3559
CORRESPONDING STATES OF ARGON AND METHANE
Corresponding States of Argon and Methane by Eugene M. Holleran and Gary J. Gerardi Department of Chemistry, St. John's University, Jamaica, New Ynrk
ll482
(Received M a y 6, 1968)
With the two characteristicconstants of the unit compressibility law as reducing constants, the compressibility factors of argon and methane are compared at equal reduced temperatures and densities. It is found that these gases depart from exact correspondence in a systematic manner. By assuming that the ratio of the deviations from ideality of the two gases is constant, the compressibility factor for methane from 0 to 250" and 0 to 12.5 M can be calculated from that of argon with an average discrepancy of 4 parts in 10,000. This represents a three-constant system of corresponding states, .with the effect of the third constant given analytically. The correspondence of the thermodynamic residual properties in this system is also discussed.
Introduction The temperatures, T,and densities, d, at which the compressibility factor, 2 = P/dRT, has the value unity have been shown to be linearly related for a number of gases over a wide range of T and d.l Thus, for 2 = 1 -T- + - d= I TB do
(1)
The Boyle temperature, TB, and the density, do, are characteristic constants which can be determined for each substance from PVT data. Upon discovery of this unit compressibility law, eq 1, the two constants were proposed as a natural set of reducing factors for a corresponding states treatment of the properties of fluids. Using reduced temperature, density, and pressure defined as
e
= T/TB
S K
= d/do
(2)
= P/d&TB
the compressibility factors of argon and xenon were compared2 at equal reduced temperatures and densities. The correspondence was found to be as good or better than that produced by other reducing factors such as the critical constants or intermolecular potential parameters. I n making this comparison, however, several difficulties were encountered, arising from the relatively low temperatures and high densities in the region of overlap of the data for the two gases. I n this region the polynomial representation of the isotherms is poorest, the required interpolations in density and temperature are least accurate, and the presently unavoidable uncertainties in the values of T Band do lead to the greatest uncertainties in the values of 2 found for a given e and S. For these reasons it was not possible to determine whether any real, systematic differences exist in the reduced equation of state of these two gases. A survey of the unit compressibility results' shows that, aside from argon, the substance whose data best
fit the unit compressibility law over a fairly extended experimental range is methane. Its constants are among the most reliable, and its data overlap with argon in a region of temperature and density where the difficulties encountered in the argon-xenon comparison largely disappear. It is therefore apparent that a comparison of argon and methane should provide a much better illustration of the degree of correspondence provided by the reducing constants of the unit compressibility law.
Argon-Methane Correspondence The methane data used for the comparison are those of Douslin, Harrison, Moore, and RlcCullough, a who tabulated 2 at densities up to 12.5 mol/l. at (mostly) 0.5 mol/l. intervals, and at temperatures from 0 to 350" a t (mostly) 25" intervals. The argon data are by Michels, Levelt, and DeGraaff below 0", and Michels, Wijker, and Wijker6 above 0", who reported isotherms from -155 to 150" up to densities of about 29 mol/l., represented by polynomials to (usually) t,he sixth power in d. The unit compressibility constants found in ref 1 were used here: TB of 408.3"K for argon and 509.3"K for methane, and do of 46.75 mol/l. for argon and 35.74 mol/l. for methane. The compressibility factors for the two gases were compared by interpolating 2 in the argon data to the temperatures and densities representing the same 0 and 6 values as the tabulated temperatures and densities of methane. Thus, T(Ar) = TB(Ar) X T ( C H 4 ) / T ~ (CH4), and d(Ar) = do(Ar) X d(CH4)/do(CH4). The density interpolation was provided by the isotherm polynomials. 