THERMAL EXPANSION OF ROCK SALT
Jan., 1961
65
THERMAL EXPANSION OF ROCK SALT' BY THORRUBIN,H. L. JOHNSTONAND HOWARD W. ALTMAN Cryogenic Laboratory of the Chemistry Department, The Ohio State University, Columbus 10, Ohio Received June S,1080
Interferometric determinations of the expansion coefficient of synthetic rock salt were made from 20 to 300°K. Similar measurements for natural rock salt were made from 90 to 300°K. Derivatives for the expansion coefficient were calculated at a variety of temperatures. Griineisen coefficientswere computed as a function of temperatures using measured compressibilities and both measured heat capacities and those computed from Kellerman's frequency spectrum. A correlation of these Griineisen coefficients with those expected from a nearest neighbor cubic closest packed structure and for a sodium chloride type lattice has been made.
Introduction The apparatus and experimental technique for determination of' expansion coefficients of synthetic rock salt were the same as those described for copper.2 The natural rock salt data were determined by a relative method using natural quartz, as a fiduciary material. Here a count of linear interference lines passing a fiduciary mark were measured over a temperature interval. Linear interpolation of the apparent position of interference line at the final temperature and at the initial temperature was made wherever the total number of fringes passing a fiducial point was not integral. The synthetic rock salt was obtained from the Harshaw Chemical Company. Natural rock salt was obtained from Cincinnati Chemical Company. No chemical analyses were made. The samples of each material were cut into pillars adjusted to the same length by careful filing. These sets of pillars served as separators between the interference plates. At the completioii of the measurements, the samples contained a few cracks which were not present a t their preparation.
Results The data for natural rock salt are shown in Table I. For calculation of the absolute values of the expansion coefficients, coefficients for quartz, determined by Buffington and were used. Alpha is the expansion coefficient determined graphically, Tavgis the mean absolute temperature and AT is the temperature interval of the measurement. These mleasurements were based on the temperatures read from a standardized thermocouple. The absolute values for the expansion coefficients of synthetic rock salt are given in Table 11. In this table r12, r2zare the squares of the apparent fringe diameter at temperature equilibrium, measured with a filar micrometer eyepiece. The optical constant is a measure of the change of the square of the fringe diameter by an increase of one fringe order, actual values of this quantity are shown in the table, column 5. Smoothed values of this quantity, marked 0', are shown in column 4, F is the number of fringes passing a fiduciary mark during the temperature change AT, a is the coeE(1) This work waa nupported in part by the Ai Material Command, Wright Field. Preaented in part at the Colorimetric Conference, Chioago. Sept. 1969. (2) T. Rubin, H. W. Altmsn and H. L. Johnston, J . Am. Chrm. So&, 78, 63 (1954). (3) R. M. Buffington and W. Latimer, J . Am. Chsm. Sw.. 48, 2305 (1928).
cient of expansion, the last column gives the temperature derivative of a. lo is the length of the sample at 25'. TOK. average is the mean of the initial and final temperature of a determination. TABLEI NATURAL ROCKSALT lo = 1.113 cm., &quartz, = 1.099 cm. AZ x 104
AT
2.417 2.186 2.401 2.296 3 * 973 4 * 435 4.522 4.499 4.595 5.160 7.021 7.280 7.358 7.120 6.939 7.280
10.99 9.10 9.34 8.89 14.36 15.03 14.66 14.59 14.46 15.91 21.74 21.72 21.64 20.60 19.79 20.10
dl x
GdT
10'
2.1987 2.3970 2.5703 2.5827 2.7663 2.9510 3.0854 3.0830 3.1784 3.2429 3.2299 3.3519 3.4001 3.4553 3.5067
....
Quarts T avg., OK. corr.
0.330 ,344 .378 .377 .414 .456 ,495 .496 .531 .567 .577 .608 .650 .699 ,747 .794
92.30 102.36 111.59 111.35 122.88 137.86 152.42 152.69 167.21 182.40 178.15 199.95 218.38 239.51 259.71 279.65
First divided diff. a X IO5
2.245 2.462 2.649 2.659 2.857 3.061 3.217 3.215 3.333 3.423 3.402 3.558 3.639 3.732 3.822 3.968
*
Measurements marked in Table I1 ivere made by using a standard thermocouple alone. The rest of the results were obtained by using the standard thermocouple in conjunction with a precise resistance thermometer. The resistance thermometertemperature data were smoothed and tabulated at equal temperature intervals above 30'K. These data then were used to calculate the temperature intervals, AT. The thermocouple was calibrated in terms of the Ohio State University Cryogenic Laboratory temperature scale.* The runs marked 93 to 104 inclusive and 107 to 118 inclusive were obtained in such a manner that the temperaturelength point of one run formed the beginning length point of the next one. The remainder of the data were separate points. Since the interpolation of the fringe number requires values of the parameter 0 be known, smoothed values, 0', were computed first in order that the effects of random errors be minimized. This was done in the following way: 0 is inversely proportional to the product of the length of the sample and the refractive index of the helium in the cell. The index of refraction is that for a pressure of 1.3 cm. of helium at room temperature. Since (4) T. Rubin, H.L. Johnstonand H. Altman, ibid., 73, 3401 (1951).
