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J. Phys. Chem. 1991,95,6971-6973
Molecular Diameters in HlgkTemperature Gases Steve H.Kang and Joseph A. K m * Departments of Aerospace Engineering and Physics, University of Southern California, LQS Angeles, California 90089-1 191 (Receiued: March 5, 1991) The kinetic molecular diameters of rotationally and vibrationally excited diatomic molecules and molecular ions arc derived. The diameters are functions of the rotational and vibrational quantum numbers of the incident and target molecules. As an example, 'typical" molecular diameters in several diatomic gases of temperature T are calculated. Introduction
Calculating some properties of gases requires' constants, called the 'molecular diameters", representing the "size" of the gas molecules. (For example, the concept of the molecular diameter is especially useful in Monte-Carlo probabilistic simulations of molecular collisions in high-temperature gases.) In low-temperature ("cold") gas the molecular diameters can be taken as the diameters of the molecules in the ground vibrational-rotational states. In high-temperature ("hot") gas, a large portion of molecules is rotationally and vibrationally excited and the values of the molecular diameters are greater than in unexcited (cold) gas. Since the molecular scattering cross sections are proportional (roughly) to the square of molecular diameters, the rotationalvibrational excitation of molecules can have a meaningful effect on the kinetic properties of high-temperature gases. This fact raises a question concerning what is the typical value of the molecular diameter in high-temperature gas. In this work, we study molecular diameters in high-temperature gases, taking into account the rotational-vibrational excitation of molecules. Two cases are considered: (a) the 'cold" gas, where the molecular diameter d, is independent of the degree of rotational-vibrational excitation of the molecules, and (b) the "hot" gas, where the molecular diameter dh depends on the degree of rotational-vibrational excitation of the molecules, that is, on vibrational ( 0 ) and rotational (J) quantum numbers of the molecules. 'Cold" cas The mean value of the molecular diameter d, in "cold" gas of diatomic molecules can be obtained from an empirical rule proposed by Hirschfelder and Eliason:'
d, = 73(r1+ rb) 1.8 (1) where all quantities are measured in angstroms, and r, and rb are mean radii of the Slater orbitals of the outer-shell electrons in the atoms forming the molecule na,b*(2%,b*+ 1) ra,b = a0 (2) 2(2a,b - Si,b) where Z is the atomic number, a, is the Bohr radius, n* is the effective principal quantum number for the orbital electrons, and S is their screening constant. Also, a review of the molecular diameters obtained from measured transport coefticients for gases of homonuclear diatomic molecules indicates that dc r* Re be; Re is the equilibrium bond length of the molecule (A) and the constant be = 2.3 A.
+
'Hot" CM The diameter dh(u,J) of a diatomic molecule excited to a uth vibrational level and a A h rotational level is proportional to the mean values of the internuclear distance (R,j) dh(u,J) r* (RvJ) + Ce (3) where it is reasonable to assume that ce = be if the degree of vibrational-rotational excitation of the molecule is not very high. (1) Himhfelder, J. 0.;Curtias, C. F.; Bird, R. B. Molecular Theory of Gases and Uqulds John W h y : New York, 1964. (2) Hirschfeider. J. 0.;Eliamn, M.A. Ann. N.Y.Acad. Scl. 1957,67,45l.
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In general, the mean internuclear distance can be found (through numerical calculations) from the eigenfunctions obtained from solution of the Schrodinger equation for a rotating anharmonic oscillator.' A simple and convenient approximation (limited, however, to levels with low and moderate values of the vibrational quantum number) to the mean internuclear distance can be obtained by using4 the following anharmonic potential, which includes the centrifugal rotational term: Uanb(S) = Uanb(0) + Bd2+ cd' + D d (4) where s = B,~/~( R4) (5)
and 4.the position of the minimum of the intramolecular rotational-vibrational potential, is R, = Re + 41,4J(J + l)/R,' (9)
12 = 16.863/~&
(10) where la is in angstroms if the reduced mass p of the molecule is in atomic units and the vibrational constant we is in cm-'. By use of the above and the second-order time-independent perturbation theory,' the eigenfunctions S, of the rotating anharmonic oscillator can be expressed as a function S,J(~ f ( B o , C o ~ D o ~ ~ , J S ~ j o ) (1 1) where SUJ"are eigenfunctions for an unperturbed rotating harmonic oscillator. Consequently,the mean intermolecular distance (Ruj) in a diatomic molecule excited to the u,Jth vibrationalrotational level is 3 8'1: Re + 5812(20 + 1) - ~ ( 1 0 9 1 22820 (&J) 192 41,' 1675~'- 12140' - 6 0 7 ~ ~+) y J ( J 1) R.
