gas over the temp. range in question, and at low pressure temp. change which occurs on isenthalpic expansion of the gas from pressure P to low pressure (which may be taken as 1 atm.)
T h e basis for this equation is readily seen b y considering a cyclic process on a P-T diagram. Starting at Pi, Ti, the gas is expanded a t constant total heat or enthalpy to PI, T f , heated at constant pressure Pj baok t o Ti, a n d then isothermally compressed to Pi. The fact t h a t AH for t h e cycle must equal zero leads at once to Equation 15.
TABLE111. COMPARISON OF OBSERVED AND CALCULATED JOULE-THOMSON EFFECTS Gas
Critical Temp. 0
K.
Critical Preasure Atm.
Initial Temp. O
Initial Reduced Temp.
c.
37.2 - 25.2 1.87 132.4 Air 37.2 150.7 132.4 3.20 Air 32.2 153 5.20 2.26 Heliumb 104.2 1.01 373.1 43.7 Propane 300 0.885 218.2 647.2 Steam 349 1.32 470.3 33.0 n-Pentane -213 1.46 12.8 33.2 Hydrogenb .~ 0 A minus change denotes a cooling effect. P b For helium and hydrogen, P R and Tri are defined as P ,and ~
Initial Pressure Atm.
Initial Reduced Pressure
Final Pressure Atm.
-Temp. Obsvd.
215.2 215.1 200 18.7 40.6 68.0 50.0
5.78 5.78 19.5 0.428 0.186 2.06 2.40
1.3 1.1 1.0 1.7 1.0 1.0 1.0
-51.7 -13.2 +12.5 -16.7 -58.4 -33.3 -17.0
Calculation of Joule-Thomson Effect and the integral effect by
($>,.
(g)T,
c. -47.7 -14.5 +11.2 -21.4 -51.4 -28.3 -18.1
Thus the temperature change may be calculated from the following equation:
(bFl n)j ~ may be taken as (AT h f) ~over
a small range of
tkmperature without serious error, .or
bT ( z--), ,
,
These are related in
simple fashion through the specific heat to
c.
T respectively c+s
tained by any one investigator a t a single temperature, the fact that all the values lie in a narrow band, for the most part close to a theoretical line based on some of the best thermal data and the third law of thermodynamics, would indicate that this method of calculating the effect of pressure on this equilibrium is satisfactory.
T h e differential effect is defined by the expression
0
Reference to ChangeSource of C a l ~ d . ~Obsvd. Value
(g)T
and
respectively. The integral effect has been directly measured for only a few gases. It may be calculated from well-known thermodynamic relationships, provided accurate P-V-T data are available for the range of pressures and temperatures in question. These data may be used graphically or in the form of a n equation of state. Cope, Lewis, and Weber ( I ) , Lewis and Luke (16, 16), and Watson and Nelson (27) have recently published meth-
b In f (-)
=
1
aT
(~n j;
- In j 2 )
=
-In AT
A-
f1
(17)
T I - T2,a small temp. interval around temp. Ti fugacity a t temp. TI, slightly above initial temp. Ti of the gas J2 = fugacity at T2, slightly below initial temp.
where AT fi
= =
Inasmuch as these fugacities are at the same pressure:
Thus, as a final equation:
MAY, 1935
INDUSTRIAL AND ENGINEERING CHEMISTRY Ti
-
=
RTia In C,(T1 - Tz)
72
T o show the application of this equation, the cooling which occurs when air a t 3.2” C. and 185.3 atmospheres expands adiabatically through a porous plug or throttle valve to 1.2 atmospheres, will be calculated. The critical pressure and temperature of air are 37.2 atmospheres and 132.4” K., respectively. The y values are both a t the initial pressure which, in this case, is a PI: of 4.98 and at two arbitrarily chosen temperatures around the initial temperature, which is a T R of 2.09. Let T I and T z be 2.20 and 2.00 in reduced units. T I - T z = (2.20 - 2.00) (132.4) = 26.5. From Figure 2 of the previous paper (19), y1 = 0.972, y z = 0.922, C, = 6.90, R = 1.987. From Equation 18, Ti - TI = 43.5”C. The measured value of Roebuck (24) is 39.6’ C. As i t stands, Equation 18 can be applied only when the final pressure is low enough so that the gas is substantially ideal. However, it is possible by a little modification to make this method applicable to the case of expansion to some intermediate pressure and also to the calculation of the differential coefficient p. An outline of the derivation of equations for this purpose follows:
over small intervals. where Hi = heat content at PI and T H 2= heat content at P2 and T Combining Equations 19, 20, and 14:
I n the same way that Equation 16 was transformed into 18, Equation 21 becomes, where T = initial temp. of the gas PI - P, = a small pressure interval T1 - T2 = same as before C, = mean sp. heat at T and the mean pressure yi = activity coefficient at Pl and T1 y2 = activity-coefficient at P, and T2 y i l = activity coefficient at P1and T z y2/ = activity coefficient at PZand Tl
581
computers. The observed values are, for the most part, taken from temperature-entropy diagrams or tables of thermodynamic properties. I n the case of air, helium, and npentane, actual Joule-Thomson measurements were used. The method is admittedly not a very accurate one, but i t is a simple and convenient way of making such a calculation for any substance whose critical pressure and temperature are known or can be estimated.
