CHLORIXE EQUILIBRIA AND T H E ABSOLUTE ENTROPY O F CHLORINE B Y A. R. GORDON AND C O L I S B A R S E S
The modern interpretation of the structure of band spectra provides a direct means of calculating the entropies, specific heats and heats of dissociation of many gases. The numbers so obtained have been used successfully in calculating various thermodynamic quantities which are not immediately obtainable from experiment. I n particular, equilibria in homogeneous gas reactions can be predicted in many instances with an accuracy which leaves little to be desired from the point of view of the chemist. I t is probably only a matter of time and of increased knowledge of molecular spectra before the method is recognised as the most natural and accurate way of obtaining the characteristics of gaseous mixtures. The absolute entropies of many gases have now been calculated for fairly wide ranges of temperature, and the numbers so obtained can be used with a confidence impossible in dealing with “third law” values. An entropy value for a gas, derived from its molecular spectrum, contains certain contributions which, in the calculation of an entropy of reaction, cancel identically with similar contributions to the entropies of other constituents; thus the presence of nuclear spins and, as will be shown below, the existence of isotopes lead to contributions of this nature. If it were only desired to calculate equilibrium constants, such terms in the entropies could be omitted from the beginning, and calculation correspondingly simplified; but t o obtain entropies which may be designated absolute, every contribution suggested by the theory must be included. In this connection the effect of isotopes on an entropy calculation can be illustrated in the case of chlorine; although the isotopes are chemically indistinguishable, it will be shown that they are without thermodynamic significance only because of a peculiar compensation. I n the present paper we shall compute the absolute entropy of molecular chlorine, and use the values so obtained to predict the equilibrium constant for the Deacon reaction. 1. The Entropy of Chlorine According t o Aston, the atomic weights of the isotopes cis, and cis? are 34.98 and 36.98 so that in ordinary chlorine they are present in the proportions 0.761 : o . z 3 9 ; if it be assumed that for the temperature range involved the formation of a chlorine molecule is statistically determined by the relative abundance of the isotopes, then in ordinary chlorine the proportions must be 35-3533 j-37:37-37 = 0.579:0,364:0.057. The molar translational entropy ST for a gas of molecular weight M at temperature T and pressure I atmosphere’ is
ST = - 2 . 2 9 5
+ 3R/z.ln A I + gR/z.ln T
(1)
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CHLORINE EQUILIBRIA AND ENTROPY O F CHLORINE
At 300°K, ST is 38.675, 38.759 and 38.841 for the three molecular varieties, respectively. The rotational or vibrational entropy is found in the usual way by S = R(Z2/Z1) R.lnZ1 (2)
+
Xi
where
=
zp,,.e-en/kT, Z2 = zpn.(en/kT).e-en/kT
E,, being the energy of the n’th rotational or vibrational state of weight pn. For temperatures above 300°, the rotational entropy assumes its classical value when the sums are evaluated:
SR’= R
+ R.ln 8x21kT/h2
(31
where I is the moment of inertia of the diatomic molecule. It is known that clss has 5 / 2 units of spin; the spin of Cia, has not been determined, but we shall assume that it is the same as for C136.1 If this be the case, the spins of each type of molecule C12 contribute a term 2R.ln 6 t o the molar entropy. Now the 35-35 and 37-37 varieties are homonuclear, which means that (if it were not for nuclear spin) alternate rotational levels would be missing, i.e. a term R In z must be subtracted from SR’in these two cases; no such correction is needed with the heteronuclear 35-37. Thus if SR denote the contribution to the entropy from rotation and spins, SR(35-35, 37-37) = sIt’(35-35,37-37) 4-2R.h 6 sR(35-37) = sR’(35-37) 2R.M
+
- R.ln 2
Themomentsof inertiaof 35-35 and35-37 are 113.7 X 10-~Oand117.4 X IO-~O respectively,2 and by simple proportion the moment of inertia for 37-37 is 121.1 X IO-~O. It is apparent from the work of Elliott,2 however, that the moments of inertia increase by about 0.0070 of their values in the ground vibrational state per unit increase in the vibrational quantum number. I n a molecule with as low a fundamental frequency as chlorine, this will introduce a n appreciable error at high temperatures where the population of the higher vibrational states is large. A rough correction for this effect can be made by A) where multiplying the numbers given above by (I
+
A
= 0.0070(Zn.e-fn/kT) / (2e-enlkT)
in which E,, is the energy of the n’th vibrational state. Since 22/21 = aln ZJa In T, this will introduce in the expression for the rotational entropy, a correcting term s o.oo.joR(T.8 A/aT A). Including this correction,s S~foi-35-35,35-37 and37-37 at300°is 22.151, 22.591 and 21.275,respectively.