+
"( R,'
138' - 278
-
+ + +
-
24 + 1)J(J + 1) - 18R,) 1,8 (95,Re R,2 189p3 848' + R,2 + %)(31 + 78u 78u2)J(J+ 1) Re R,) +
-
(645
+
+
+ 20860 + 26930' + 12140' + 607d)J(J+ 1) (12)
(3) Kunc, J. A. J . fhys. B 1990,23, 2SS3. (4) Bonham, R. A.; Peacher, J. L. J. Chem. fhys. 1963,38, 2319.
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6972 The Journal of Physical Chemistry, Vol. 95, No. 18, 1991
where 8 is the Morse constant in the Morse function approximating the intramolecular potential. Neglecting small quantities and reducing (using the symbolic algebra) the second and the third / ~ ( u one terms in relationship 12 to 9plu2 e ~ p [ 2 ( ~ l ~ )-~ 1)]/2, obtains 9 412 Re + -812 exp[2(~lu)1~z(u - l)] 7 J Z (13) 2 Re If the molecule is not excited (u = 0, J = 0) then the mean internuclear distance is slightly different from Re because the u = 0 vibrations are slightly anharmonic. One should also remember that some vibrational-rotational levels do not exist in stable diatomic molecules even though the rotationless vibrational states with the same vibrational quantum number do exist. (R,,)
H
+
"Typical"Molecular Diameters in Higb-Temperature Gas The value of the "typical" molecular diameter for a diatomic gas of temperature T can be taken as one for the molecule excited to the vibrational-rotational level with "typical" quantum numbers u and Jp' (An attempt in ref 4 to average eq 12 over equilibrium dstributions of vibrational and rotational levels produced long and cumbersome formulas containing some misprints.) We choose the values upand Jpas the gas mean vibrational quantum number and the most probable rotational quantum number, respectively. Such choice is well justified, as representationof the corresponding distributions of rotational and vibrational levels, in gases of homonuclear diatomic molecules. The most probable rotational quantum number Jpcan be estimated from the equilibrium distribution gxJ) of the rotational levels because the rotational temperature is usually very close to the gas translational temperature while typical relaxation times for translational energy are usually very short. This distribution is gdJ) d J = C x 2 J + 1) exp[-BJ(J + l)hc/kT] d J (14) where C, is a normalization constant and Be (in cm-I) is the rotational spectroscopic constant. Since the rotational energy levels in diatomic molecules are closely spaced, the distribution (14) can be approximated by a continuous function of J with Jpbeing an integer close to
In stationary gases the value of u can often be obtained from the local thermal equilibrium (LTh) distribution (or from the Treanor distribution) of vibrational states at translational temperature T. It should be emphasized that in a gas where low and medium vibrational states are in the LTE with translational temperature other modes of energy distribution (including dissociation) does not have to be in the LTE. For example, the population of the higher vibrational levels (the most important for the process of dissociation) can strongly deviate from the LTE pop~lation;~ as a result, the overall dissociation process can easily maintain a state of nonequilibrium in the gas. The LTE distribution g,(u) of vibrational states has the following form if the vibrational energy is measured from the ground vibrational state where E, is the energy of the oth state, and Q, is the vibrational partition function
where u, is the number of vibrational states taken into account. Since the statistical weights of these states are equal to 1 and since (5) Nikitin, E.E.Theory of Elementary Atomlc and Molecular Processes In Gases: Clarendon: Oxford, U.K.,1974.
