THE ENTROPY OF STE-411, AYD THE IYATER-GAS REACTION ny
A R . GORDOS A K D COLIX BARSES
The Entropy of Steam and the Water-Gas Equilibrium A knowledge of entropies provides the most direct method of obtaining free energies of react’ion, equilibrium constants, etc. from purely thermal data. The entropies of many diatomic gases at moderate temperatures have now been calculated from the formulae of quantum mechanics and the data of spectroscopy, and a few polyatomic molecules have been treated as symmetrical rotators, a strucrure of limited applicability. The entropy of steam is of fundamental importance in many thermodynamic problems, but so far the calculation has not been made for such an asymmetrical top. In this paper we shall compute the entropy, and check the result against the experimental values of the free energy and heat of formation of water. A second check on our value can be obtained from the watergas reaction whose equilibrium constant is now accurately known. Values of the entropy obtained by using an “effective” moment of inertia‘ have been previously employed to determine the “integration constant” for this reaction, but a more exacting test’ is to calculate the equilibrium constant itself, which, after all, is the qaantity given directly by experiment. This latter calculation requires also the entropies and heat capacities of hydrogen, carbon monoxide and carbon dioxide for temperatures from 600°K. to rzoo”K. These three gases are treated by a uniform method applicable t,o diatomic (and pseudo-diatomic) gases, and since t’he process’ is now a familiar one, only a very brief description of the essential points in the calculation is given. The heat capacities are comput,ed and tabulated with the accuracy necessary for the calculation of the water-gas equilibrium; the so-called experimental values3 are of no use; they still need a critical revision and a sharp dist’inction between laboratory values and numbers derived by extrapolation. As is usual, we get the contributions to the entropy of steam from the translational, vibrational and rotational parts of the energy. We believe that for this molecule the process gives reasonably good results up to rzoooK., though of course the interactions must eventually be taken into account. If the energy state en occurs with the weight pn we define the “state-sum” SI and the “energy-sum” 52 by
___
Zl
=
Z p n .e-e,,/kT
(1)
See for example Eucken: Physik. Z., 30,818 (1929).
* See for example Rodebush: Chem. Rev., 9,319 (1931). 3 Inter. Crit. Tables, 5, 79. This point has already been noted by Bryant: Ind. Eng. Chem., 23, 1019(1931).
1 I44
A. R. GORDON AND COLIN BARNES
22 = Zp, . (r,/kT) e-cn/kT
(2)
n
The entropy is then
S
+ R In Zl
= R(ZZ/Z~)
(3)
Here E , is the energy of a vibrational or rotational state. The translational entropy is given by the usual Sackur-Tetrode expression corrected where necessary for the deviation of the gas from ideality. We have used the following values of the constants:i R = 1.9858 cals./deg.; k = 1.372X10-l~ ergs/deg.; h = 6.55X10-*' ergs/sec.; N = 6.o6X1oz3; I atmosphere = 1,0133 X 106 dynes/cm.2 All temperatures are in degrees Kelvin, pressures in atmospheres, and entropies and heat capacities in cals./deg.
8 I : T H E TRANSLATIONAL AND VIBRATIONAL ENTROPY OF STEAM. The translational entropy is given by the familiar expression
+
+
+
ST = R l n [ ( ~ s / h ~ N 2.k5'Z.eS'2] )~ 5R/2 In T-R In P 3R/2 In 31 s (4) where M is the molecular weight in grams, s is the correcting term for deviation from ideality, and the other symbols have their usual significance. To compute s, we used the equation of state for water vapour given by Steinwehr;2 the magnitude of s is obvious from a comparison of the entries in the second and third columns of Table 11; at 300' for saturated steam it amounts to -0.003 in the entropy, while for 400' and I atmosphere it is about ten times as great. There must be three fundamental frequencies in the vibration spectrum of the H20 molecule; two of them are reasonably certain, the third is doubtful. Fortunately, the smallest frequency, which is by far the most important, is fairly accurately known. We adopted the values of Mecke3 w1
= 3930
- 95",
w2
= 3950 - ;oo,,.
The corresponding energy levels are
+
u3
= 1600
cm-1.