Literature Cited (1) Cope, J. Q., Lewis, W. K., and Weber, H. C., IND.ENG.CHEU., 23, 887 (1931). (2) Dodge, B. F., Ibid.,24, 1353 (1932). (3) Gibson and Sosnick, J . Am. Chem. Soc., 49, 2172 (1927). (4) Gillespie, L. J., J . Am. Chem. Soc., 47, 305 (1925). (5) Ibid.,48, 28 (1926). (6) Gillespie, L. J., J . Math. Phys. Mass. Inst. Tech., 4, 84 (1925). (,7) Gillespie, L. J., and Beattie, J. A., Phys. Rev., 36, 743 (1930). (8) Keenan, J. H., Mech. E ~ Q 48, . , 144 (1926). (9) Keesom, W. H., and Houthoff, D. J., Communications Phys. Lab. Univ. Leiden Suppl., No. 65d to Nos. 181-192 (192628), (10) Keyes, F. G., J . Am. Chem. SOC.,49, 1393 (1927). (11) Kuenen, J. P., “Theorie der Verdampfung und S‘erflussigung von Gemischen und der fraktionierten Destillation,” LeipBig, Johann Barth, 1906. (12) Larson, A. T., J . Am. Chem. Soc., 46, 367 (1924). (13) Larson, A. T., and Dodge, R . L., Ibid.,45,2918 (1923). (14) Lewis, G. N., and Randall, M.,“Thermodynamics and the Free Energy of Chemical Substances,” New York, McGraw-Hill Book Go., 1923. (15) Lewis, W. K., and Luke, C. D., IXD.ENO.CHEX.,25, 725 (1933). (16) Lewis, W. K., and Luke, C. D., Trans. Am. Soc. dfech. Engrs.. 54, No. 17 (1932). (17) Masson and Dolley, Proc. Roy. Soc. (London), 1038, 524 (1923). (18) Mere, A. R., and Whittaker, C. W.,J. Am. Chem. SOC.,50. 1522 (1928). (19) Newton, R. H., IND. EKQ.CHEM.,27, 302 (1935). (20) Newton, R . H., and Dodge, B. F., J . Am. Chem. Soc.. 56, 1287 (1934). (21) Pattee, E. C., and Brown, G. G., IKD.ENO.CHEM.,26, 511 (1934). (22) Roebuck, J. R., Phys. Rea., 43, 60 (1933). (23) Roebuck, J. R., Proc. Am. Acad. Arts Sci., 60, 537 (1925). (24) Ibid., 64, 287 (1930). (25) Sage, R. H., Schaafsma, J. G., Lacey, W. N., IND. ENG.CHEM., 26, 1218 (1934). (26) Waals, J. D. van der, and Kohnstamm, P. H., “Lehrbuch der Thermodynamik,” Leipeig, Johann Barth, 1912. (27) Watson, K. hl., and Nelson, E. F., IND.ERG.CHEM.,25, 880 (1933). RECEIVED December 29, 1934
C, may be estimated from the curves published by Dodge (9). The calculation of temperature change for any finite
pressure change is accomplished as follows: Since
Pav.
=
(g)H
Equation 23 may be used to calculate the temperature change for a pressure change Pi - P2where Pz is an intermediate pressure at which gases would be far from ideal. C p is the mean specific heat at the constant pressure Pzfor the temperature interval (Ti - T,). The results obtained in the application of Equation 18 t o a wide variety of substances and conditions are shown in Table 111. The agreement is remarkably good considering the fact that rather small differences are read on the activity coefficient chart, and the fact that i t is difficult to calculate accurately Joule-Thomson effects even with a good equation of state for a particular substance. Each calculated value is the average of two values obtained independently by two
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