Kang and Kunc E, >> k T when u is high, the contribution of the upper states to the partition function Q and to the first moment of the distribution g,(u) is very small. Consequently, up is much smaller than u,. Thus, one can assume (but only for the purpose of calculating u ) that the partition function Q, and vibrational eigenvalues are cfose to their values given by the model of harmonic oscillator. Then
Q, N :Q = [ l - exp(-hcw,/kT)]-l
(18)
where h, c, and k are Planck's constant, the speed of light, and Boltzmann's constant, respectively. The number of the vibrational states of the anharmonic oscillator can be given as
where w&x, is the anharmonicity constant (cm-I), and Do is the dissociation energy (cm-') measured from the ground vibrational state. A simple and acceptably accurate expression for u can be obtained by assuming that u, = D/hco,, where D is t i e dissociation energy (ergs) measured from the bottom of the intramolecular potential curve. Then, neglecting small quantities one obtains
where the energy of the first excited vibrational state is El
hew,
(21)
and y = [l
- (D/kT) exp(-D/kT)][l - exp(-hcw,/kT)]
(22)
In most cases, the ratio D/kT is much greater than 1. (The case when the ratio D/kT is less than 1 corresponds to the situation when degree of dissociation of the gas is high. Then, atom-atom, not molecule-molecule, interactions dominate kinetic properties of the gas.) The assumption that D/kT is significantly greater than 1 gives y = hcw,/kT (if hcw,/kT < 1) or y = hcoe/2kT (if hcw,/kT 5 1); we used the expansion e* = Cx"/n!(neglecting the higher order terms) in the former relationship. Consequently up = q(kT/hcw,)
(23)
where in the case when the vibrational states have the LTE distribution when hcw,/kT < 1 7= 1 = 1/2 when hcw,/kT 5 1 (24) As said above, the "typicaln molecular diameter d{(T) in a gas of temperature T can be taken as the diameter of a molecule in the u th vibrational and Jpth rotational level. Then, using relationsRips 13, 15, and 23, one obtains d,,'(T) = d,
9 + ,812
exp[2fi(qkT/hcwe
21,4k~ - 111 + -
BehcR,' (25) where d, = Re + c,, and we and Be are in cm-I if h, c, and k are in centimeter-gram-second units. Equation 25 is of course valid for anharmonic molecules too (as is eq 12) because the molecular anharmonicity is contained in the Morse constant 8. As can be seen from eq 25, heating of a gas with a distribution of vibrational states close to the LTE distribution can have a significant effect on molecular diameters in gases with molecules having low values of the spectroscopic constants we and Be. At low and moderate temperatures with the V-V exchange processes dominating at the lower vibrational states, the distribution of the states is close to the Treanor distribution? Then
J . Phys. Chem. 1991,95,6973-6978
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Thus, in gases with the Treanor distribution of vibrational states, the value of the typical molecular diameter depends on the rotational constant B, and the ratios w , / ~ & ,and T/Tu;at higher temperatures, the diameter can be a strong, increasing function of these two ratios. As an example illustrating the impact of the rotational-vibrational excitation on the molecular diameters we calculated T dependence of the ratio
‘‘20m--7--7
X = (d,,’/dc)2
1.00 “ 1000
3 w00
2000
4000
5000
5
6000
7000
8wO J
Temperature (K)
Figure 1. T dependence of the ratio X (eq 28) in nitrogen, oxygen,
chlorine, and iodine when the low and intermediatevibrational states are in local thermal equilibrium with translational temperature. The ‘bump” on the N2and O2curves are in vicinity of q = 1, and they result from approximation 24.
upcan be taken as the vibrational quantum number corresponding to the minimum of the distribution. If T C To up
= ( E , /Zhcw&,)(T/Tu)
(26)
where Tuis the vibrational temperature. In the same case, but with T > Tu,the quantum number up can be obtained from eq 23 by replacing T with To. Comparison of expressions 26 and 23 indicates that relationship 25 also can be used when the distribution of the vibrational states is the Treanor distribution. In such a case the factor r] is t = (w,2/2w&,)(hc/kTu) (6)
(27)
Treanor, C.; Rich, J.; Rehm, R. J. Chem. Phys. 1968,18, 1798.