+
hc(nlw1 nzwz n3wd There are no weight factors when the summations involved run over all possible zero and integral values of nl, nz and n3. The contributions at the various temperatures are listed in the column headed SV in Table TI. At 1200' the vibrational entropy amounts to 1.215 with perhaps an error from the uncertainty in the frequencies of five per cent, while a t 300' the vibrational entropy is only 0.008.
0 z : THEROTATIONAL ENTROPY OF STEAM. The HzO molecule, so far as rotational states are concerned, is an asymmetrical top; its three principal moments of inertia are not known very accurately, and we selected the values 0.98 X I O - ~ O , I .80 X IO+O and 2.80 X IO-~O, given by h l e ~ k e .Fortunately, ~ a moderate change in the moments of inertia
_ _ _ _ ~ 3
Standard table (1926) Handbuch der Physik. Steinwehr: Z.Physik, 3, 466 (1920). Mecke: Physik. Z.,30,907 (1929). Mecke: Trans. Faraday. SOC.,26,214 (1930).
I745
THE ENTROPY O F STEAM
causes only a relatively slight change in the entropy; for example, the use of Bailey’s values’ (0.97 X I O - ~ O , 1.91 X I O - ~ O , 2.91 X I O - ~ O ) would lead to a rotational entropy only 0.09 higher than that entered in Table I. The energy levels of an asymmetrical rotator are now well known; the equations for the first nine sets J = o to J = 8 have been conveniently summarized by Dennison,* and need not be given here. The energy depends not only on J but also on a subsidiary number T which takes the 2 J + I values - J , - J + I , - J+z, . , . J- I , J. As regards the symmetry of the rotational states, we have (for even values of J ) symmetrical levels T = - J, - J+2, . . . and antisymmetrical levels T = - J + I , - J+3, . . . while for odd values of J, the levels T = - J, - J+z, . . . are antisymmetrical and the levels T = - J + I , - J+3, . . . are symmetrical? Moreover each level has the usual weight 2 J + I . In addition, the symmetrical and antisymmetrical rotational states must be weighted differently on account of the two hydrogen atoms, each with nuclear spin, in the molecule; the relative weights here are 3 and I . I t makes no practical difference (less than 0.001 in the entropy) whether the symmetrical or antisymmetrical states are given the weight 3, a circumstance which justifies the separate treatment of the vibrational contributions, for the symmetry of the vibrational part of the wave function changes for every change by unity in the quantum number n3. In our calculation, we gave the complete weights 3 ( 2 J + I ) to the symmetrical rot,ational levels and 2 J + I to the antisymmet,rical. The rotational energy levels E J for ~ J = o to J = 8 mere found by solving the exact equations;? for higher J values asymptotic formulae were used.j I t may be of interest for subsequent calculations to sketch briefly the use of these formulae. From the levels J = 8, the quantity A, = 8r’.ej,/h”J(J+1)
+ + 2p)/(J +
is plotted against ( J T . . . and is equal to 1 / 2 for .-
T
=
J -
I,
1/2)
(5)
where /3 = o for T = J, J - 2 , . . . From this graph we can
J - 3,
~.
Bailey: Trans. Faraday. S O ~ 26, . , 203 (1930). 2 Dennison: “Reviews of Modern Physics,” 3, 280 (1931). a This assignment of symmetry and antisymmetry to the rotational part of the wave function corresponds to Dennison’s case b (loc. cit. p. 32) and assumes that the intermediate moment of inertia lies along the axis of symmetry of t i e molecule (cf. Mecke:Trans. Faraday Soc. 26, 214). If the smallest moment of inertia were assumed to lie along the axis of symmetry, as in Eucken’s early model for the water molecule, Dennison’s case a would apply, . would be symand for either odd or even J , the levels r = - J, 2, i69.9, 769.9, J = 6: 473.6, 474.0, 574.01 582.1, 637.3, 679.4, 694.5, (88.3, 789.7, 925.0, J = 7: 622, 625> 731, 745, 823? 858, 889, 980, 1001, 1106, 1107, 1 2 7 2 , 1 0 ~ 3 . 1 1093.1. , I2i3, 1473, 1473, J =.8: 790, 790,934,937, 1042, 1061, 1x13, 1179, 1194, 131% 1317, 1478, 1482, 16797 1679, 1909, 1910. Kramersand Ittmann: 2. Physik, 58,217 (1929). I
+
J2.f’+
.