+
~~
If it should prove that CIS,have a spin different from this, the change in the numbers obtained can be readily calculated; for example, if the spin should be 7/2 units instead of 5/2, the entropy of 37-37 will be increased by zR In 8 / 6 and that of 35-37 by R In 8/6 so that the numbers for ordinary chlorine in Table I will be increased b o 273. The molar entropies of hydrogen chloride in the same Table will be increased %y ’exactly half this amount. %Elliott:Roc. Roy. SOC., 123A, 629 (1929);127A,h38 (1930). The numerical values of s are as follows: T“K 300 400 500 600 700 800 8
0.004
0.007
0.011
o 014
0.018
0.021
goo 0.025
1000
0.029
2294
A.
R. GORDON AND COLIN BARKES
+
+
The vibrational energy levels are given by E , = (n I / Z ) . W ~ - (n I/2)2,b and the weights pn are unity. Elliott? gives wo = 564.9 and b = 4.0 for 35-35; if it be assumed that WO is inversely proportional to the square root of the reduced mass of the molecule, wo for 3 5-37 and 37-37 will be 5 5 7 . 2 and j 4 9 . 4 , respectively. The vibrational entropy Sv for the three varieties is o . j 4 j J 0.560 and 0.577 a t 300°, so that the molar entropy a t the same temperature is 60.37, 61.91 and 60.69. Therefore the entropy of one mole of chlorine a t 300' is 0.j79 X 60.37
+ 0.364 X 61.91 t o.oj7 X 60.69 - i.98j8 (o.j;9 In 0,579 + 0.364 In 0.364 + In 0.057) = 62.63 0.05 j
The last item ( = 1.683) is the entropy of mixing. The values of thc entropy and heat capacity of ordinary chlorine for various teinperatures are listed in Table I ; the vibrational part, of the heat capacity \vas computed from the vibrational energy levels in the usual way, and a term R.L3(T2dA/dT)/dT was included to take into account t,he increase in the "average" moinent of inertia through rise in temperature; the numerical value of this correction varies from 0.006 at 300' to 0,035 at 1000'. 2. The Entropies of Hydrogen Chloride and Oxygen up to 1000° The translational entropy of HClZs and HC137 is given by Eq. I with hl = 35.99 and 37.99, giving for 300' ST = 36.695 and 36.857 respectively. The rotational entropy is given by Eq. 2 wit,h €1 = J ( J I)h2/8.rr21,pJ = 1 2 (zJ I ) , a n d 1 = 2.617 X ~o-~~forbothvarieties;'t,hefactor I Z int,heweight arises from the spins of the hydrogen and chlorine atoms. For 300' SR = 12.852. The vibrational ent,ropy2is given by Eq. 2 with pn = I and en = n For 300' Sv is negligible and for 1000' (2990 - j3.4n), where n = 1 / 2 , 3/2, is only 0.166, the isotope affect can be neglected in SR and SV. The entropy of ordinary hydrogen chloride as entered in Table I is given by
+
+
+
+
S = 0 . 7 6 1 S ~ ~ l , , O . Z ~ ~ S-~R(o.761 C , ~ ~lno.761 o.2391no.239) the last term ( = 1.093) being the entropy of mixing. If from the value for 300°, we subtract the spin entropy (R In I Z = 4,935) and the entropy of mixing, t,heresult, is 43.65, corresponding t o 44.61 a t 298'; Giauque and Wiebe? give 44.64 for this temperature; the difference is exactly accounted for by our use of R = 1.98j 8 instead of R = 1.9869. I n the case of oxygen, Giauque and Johnston4 have computed the entropy for 298.1'; for higher temperatures, ST is obtained from Eq. I with M = 32.00, SR by setting5 I = 19.27 x I O - ~ O in Eq. 3 and then adding R In 3/2 to the result, while Sv is computed from Eq. 2 with pn = I and E , = I j65n - 11.411~ where n = 0,1,2,3, . . Giauque and Johnston have shown that this method of
.