T. RUBIN,H. L. JOHNSTON AND HOWARD W. ALTMAN
66
Vol. 65
TABLEI1 SYNTHETIC ROCKSALT,lo = 0.5570 cm. Run
AT
T,avg., OK.
93 94 95 96 98 99 100 101 102 103 104 106 107 108 109 110
13.215 13.720 14.414 14.530 16.212 20.248 20.082 25.183 29.746 29.746 29.841 10.48 10.354 9.512 7.396 8.364 8.421 8.440 7.167 12.271 11.685 8.48 6.92 9.69 6.44
84.342 97.81 111.88 126.35 141.72 159.97 180.12 202.80 230.21 259.95 289.73 95.57 26.473* 36.41 44.86 52.74 61.13 69.56 77.37 74.83 86.80 19.84* 27.49* 29.05* 34.71*
111
112 113 114 115 49 50 43 51
&a
0’ Smoothed
0
r12
r9
F
28.3 28.3 28.25 28.20 28.20 28.10 28.10 28.10 28.0 27.9 27.9
28.1 27.8 28.1 28.7 28.3 28.2 27.7 27.4 27.6 27.9 27.9
24.651 37.515 48.651 33.062 35.880 25.150 44.689 47.886 50.268 49.491 37.210
37.515 48.651 33.062 35.880 25.150 44.689 47.886 50.268 49.491 37.210 48.650
5.4563 6.3935 7.4482 8.0999 9.6195 12.6953 13.1138 17.0848 20.9723 21.5598 22.4100 4.7757 3.3154 0.7654 1.0181 1.6741 2.2270 2.7351 2.6396 4.3714 4.9139 0.0794 0.2295 0.3767 0.5447
...
28.30 28.30 28.30 28.30 28.30 28.30 28.30 28.30
..
....
....
28.0 28.2 28.2 28.5 28.7 28.3 28.0 28.3 28.0 28.4 28.5 27.6 28.4
23.620 32.547 25.908 26.419 45.495 23.619 44.422 19.448 29.921 6.864 5.808 11.760 12.350
32.547 25.908 26.419 45.495 23.619 44.422 34,222 29.921 27.510 9.120 12.350 22.180 27.820
x 10s 2.184 2.455 2.735 2.949 3.137 3.315 3.455 3.588 3.728 3.838 3.970 2.411 0.156 0.422 0.728 1.089 1.398 1.715 1.949 1.886 2.226 0.048 0.176 0.201 0.375
a
x
108
20.6 21.1 16.5 13.4 11.8 7.3 5.3 6.2 4.4 0.7 5.7
..
20.9 29.1 43.5 42.2 36.0 36.3 24.9
..
.. .. .. *.
..
the cell volume contains most of the gas, the refractive index is thus independent of temperatures and is almost equal to 1.0000. This is because the amount of helium is so small that the refractive index is essentially equal to the value for a vacuum. Thus the optic constant varies with the inverse of the sample length. Values of 0 were picked a t values of the reciprocal of the sample length corresponding to each mean temperature. These are the 0’ values shown in the table. Since most of these data may be regarded as continuous sets of length (on an arbitrary scale)temperature measurements a t unequal temperature intervals-first and second temperature derivatives of length were computed from them by the method of divided diff erences.6J These derivatives were computed a t the mean temperature of each interval. Division of the derivatives by Eo yields the values of a and b a / b ~previously mentioned. For a, calculations of divided differences to the fourth order were used. Calculation of ba/br required divided differences up to the sixth order. Sodium D radiation was used for all the interference measurements. Errors.-The absolute temperatures are accurate to about 0.03’K. Temperature intervals measured by the thermocouple alone are precise to about 0.02’. Those intervals measured in terms of the resistance thermometer are precise to a few thousandths of R degree. For the synthetic rock salt, the error in a length measurement is 0.007 fringe order which is the error in values of F.