2 was calculated at the desired density on the two isotherms straddling the desired temperature, and then 2 was found for this temperature by (1) E. M. Holleran, J . Chem. Phys., 47, 5318 (1967). (2) E. M. Holleran, J . Phys. Chem., 72, 1230 (1968). (3) D. R. Douslin, R. H. Harrison, R. T. Moore, and J. P. McCullough, J . Chem. Eng. Data, 9,358 (1964). (4) A. Michels, J. M. Levelt, and W. DeGraaff, Physica, 24, 659 (1958). (5) A. Michels, H. Wijker, and H. Wijker, ibid., 15, 627 (1949). Volume 78, Number 10 October 1988
3560
EUGENE M. HOLLERAN AND GARYJ. GERARDI
Table I : Argon Interpolation in Inverse Temperature [Z(direct) - Z(interpo1ated)l X lo4 t,
oc
- 110
0
0.3998" 100 0.4241 - 85 0.4608 - 70 0.4976 _-_-_______ - 50 0.5466 25 0.6078 0 0.6690 25 0.7302 50 0.7915 0.8527 75 100 0,9139 0.9752 125
-
-
a
0.05
0.10
0.15
0.20
5 3 4
6 9 7
12 10
14 8 5
3 2 2 1
6 3 3 2 2 1 0
1 1 0
8
7 4 3 3 3 2
6 5 3 5 3 2
1
1
0.25
0.35
0.30
5 2
-9 -5 -2
2
6 7 3 5 : -- -2- - - __ : j 3 j 2
-21 -9 -5
6 1 7 : - - -3- - -.-.i 7 3 4 3
0.45
0.50
0.55
0.60
-21 34
- 10
2 2 9 15 22 26 20 14 18 11 13 10
5 12 18 23 36 39 26 19 24 11 21
12 24 44 24 61 61 24 23 14 11 34
12
11
-
-6 0 6 14 19 14 10 14 9 8 7
--o
12 10 6 10
8 8 5 8 5 5 4
_--_
0.40
7 7 6
The differences on this isotherm differ slightly from those given in ref 2 where corrections to the polynomials were included.
interpolating between the inverses of the two isotherm temperatures. The accuracy of the density interpolation is verified by the fact that, in our range of interest (above -50°), the isotherm polynomials reproduce the experimental points to within 2 parts/10,000. The validity of the inverse temperature interpolation is verified by the results shown in Table I, which lists the differences in the fourth decimal place between 2 as given along an argon isotherm by the polynomial and 2 interpolated to this temperature from the two adjacent isotherms. In the outlined region of overlap with the methane data there is only a hint of the beat wave pattern between the isotherm polynomials which is seen at the lower temperatures and which tended to obscure the argonxenon comparison. The interpolation error is relatively small in the methane overlap region, and since the temperature span of the interpolations performed in the correspondence calculations was only half that needed in Table I, the interpolated compressibility factors are too low on the average by less than two units in the fourth decimal, and no correction was made. The above procedure therefore permits a very accurate comparison of the compressibility factors of argon and methane at equal reduced temperatures and densities. Upon beginning the comparison, it was immediately evident that these two gases do not correspond exactly. This is not particularly surprising, because two-parameter systems of corresponding states, like two-constant equations of state and two-parameter intermolecular potentials, are not generally adequate. However, the nature of the deviation from exact correspondence is very interesting and is illustrated in Figure 1. Compressibility isotherms us. 6 are shown with exaggerated differences for the two gases at equal 8. The isotherms coincide of course at 6 = 0 and also at 6 = 1 - e, where 2 = 1, as required by the unit compressibility law. The curves are of similar shape, with minima apparently at the same 6. For Z < 1, 2 for The J o t ~ r n a of l Physical Chemistry
I
d Figure 1. Typical compressibility isotherms for argon and methane at the same reduced temperature, T / T B . Differences are greatly exaggerated.