Czerny: Z. Physik, 45, 476 (1927). Kemhle: J. Opt. Soc. America, 12, I (1926). Giauque and Wiebe: J. Am. Chem. SOC.,50, I O I (1928). Giauque and Johnston: J. Am. Chem. Soc., 51, 2300 (1929). 5 Dieke and Bahcock: Proc. Nat. Acad. Sci., 13, 670 (1927); Rasetti: Phys. Rev., (2) 34, 367 (1929); Birge: Int. Crit. Tables 5 , 411. 2
2295
CHLORINE EQUILIBRIA AND ENTROPY O F CHLORINE
approximation leads to the same numerical result a t 298.1' as their more exact method based on the band spectrum formula for oxygen. The resulting values for the entropy and heat capacity are entered in Table I; for 298.1' the entropy is 49.00, which is 0.03 less than Giauque and Johnston's value; as in the case of hydrogen chloride, this discrepancy is exactly accounted for by our use of a smaller value of R.
TABLE I Molar Entropies and Heat Capacities of Chlorine, Hydrogen Chloride and Oxygen T"K
300 400 500
600 700
800 900 I000
Chlorine S CP
62.63 65.02 66.92 68.50 69.85 71.03 72.08 73.02
8.12 8.43 8.62 8.73 8.81 8.87 8.91 8.94
Hydrogen Chloride S CP
50.68 52.68 54.23 55.51 56.61 57.57 58.43 59.21
6.95 6.96 6.99 7.05 7.14 7.26 7.39 7.52
+
Oxygen
S 49.04 51.08 52.71 54.09 55.28
56.34 57.30 58. I7
CF
7.02
7.19 7.42 7.65 7.86 8.03 8.17 8.29
+
3. The Deacon Equilibrium 2C12 2Hz0= 4HC1 O2 If Q be the heat of reaction at temperature T and if ZS be defined by
2s = 2SCl1 f
2SHa0
- 4SHCI
- SO2
the entropies being all for temperature T and pressure I atmosphere, then at equilibrium R In K = R In (PcI,)~(PH,o)~//(PHcI)~(Po~) = - Q/T ZS
+
The heat of formation of hydrogen chloride1 at 291' is 22030 cals; the heat of formation of waterZa t 298' is 68313 cals. and the heat of vaporization at the same temperature4 is 10485 cals.; hence a t 298', Q = -27,500 cals. From this value, and the heat capacities of steam4 and the other gases (Table I), the values of Q/T entered in Table I1 are found by tabular integration. ZS, entered in the same table, is obtained from Table I above and from the known entropy of steam.4 The resulting R In X were used to construct the curve in Fig. I ; the experimental values of this quantity, as obtained from International Critical Tables: together with some additional experimental results of von Falkenstein6 and Neumann' are indicated on the figure. The following values are suggested for the reaction: AK'z98.1 = 27,500; AS'zss.~ = 30.99; AF'2ss.i = 18260. Int. Crit. Tables, 5, 176.
* Rosdni: Bur.Standards J. Research, (6) 1, 36 (1931). Int. Crit. Tables, 5, 138. J. Phys. Chem., 36, 1143 (1932). 6 Int. Crit. Tables, 7, 233. evon Falkenstein: Z.physik. Chem., 65, 371 (1909). Neumann: Z. angew. Chem., 28, 233 (1915).
* Gordon and Barnes:
2296
A. R . GORDON A N D COLIN BARNES
TABLE II The Equilibrium Constant for the Deacon Reaction T"K
- QP
-
s
RInK
600
700
46.70
40.17
32.21 $14.49
800
35.27
32.40
32,52
+7.77
+2.7j
900
IO00
31.46
28.42
32,60
32.71 -4.29
-1.14
+15 .Q 910 .o +5.0
Rln K 0.0
00 2 FIG.I
4.