Buffington and Latimer3 have determined the expansion coefficient of natural rock salt from about 120 to 3OOOK. The datta reported here for natural rock salt agree with theirs within 0.3%. The results from this research are always smaller in magnitude. The expansion coefficients for natural rock salt determined in this work are consistently about 0.6y0lower above 120’K. than the values of synthetic rock salt. Below 120°K. the values for natural rock salt are progressively lower, becoming about 4y0 a t 90’K. relative to the values for synthetic rock salt. 1. Theory of Griineisen’s Relation and Comparison with Experiment.-The approximate proportionality between thermal expansion and heat capacity a t constant volume (Griineisen’s relation) is justified by others’ on the basis of a simple model which is described below. It is assumed that the partition function for a solid may he obtained with sufficient accuracy on the basis of a model in which (i) the lattice vibrations are harmonic, and (ii) the vibration frequency depends on the volume. Then the derivative of the Helmholtz free energy with respect to temperature gives a formula for the heat capacity a t constant volume
(5) F. A. Willers, “Practical Analysis,” translated by R. T. Beyer, Dover Publications, Inc., 1947,p. 77. (6) J. B. Soarbrough, “Numerical Mathematical Analyses,” Johns Hopkins Press, Baltimore, Md., 1930,p. 115.
I n these formulas, B i = (hyi/lcT)2/sinha(h y i / 2 k T ) , yi = - 3 In vi/b ln V , Vi is the frequency of the
B,
Cv =
(1)
z
while the second derivative with respect to temperature and volumes gives the ratio of the thermal expansion coefficient (a)to the compressibility (8)
(7) D. Bilz and H. Pullan, Physica, 21, 285 (1955).
THERMAL EXPANSION OF ROCK SALT
Jan., 1961
67
0
0.10
0.20
2a 0.30
P
I i-
0.40
0.50
I
0.60
0.03
0.23
Fig. 1.-Variation
0.43 0.63 0.8d Ti3 m . of the Gruneisen parameter with temperature.
1.03
ith normal mode of vibration, R is the gas constant, V is the molar volume and the other symbols have their usual meanings. If we define a quantity y by the equation
pressed as the difference 7-y289.7. The temperature parameter, T/8, includes a value of 8 = 280°K.10 The value of R is taken as 82.054 cc. atm./mole, the density of rock salt is the room temperature value of 2.165 g./cc. YiBi For comparison, heat capacities were calculated i by graphical integration using Kellerman's frey = m quency distribution and using Einstein functions. l 4 i The difference y-y289.7 obtained from these also are shown in Fig. 1. (Dotted line with crossed points.) then from (1) and (2) it follows that Most of the scatter results from uncertainties in the (3) compressibilities. The function y has been expressed as a power and K = R y / V . series in T-l by Barrong~l6 on the basis of two If K were constant, this equation would be the approximate models. In the first model, the freGruneisen relation. It has been found experimen- quency distribution is assumed given with sufficient tally that K is very nearly constant at room tem- accuracy by considering only the interaction between perature and higher8 but decreases a t low tempera- nearest neighbors in a cubic crystal whose atoms are tures.2 A theory of the variation of y with tem- all of equal mass. In the second model, an interperature has been. developed by Barrong and it is of ionic potential for NaC1, consisting of a coulomb interest to compare this theory with experiment. term and an inverse 8th power repulsion was For this purpose it is necessary first to calculate y assumed but the positive and negative ions were from the experimental data. taken to have equal masses. For the first (nearest The results of this work together with the heat neighbor) model Barron found exprewion (4) for capacity results of Clausius, Goldman and Perlicklo y - y.., where y mis the high temperature limit. and the compressibilities of Rosejll Durand12 and y - y m = - 0 . 1 7 7 ~ ' ~ 0 . 0 6 5 ~ ' ~ 0.0270+8 + 0.013u+s Galt13 is sufficient for this computation over the (4) temperature range from 20 to 300'K. The results where Q = hvma,/2akT where vma, is the maximum of this calculation above 40'K. are shown in Fig. 1 as the solid line with circled points. They are ex- lattice vibration frequency. This constitutes the first four terms of an infinite series which converges (8) E. Griineisen, "Himdbuch der Physik," Vol. 101, Springer, Berlin, for u < 1 and may converge for larger values of u 1920.
+
(9) T. H. K. Barron, P h i l . Mag., I71 46, 720 (1955). (10) IC. Clausius, J. Goldman and A. Perlick, Z.Natusforschung, Ca, 424 (1949). (11) F. C. Rose, Phyr. Rev., 49, 50 (1935). (12) M. A . Durand, ibid., 60, 449 (1936).
(13) J. K. Galt, Phys Rev., 75, 1460 (1948). (14) "Contributions t o the Thermodynamic Functions by a PlanckEinstein Oscillator in One Degree of Freedom," H. I,. Johnston, L. Savedoff and Jack Beltzer. (15) T.H. K. Barron, Ann. Phusik, 1, 77 (1957).