(28)
for several gases of homonuclear diatomic molecules with an LTE distribution of low and intermediate vibrational states. Ratio 28 is roughly proportional to the ratio of the “typical” kinetic cross section Qbin “hot” gas to the corresponding cross section Q, in “cold” gas. Examples of T dependence of ratio 28 are given in Figure 1. As can be seen from the figure, the rotational-vibrational excitation of molecules can have a meaningful effect on the collisional properties of gases. This effect is stronger in gases of weakly bound molecules (molecules with low values of the dissociation energy). (The dissociation energies of the molecules considered in Figure 1 are as follows: 9.76 (N2), 5.12 (02), 2.48 (Clz), and 1.54 eV (121.)
The assumptions made during the evaluation of the analytical approximation 25 limit its validity (if accuracy better than a few percent is required) to temperatures T S 9000 (N2), 8000 (02), 6000 (C12),and 3000 K (I2). However, because all diatomic gases are usually well-dissociated and atom-atom, not molecule-molecule, collisions dominate the gas properties at temperatures higher than those temperatures, this limitation is not important.
Acknowledgment. This work was supported by the National Aeronautics and Space Administration, Grant NAGW-1061, by the Air Force Office of Scientific Research, Grant 88-01 19, and the URI Grant 90-0170. Registry NO. 12, 7553-56-2; CIz, 7782-50-5; 02,7782-44-7; NZ, 7727-37-9.
X-ray Absotptlon Spectroscopy Study of the Titania- and Alumlna-Supported Tungsten Oxlde System Frank Hilbrig,”: Herbert E. Cabel,# Helmut Knozinger,*lt Helmut Schmelz,: and Bruno Lengeler*J Institut fur Physikalische Chemie, Universitat Miinchen, Sophienstrasse 1I, 8000 Miinchen 2, Germany, Power Generation Group KWU, Siemens AG, Otto Hahn Ring 6, 8000 Miinchen 83, Germany, Corporate Research and Technology, Siemens AG, Otto Hahn Ring 6, 8000 Miinchen 83, Germany, and Znstitut fiir Festkarperforschung, Forschungszentrum Jiilich, Posrfach I913, 5770 Jiilich I , Germany (Received: August 9, 1990; In Final Form: March 28, 1991) Tungsten oxide supported on titania is an important material for the selective catalytic reduction of nitrogen oxides NO,. A structure analysis by means of X-ray absorption spectroscopy (XANES and EXAFS) is reported for materials prepared by spreading of W03in physical mixtures with the support oxide as well as by impregnationfrom aqueoussolution. A comparison is also made with alumina-supported materials. Analysis of the XANES region at the W LIand W L3 edges indicated that tungsten is hexavalent and anchored to the surface as W05 and W04 units, the relative proportion of which increases with loading. When water is absorbed, pseudooctahedrally coordinated species are formed in both cases. The analysis of the EXAFS provides additional support for the existence of these structures, which contain oxo groups W 4 and W 4 W bridges. A tentative structure model is proposed, in which islands of surface tungstate species are formed by branched chains of WO, units. The chains are assumed to be terminated by W04 units, the W05/W04 ratio thus increasing with chain length or island size, which obviously increases with loading.
Introduction Titania-supported tungsten oxide catalysts are very efficient for various acid-catalyzad heterogeneous reactions,’ preferentially Universitit MBnchen. $PowerGeneration Grou KWU, Sicmen8 AG. I Corporate Rcaearch ant!Technology, Siemens AG. I Inititut far FcrtkCperfonchung.
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for the disproportionation of pr0pene.u For the selective catalytic reduction (SCR) of N O by NH3. WOdTi02 catalysts are of (1) Ai, M.J . Carol. 1977,19, 305. (2) Yamaguchi, T.; Tanaka, Y.; Tanabe, K.J . Coral. 1980, 65, 442.
(3) Yamaguchi, T.; Nakamura, S.;Nagumo, H.In Proceedings of d e International Congre8s on Catalysis, 8th; Vcrlag Chemic: Weinheim, Germany, 1984; Vol. 5, p 579.
CP 1991 American Chemical Society