1)
I
146
A. R. GORDON AND COLIN BARNES
read off values of X, and hence of t J T ,for higher values of J and for positive values of r not too near zero. Another graph of the same two quantities but with p = I / ? for T = - J, - J 2, . and = o for T = - J I , - J + 3, . . . gives the value of X, for negative values of T not too near zero. Finally, the values of X, for r equal to or near zero are found directly from Eq. 47 of Kramers and 1ttmann.l The graphical method gives the mean positions of the pairs into which the rotational states fall for high J's, and for the present purpose it is unnecessary to compute the separations. Checks on the accuracy of the process were obtained by getting the sums for J = 8 from the levels J = 7; the exact method for J = 8 gives
+
2.688 (sym) . I ,
+
..
0.805 (anti)
12.169 (sym)
3.506 (anti)
as the contributions to the state and energy sums respectively (see Table I ) , while the approximation gives'the corresponding values 2.667 (sym)
0.828 (anti);
12.083 (syrn)
3.684 (anti).
Since the contribution to the energy sum from states with J > 8 is only about 5% of the whole sum, and the contribution to the state sum from the same levels is less than 2%, we see that the error introduced into the entropy by the approximation cannot exceed 0.005.
TABLE I Rotational Entropy of Steam at 300'K State Sum Sym. Terms Antisym. Terms
J
3.000 7 ' 502 27.555 23.016 27,591 15.555 11 ,417 5 094 2.688
0
I 2
3 1
0.000
0.000
0.000
5,095 5.947 I O , 602
1,365 13.295 22.322 41.568 33.665 32.971 18,495 12.169 5.514
0.828 3,040 9.966 10.840 13 .go6 9 ' I33 7.247 3.586 1.933 0.826 0.338
7.072
I1
0.396 0 .I34
I2
0.042
6.431 3.190 I ,982 0.805 0,355 0.129 0.045 0.014
I3
0.012
0.004
14
0.003
0.001
'5
0.001
0.000
5 6 7 8 9
1.025
IO
125.03
Energy Sum Sym. Terms Antisym. Terms.
2.550
0.997 0.360 0.117 0,033 0.006
+
22
SI, = R(z,/Z,) R In 21 = 1.9858 X 247.25/166.70 = 13.104cals./deg.
1
+ 4.5725 X
Kramers and Ittmann: loc. cit.
2.2219
0.039 0.011 0.002
61.82
185.43
41.67
2, = 166.70
0.120
247.25
1147
THE ENTROPY O F STEAM
The calculation of the rotational entropy was carried out for 300' and 400'; the former calculation is summarized in Table I. The rotational entropy for 400' exceeds that for 300' by (3Rjz).ln 4 / 3 so that we are justified in using the complete "classical" excitation of the rotational energy to get t,he rotational entropies at higher temperatures from our values at 300'; the numbers so obtained are entered in the column headed Sn in Table 11. Classical methods,' using an "effective" moment of inertia, the Ehrenfest "symmetry number" and an added constant ZRIn 2 (which allows for spin), lead to a rotational entropy a t 300' only 0.007 greater than the value in Table I ; if this agreement is not fortuitous, it suggests that for thermodynamic calculations the older formula should be quite satisfactory for asymmetrical rotators, and that the very laborious calculations sketched above could be avoided.
9 3 : THEESTROPY OF SATURATED STEAM AT 300'. Since the vapour presriure of water at 300' is 26.739/760 atm., the translational entropy of saturated steam a t this temperature, including the correction from the equation of state, is 41.278. From Table I, SR = 13.104 and Sr = 0.008; hence the entropy of saturated steam at 300' is j4.39. The heat of formation per mole of liquid water at 298.1' and I atm. is 682;o cals.2and the free energy of formation, under the same conditione, is 56560 cals. Hecce SHI()(,)
-
-
I/Z
Soi
= -11;10/298.1
= -39.29
Since SH,is here3 33.98 and Sol is4 49.03, this gives for 298.1' SI12(, = 19.21 and for 300°, S H ? ~ , = ~ , ,19.33. The latent heat of vaporization5 at 300' is 2432 joules/gram = 104;o cals./mole; hence the molar entropy of satu10470/300 = j 4 . 2 3 , in satisfactory agreement rated steam at 300' is 1 y , 3 3 with our calculated value j 4 . 3 9 ,
+
Had we worked from our spectroscopic entropy value for steam at 3 0 0 ° , using the heat of formation, the latent heat of vaporization, and the spectroscopic entropies of hydrogen and oxygen, we should have arrived at a value for the free energy of formation of liquid water at 298.1' only j o calories higher than the accepted j 6 j 6 0 ; this difference is well within the limits of possible error. For higher temperatures and for I atmosphere pressure, the entropy of steam is entered in Table I1 for reference. 9
5
See, for example, Eucken: Physik. Z., 30, 818 (1929). Inter. Crit. Tables, 7,2 j 6 . Giauque: J. Am. Chem. SOC.,52,4816(1930). Giauque and Johnston: J. Am. Chem. SOC.,51,2300(1929) Inter. Crit. Tables, 5, 136.