68
P. E. EBERLY, JR.
Vol. 65
The second model used by Barron also leads to a series expansion similar to (5) but contains fewer terms since fewer moments were evaluated. This series is y
-
ym
-0.62~’
(6)
There is a notable difference (6) and (8) in that the term linear in y is missing from equation 6. This arises because two of the moment coefficients, r(2), ymlbwere taken by Barron to be equal and this makes the term in y vanish. y is computed from u by means of the relation given below.” A comparison between (5) and (6) with the’? experimental data, is shown in Fig. 2 (curves 1 and 2, respectively). The variation in y - y m predicted by either model is smaller than that observed 0 0.2 0.4 0.6 0.8 experimentally. However, the maximum values of y-ym for the nearest neighbor model has been Y. Fig. 2.-Comparison of theoretical and experimental greatly exceeded by the experimental values. This Griineisen parameters: 1, nearest neighbor model; 2, discrepancy cannot be remedied by considering sodium chloride model. the effects of more neighbors or by adjusting the values of the parameters’s within reasonable limits. It has been pointed out recently by Marruddin, The NaCl type model seems to be the more proWeiss and Sacklo that the expansion of Cv in a mising one since the curvature of the theoretical similar series which converges for large values of u can be made by an application of Euler’s trans- curve qualitatively appears to be about correct. It would seem worthwhile to compute some of the f~rmation.’~In view of this result, Euler’s transformation has been applied to the series of equation higher moments for the NaCl structure so that a more reliable value of the high temperature limit, 4 giving the series y m , could be estimated as well as furnishing more y - y m = -~(0.177 0.107~ 0 . 0 6 9 ~ ~0 . 0 4 5 ~ ~ )(5) terms for the series.7,’8~1g*20 where y = u2/(1 u2). This new variable is a It is a pleasure to acknowledge the help and enconvenient one. It is employed in the subsequent couragement offered by E. N. Lassettre of this dediscussion of the data. partment.
+
+
+
+
(16) A. A. Marruddin, G. H. Weiss and R. Sade, Bull. Am. Phys. SOC. in Detroit, 1960. (17) Bromwich, “Theory of Infinite Series,” The Macmillan Co., New York, N. Y.,1947,pp. 62. 2nd ed.
(18) G. Liebfried and W. Brendig, Z. Phyaik, 134, 451 (1953). (19) E. W. Kellerman, Phil. Trans., A.235,613 (1940). (20) E. W.Kellerman, Proc. Roy. SOC.(London). 1188, 17 (1941).
HIGH TEMPERATURE ADSORPTION STUDIES ON 13X MOLECULAR SIEVE AND OTHER POROUS SOLIDS BY PULSE FLOW TECHNIQUES’ BYP. E. EBERLY, JR. Esso Research Laboratories, Esso Standard, Division of Humble Oil & Refining Co., Baton Rouge, Louisiana Received June 0, 1960
The adsorptive properties of solids a t high temperatures are conveniently studied by allowing a pulse of adsorbate to be transported through a packed column of adsorbent by an inert carrier gas stream. The concentration of adsorbate in the effluent stream is continuously measured by a sensitive thermal conductivity cell. Modifications of such a technique permit the detection of an adsorption process and enable one to determine its degree of reversibility. In addition, heats of.adsorption can be determined by measuring the pulse retention times a t a series of temperatures. This flow method is particularly useful for studying adsorption at high temperature conditions where static methods cannot be used because of the long contact times involved which lead to decomposition of the adsorbates. In this manner, it has been found that materials such as 13X molecular sieve, silica gel, platinum on alumina and alumina itself adsorb significant quantities of benzene at temperatures as high as 427’. The adsorption capacity of 13X molecular sieve (0.632 mmole/g. a t 91 mm.) at these conditions.is markedly greater than those of the other adsorbents. With the exception of platinum on alumina, the adsorption is readily reversible and the heats of adsorption are typical of those ordinarily associated with a physical adsorption process. The heats of adsorption of benzene on 13X molecular sieve, silica gel and 7-aiumina were determined to be 15.5,Q.Sand 6.8 kcal./ mole, respectively, in the range of 260-454’.
I. Introduction I n the petroleum and allied industries, many chemical reactions and separation processes involving hydrocarbons frequently are conducted in the (1) Presented a t the Southwest ACS Regional Meeting, Capitol Home, Baton Rouge, Louisiana, December 3,4 and 6. 1959.
presence of solid materials a t elevated temperatures. Adsorption measurements under such conditions become of interest in elucidating the kinetics and mechanism of the reaction. Also, they are important in determining the various processes which must be taken into account in describing the mass