argon exceeds Z for methane, but for 2 > 1, 2 for methane is the greater. In fact, it appears that the deviations of the two isotherms from ideality, that is from unit Z, are proportional. I t is as if the molecular interactions for the two gases are proportional at equal 0 and 6. At 6 = 1 - 8, the effects of the attractions and repulsions on 2 are balanced, but presumably with stronger forces in balance for methane than for argon. At other reduced densities then, the imbalance of forces and the deviation from ideality are greater for methane than for argon. Whether or not such speculations are correct, we can proceed on the working hypothesis that the ratio of the deviations from ideality for the two gases is constant. Thus we assume that (2 -
1)CHa
k(2
- 1)Ar
(3)
for equal 8 and 6. We then optimize k by finding the value which gives the smallest average absolute difference between the 2 values observed for methane and those calculated from argon data by Z C H ~ = kZA, (1 - k ) . The value of IC found in this way is 1.044. ' The accurate correspondence obtained between the observed and calculated compressibility factors for methane is shown in Table 11. The experimental vaIues of 2 are listed at intervals of 25' and 1 mol/.,
+
CORRESPONDING STATES OF ARGON AND METHANE
3561
Table 11: Compressibility of Methane and Difference [Z(calcd from Argon) Density, mol/l. t,
- Z] X lo4
-
-
oc
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
0
0.9492 2
0.9038 3
0.8636 4
0,8290 2
0.7996 0
0.7755 3
0.7579 -5
0.7453 -3
0.7392 -2
0.7396 1
0.7477 2
0.7640 7
25
0.9595 0 0.9679 0
0.9239 2 0.9404 0
0.8931
0.8672 1
0.9172 0
0.8987
0.8763 1
0.8207 -2 0.8731 -1
0.8163 -2 0.8753
1
0.8464 -2 0.8852 .- 2
0.8309
1
1
0.8179 0 0.8840 0
0.8263 7 0.8990 6
0.8423 7 0.9222 5
0.8672 9 0.9539 8
0.975'0 -1
0.9539 2
0.9373
0.9250 2
0.9174 -2
0.9149 -1
0.9173 -1
0.9251 4
0.9395 3
0.9606 5
0.9894 7
1.0268 13
0.9809 -1
0.9657 0
0.9544
0.9474 2
0.9449 3
0.9471 4
0.9547 3
0.9673 9
0.9864 10
1.0129 7
1.0469 9
1.0903 4
0.9859
0.9754 1
0.9689 2
0.9666 1
0.9687 3
0.9752 5
0.9869 7
1.0045 7
1.0278 10
1.0589 5
1.0971 7
0.9902 0
0.9841 0
0.9816
0.9832 2
0.9890 5
0.9995 6
1.0150 8
0.0361 10
0.0632 13
1.0976 13
0.994.1
0.9915 0
0.9927 1
0.9979
1.0069 6
1.0211
1
4
1.0399 7
1.0646 6
1.0951 -3
0.9981 0
1.0025 0
1.0105 3
1.0228 8
1.0396 8
1.0617 7
1.0894 6
1.1228 8
1.0003 0
1,0039 0
1,0110 1
1.0220 1
1.0372 12
1.0567 4
1.0813 6
1.1113 6
1,0030 0
1,0091 0
1.0186 4
1.0321 3
1.0498 5
1.0718
1.0987 4
1.1310 7
50
75 100 125
-1
150 175
-1
200
0.9974 -1
226 250
1 1
1
-2
together with differences in the fourth decimal place between these and the calculated Z values. The agreement is very good, with a maximum deviation of 0.0013 and an average absolute deviation of 0.0004. The preponderance of positive differences, which cannot be eliminated by adjusting k since they occur both above and below Z = 1, indicates that refinements in the values of T B and do could produce a somewhat improved agreement. However, the agreement is already within the estimated maximum experimental errors in the methane data, which range from 3 parts/ 10,000 at low densities and temperatures to 30 parts/ 10,000 at high densities and temperatures. The use of the constant k in conjunction with T Band do constitutes a three-constant system of corresponding states. Its possible generality will be investigated, although data of the necessary quality and extension are rare. If other gases in addition to methane should prove to be related in this way to argon, each with its own constant k , then, taking argon arbitrarily as a standard with k = 1, the quantity (Z - l ) / k would correspond at equal 0 and 6 for all such gases. This three-constant system would have a great advantage over others because the effect of the third constant is given analytically.
Thermodynamic Residual Properties Knowing that the property (2 - l)/IC of methane and argon correspond a t equal e and 6, we can determine
4
how the Oherrnodynamic residual properties of these gases are related. A residual property, X * , may be defined as6
x* = JI[(dX/dV),
- (dX/dV),O]dV
(4)
where the superscript zero indicates the ideal gas. By writing 6 in place of V , 0 in place of T , and 0 in place of , eq 4 becomes the equation for X * (e, 6). Letting X represent Z, we see that Z* is 2 - 1, so that Z*/k for the two gases correspond. Letting X represent T and noting that P = 206, we find that n* = P - 06, so that a*/k is also a corresponding property a t equal 0 and 6. Employing the thermodynamic relation, (bX/dV), = (dP/bT)v, we find that the property S*/kR corresponds. Similarly, using (dE/bV)T= T(dS/dV), - P, we learn that E*/LRTBis a corresponding property. Others include H*/kRTg = E*/kRTg eZ*/k, A*/kRTB = E * / ~ R T B- OX*/lcR, and G*/kRTg = H*/kRTg BX*/kR. These results can be summarized by the observation that all these thermodynamic residual properties, reduced to dimensionless form in the normal way ( P by doRTB, 1.5' by R, all energies by RTB),become corresponding properties upon division by IC. Q)
+
(6) A. Miohels, M. Geldermans, and 9. DeGroot, Physica, 12, 105 (1946).