I 148
A. R. GORDON AND COLIN BARNES
TABLEI1 Molar Entropy of Steam a t ST (ideal)
T"K.
400
36.062 37,170 38.075 38.840 39.503 40.088 40.611 41.084 41.516
500
600
7 00 800 900 1000 I IO0
I200
Sr
(actual)
36.025 37.152 38.066 38.835 39,501 40.088 40.611 41.084 41.516
I Atmosphere SR SY 13.959 0.043 14.626 0.115 15.167 0,222 15.626 0.346 0.494 16.024 16.375 0.657 16.688 0.831 16.972 1.020 17.231 1.215
5
50.03 51.89 53.46 54.81 56.02 57.12 58. I 3 59.08 59.96
§ 4 : THEWATER-GASEQUILIBRIUM. In Tables 111, IV and V, are tabulated the entropies of hydrogen, carbon monoxide and carbon dioxide for temperatures ranging from 300' to IZOO' and for I atmosphere pressure. In the case of hydrogen, the rotationalvibrational entropy was calculated by the method of Giauque', though a direct calculation in the standard way using a moment of inertia 0.48 X I O - ~ O and a fundamental frequency w. = 4262 - 113.511 gives entropy values differing from those obtained by his method by 0.04 a t most. For carbon monoxide,* the moment of inertiaused is 14.9X10-~0 and wo = 2155 - I2.7n, while for carbon dioxide, the moment of inertia3is 70.2 X I O - ~ O and w1 = 2295, w2 = 672.5 and w 8 = 1223.5 cm-I (Eucken's freq~encies).~I n the case of carbon dioxide, alternate rotational levels (J = I , 3, . . .) are missing, and
TABLEI11 Molar Entropy of Hydrogen a t T"K.
300 400 500
600 7 00 800 900 1000
I100 I200
I
Atmosphere
ST
SRY
28.110 29.538 30.646 31.551 32.316 32.979 33 ' 564 34.087 34.560 34,992
5.894 6.461 6.910 7.279 7.594 7.872 8.123 8.354 8.572 8.778
5 3 4 . 005 36.00 37.56 38.83 39.91 40.85
41.69 42.44 43 ' '3 43.77
See Giauque: J. Am. Chem. Soc., 52,4817 (1930),Table I. (2) 32, 206 (1928). a Houston and Lewis: Proc. Nat. Acad. Sci., 17, 231 (1931). Eucken: 2. Physik, 37, 714 (1926). 6 Giauque's value vis. 33.98 for 298.1' would correspond to 34.02 for 300"; the slight discrepancy arises from our use of R = I ,9858while he used R = I ,9869.
* Mulliken: Phys. Rev.,
THE ENTROPY O F STEAM
I I49
TABLEI V Molar Entropy of Carbon Monoxide a t T"K. Sr SR ,369 11,940 12.383 12.745 13.051 13.316 13.550 13,759 13.948
35.948 37.376 38.484 39.389
3 00 400 500
600 7 00 800 900
11
40,154
40.817 41.402
IO00
41,925 42.398 4 2 ,830
I IO0
I200
I
Atmosphere S"
5 47.32 49.32 50.90 52.21 53.34 54.35 55.26 56.08 56.84 57'54
0.000
0.008
0.031 0.075 0.137 0.215 0,303 0,397 0.495 0.591
14.121
TABLEV Molar Entropy of Carbon Dioxide a t ST (ideal gas) Sr (actual gas) SI