Isotopes in Entropy Calculations
The entropy values of chlorine and hydrogen chloride as derived in Sec. I and 2 may be used to illustrate the general question of the effect of isotopes on an entropy calculation. Earlier calculations of the entropy of these two gases ignored the existence of isotopes in chlorine, and it is important to show the effect of this on the numbers obtained. To a first approximation, we may disregard the effect of isotopes on the values of SR and Sv,and consider ST as a linear function of the molecular mass LX. With isotopes considered in chlorine, there occurs an entropy of mixing of 1.683, which is absent when these are disregarded. In addition, in a non-isotopic chlorine, all molecules would be homonuclear, and the extra entropy due to the heteronuclear nature of 3 5-3 7, viz. 0.364R In 2, would not be present in the final entropy expression. Thus to ignore the existence of isotopes is to decrease the entropy of chlorine by 1.683 0.364R. In 2 = 2.18. If for non-isotopic chlorine h1 = 70.92, I = 116 X I O - ~ O , and wg = 560 cm-', the entropy can be readily evaluated and is found to be (at 300') 60.46, which is just 2 . 1 7 less than the number in Table I. Now to disregard the chlorine isotopes in hydrogen chloride is practically equivalent to dropping the entropy of mixing, via. 1.093, so that as far as the calculation of A S for the Deacon reaction is concerned, the effect of the isotopes cancels. By writing down the general expression for the entropy of mixing for a diatomic gas Xz having any number of isotopes in any relative proportions, and taking account of the -R In 2 which will occur in the molar entropy of
+
CHLORINE EQUILIBRIA AND ENTROPY OF CHLORINE
2297
any homonuclear combination, we can get the extra entropy due to isotopes. If we compare this with the entropy of mixing for molecules of the type AX, then in any reaction of the type
. . , + z A + X ~ = ~ A X +. .. AS for the reaction is the same (to a first approximation) whether we consider the presence of isotopes or disregard them. Of course the temperature must always be so high that the rotational energy is classical (Eq. 3). The indifference of AS to isotopes appears to be general, as would be expected for chemical reasons. The above type of cancellation can be shown to take place, for example, in the reaction
. . . A+zXz=AX4+ . . . For simplicity, suppose that X has two isotopes X‘ and X” in the relative proportions a,P = I - a. Then the entropy due to isotopes on the left hand side of the reaction is
- zR[a2.1naz + P2.1n Pz + zap.ln = - qR(a.111 a + P.ln P).
2 4 1
+ 4Rap.h
2
Now if AX4has the symmetry properties of methane, as may be assumed for the illustration, there is the following distribution of molecular types Type
AX4’ AX3’X’ AXz‘Xi‘ AX‘XC AX;’
Proportion
4
Symmetry Number 12
a4
4
3
6a2Pz
2
4aP3
3
P4
I2
The column headed “Symmetry Number” determines the correcting term to be taken from the classical entropy expression to allow for the absence of a certain fraction of the energy states. Thus the symmetry number for the homonuclear varieties of chlorine is 2 , and R In 2 is to be subtracted from SR’in Eq. 3 ; similarly the entropy of a mole of AXz’XZ”will be R In I Z / Z greater than that of a mole of AX4’ or AXI“, and so on. Therefore, the isotope entropy on the right hand side of the reaction in question is
- R[a4.1na4+ 401SP.ln 4 4 3 + 6dP2.1n 6a2P2+ 4001S.h 4aP3 + P4.1np4] f R[4a3/?.ln4 =
+ 6a2f12.1n6 + 4aps.ln 41
- 4R(a.ln a + @ln P).
This latter quantity is identical with that obtained for the left-hand side of the reaction. Of course, one cannot assert that this type of cancellation will necessarily take place in every reaction, since the “symmetry number” is known for only the simplest types of molecules; in all the simple cases that can be treated by the method sketched above, however, it is found that the part due to mixing
2298
A . R. GORDON .4ND COLIN BARNES
together with the part due to "symmetry" gives, on either side of the reaction, the same numerical result. To this extent, then, the effect of isotopes may be disregarded in questions of chemical equilibrium. The compensation is not exact, of course, for the mass effect of the isotopes has been disregarded, and this in general will not cancel, e.g. in the terms Sy of the entropies: this is a "second order" effect, however, amounting to (at most) a few hundredths of an entropy unit in the entropy of reaction.
Summary The ent'ropy of chlorine from 300' t o 1000' is computed from the spectroscopic constants of Elliott; in conjunction with the known entropies of hydrogen chloride, oxygen and steam, the equilibrium constant of the Deacon reaction is calculated for the range 6 0 0 ° - ~ 0 0 0 0 . The agreement b e t w e n the calculated and observed values of the equilibrium constant is satisfactory. It is shown that to the first order, the entropy of reaction is not affected by disregarding the existence of isotopes in the calculation; the rule appears to be general. A d d e d in proof, June 2 8 . 1932: Since this paper was sent to the Journal, Giauque and Overstreet (J. Am. Chem. Soc., 54, 1731 (1932) ) lt,ive computed the entropy of chlorine by a method which takes exact account of the change in the spectroscopic constants in the higher rotational-vibrational states; they find 53.310 as the entropv of chlorine at 298.1". If from o u r value at 3o0°, the spin entropy (7.116) and the isotope entropy (2.186) be subtracted, the result is 53.33, corresponding to 53.28 at 298.1'; with R = 1.9869, this last value becomes 53.31. A calculation by their method for 1000' shows that the entry in Table I is less than 0.02 in error.
The Cniversity of Toronto, Toronto, Canada. .'lpril, 1992.