Volume 72. Number 10 October 1968
EUGENE M. HOLLERAN AND GARYJ. GERARDI
3562
Table 111: Negative Residual Entropy (cal/mol) of Argon and the Difference [X* , j ,
t , "C
80
40
160
320
360
0.941 9
1.133 10
1.331 10
1.535 10
0.749 9
0.711 3
0.893 3
1.081 2
1.274
1.476 0
1.687
1
0.682 0
0.860 -1
1.044 -3
1.234 -5
1.432 0
1.640 -5
0,490 0
0.659
0.833 -3
1.013 -5
1.200 -7
1.395 5
0.475
0.640 -1
0.811 -3
0.987 -4
0.170 -4
0.464 -2
0.626 -3
0.793 -3
0.966 -4
0,301 -3
0,456 -4
0.615 -5
0.779 -5
0,950 -5 0,938 -8
0,376 6
0.564 6
0.752 7
- 25
0.176 2
0,353 2
0.531 2
0.168
0.337
0.508
1
1
1
0.161 5
0.324 0
0.155
0.313
1
1
75
0.152 -1
0.307 -2
100
0.149
50
-1
_ _ _ _ _
280
0.188 4
25
amagatsa 200
108
240
- 50
0
a
120
- S*(calcd from Methane)] x
-1
-1
125
0.147 "-3
0.296 -5
0.450 -8
0.607 -8
0.770 - 10
150
0.145 -4
0.293 -8
0.444 - 10
0.600 - 11
0.761 - 13
-1
For argon, 1 amagat unit of density is 0.044647 mol/l.
Table IV: Negative Residual Energy (cal/mol) of Argon and the Difference [E* - E* (calcd from Methane)] oc
40
80
120
160
200
240
- 50
70.4 0.8
138.8 2.2
205.1 1.5
269.4 2.7
331.9 1.8
392.9 2.0
452.8 2.1
512.0 1.4
570.7 3.6
- 25
67.8 0.2
133.5 1.2
197.5 0.5
259.8 0.6
320.7 -1.7
380.6 0.2
439.6 0
498.0 -1.6
556 1 1.8
0
65.6 0
129.4 1.5
191.6 0
252.4 -0.3
312.2 0.9
371.1 -1.1
429.2 -0.8
486.8 -2.4
544.0 1.2
63.4 0.2
125.5 0.6
186,l 0
245.8 -0.4
304.5 -1.0
362.3 -1.4
419.4 0.9
476 0 -0.9
61.9
122.4 0.2
181.5
1.1
0
239.7 1.3
297.1 -0.5
353.8 -0.9
409.9 -2.0
75
60.5 3.7
119.6 -0.2
177.5 -1.4
234.5 0.7
290.7 -2.7
346.3 -2.2
100
59.6 0.3
117.7 -1.2
174.7 -1.2
230.7 -2.2
286.1 -1.3
340.8
58.9 -1.4
116.3 -2.8
172.7 -2.8
228.1 -3.0
282.7
336.6 -2.7
58.2 -1.9
115.1 -4.6
170.8 -4.6
225.7 -5.1
279.7 -3.2
1,
25 50
125 150 a
-3.0
280
320
360
I
I
-1.1
For argon, 1 amagat unit of density is 0.044647 mol/l.
The correspondence of the residual internal energy and entropy is shown in Tables I11 and IV. The The Journal of Physical Chemistry
tabulated values for argon are those reported by Michels, Lunbeck, and Wolkers7. Also listed are the
MOSSBAUER
SPECTROSCOPY OF SUPPORTED GOLDCATALYSTS
differences between these values and those calculated for argon from methane by interpolation in the methane data of Harrison, Moore, and Douslin8 and application of the above correspondence relation. Understandably, the percentage accuracy for these residual properties is not good at low densities, but it improves with
3563
increasing density and reaches agreement to within a few parts per thousand a t the high densities. (7) A. Michels, R. Lunbeck, and G. Wolkers, Physica, 15,689 (1949). (8) R. Harrison, R. Moore, and D. Douslin, “Thermodynamic Properties of Compressed Gases,” Bartlesville Petroleum Research Center, Bartlesville, Okla., A.D. 647 893.
Mossbauer Spectroscopy of Supported Gold Catalysts1 by W. N. Delgass,2M. Boudart, and G. Parravano Department of Chemical Engineering, Stanford Universitg, Stanford, California 94506
(Received M a y 7 , 1968)
Supported gold catalysts have been studied by Mossbauer spectroscopy, electron microscopy, and X-ray diffraction to determine the effect of the impregnation compound, the support, and heat treatment on the nature of the catalyst formed. MgO was found to be a more inert support than ~-A1203toward H.4uC14, and HAuC14 was found to be more easily decomposed than KAu(CN)z to give gold particles on 7-A1203. Chemical changes in the gold complexes on alumina after different heat treatments were observed in the Mossbauer spectra of 197Au, thus illustrating the utility of the Mossbauer effect for the study of the genesis of a supported catalyst.
Introduction States of matter with a high surface-to-volume ratio often exhibit high catalytic activity as a consequence of the special environmental conditions and large number of atoms present at the surface. Metals and metallic salts dispersed on high surface area supports are important examples of such catalysts. The genesis of these catalysts is a complex process, usually involving impregnation of the support with a metal salt, drying, and final treatment to form the desired metal species. Since the final state of the metal atoms is determined in some measure by the detailed nature of the chemical interactions present during pretreatment, it is of interest to investigate those interactions with the aim of establishing a relationship between the chemical state of the metal during the pretreatment stages and the final state of the metal in the catalyst. I n addition, the investigation of such chemical states may reveal the formation of unusual chemical environments which may be catalytically active. RIeasuring the chemical state of a supported metal atom is, however, a difficult task. X-Ray diffraction and line broadening and electron microscopy give information primarily concerning the physical properties of small particles. The chemical properties are obtained most directly by spectroscopic methods when applicable. It is the purpose of this paper to present some physical and chemical measurements on a supported-metal system and, particularly, to explore the use of Mossbauer spectroscopy as a tool for studying the chemical states just described.
The use of -1Iossbauer spectroscopy in catalytic and surface studies has been reviewed in the literat~re.~-6 The specificity of the effect for a single type of atom in a heterogeneous system and the sensitivity of the effect to chemical environment make it well suited for such investigations. Gold was chosen as the supported metal since it catalyzes hydrogenation and oxidation reactions and because its resonance absorption of the 77-keV radiation, emitted during the decay of the first excited state of lg7Auto the ground state, has been successfully measured on some gold alloysa and compound~.~ The experimental variables investigated in this study were (a) the nature of the support and of the gold compounds used in the preparation of the samples and (b) the temperature and time of pretreatment. I n addition to being analyzed by Mossbauer spectroscopy, the samples were examined by X-ray diffraction and electron microscopy. (1) This work was supported in part by the U. S. Army Research Office under Contract No. DAHC0467C0045. (2) National Science Foundation Graduate Fellow, 1964-1968. (3) J. T.V. Burton, H. Frauenfelder, and R. P. Godwin, “Applications of the Mossbauer Effect in Chemistry and Solid State Physics,” International Atomic Energy Agency, Vienna, 1966, p 73. (4) M. J. D. Low in “The Solid-Gas Interface,” Vol. 2, E. A. Flood, Ed., NIarcel Dekker, Inc., New York, N. P., 1967, p 947. (5) W.N. Delgass and M. Boudart, Catal. Rev., 2, 129 (1968). (6) For example, P. H. Barrett, R. W. Grant, M. Kaplan, D. A. Keller, and D. A. Shirley, J . Chem. Phys., 39, 1035 (1963). (7) For example, D. A. Shirley, R. W. Grant, and D. A. Keller, Rev. Mod. Phys., 36, 352 (1964). Volume 72,Number 10
October 1968
