J. Phys. Chem. 1994,98, 13123-13130
13123
The Complex Refractive Index h(a,T,p) of N02/N204 in the Visible and Static Dipole Polarizability of N204 Dirk Goebel, Uwe Hohm,* Klaus Ked, and Ulf Triimper Institut fiir Physikalische und Theoretische Chemie der Technischen Universitat Braunschweig, Hans-Sommer-Strasse 10, 0-38106 Braunschweig, F.R.G. George Maroulis*7+ Department of Mathematics, University of the Aegean, GR-83200 Karlovassi, Greece Received: May 2, 1994; In Final Form: August 25, 1994@
Dispersive Fourier transform spectroscopy in the visible (DFTS-VIS) is used to study the complex refractive index A of equilibrium mixtures of N O D 2 0 4 in the gas phase between 290 and 350 K and in the wavenumber range between 11 000 and 19 000 cm-'. With the aid of quantum-chemical MBPTMP2 calculations of the components of the static dipole polarizability tensor of N2O4, the quasi-continuous spectra of the complex mean dipole polarizability = a - ia' of the neat components NO2 and N204 are obtained. On the other hand, for these neat components a(uo)at wavenumber 00 = 15 798 cm-' is experimentally determined to be m3. The static limit of the mean dipole ano,(uo)= 3.0233(37) x m3 and aN,o,(uo)= 6.69(13) x polarizability a of N204 is calculated to be a(0) = 6.43(19) x m3. In addition, we report experimental results on the equilibrium constant Kp of the dimerization equilibrium N204 === 2N02, obtained from the temperature and pressure dependence of iz, as well as quantum-chemical calculations of the quadrupole and hexadecapole moment tensors of N2O4.
1. Introduction The propagation of an electromagnetic wave through an isotropic medium is fully characterized by the complex refractive index A(u) of this medium:'
A(u) = n(u)- -zP ( 4 4nu n(u) means the ordinary (real) refractive index, a(u) is the absorption coefficient of the Lambert-Beer law, o = l/A is the wavenumber of the electromagneticradiation of wavelength A, and i2 = -1. As A(@ is one of the most important macroscopic electrooptical quantities, it is worth studying the dispersion of A(o) experimentally without explicit use of the Kramers-Kronig relations.2 The best experimental technique for this examination seems to be dispersive Fourier transform spectroscopy (DFTS), which allows for the simultaneous determination of both n(u) and a(u) of eq 1. Therefore, at present DFTS experiences considerable interest, as can be seen from recent review^.^,^ However, in the visible wavenumber range, the suitability of this technique has been demonstrated so far only for nonabsorbing gases (a(a)= In the present work we will extend this technique to the case of absorbing gases. In this context we have chosen the gaseous equilibrium system N204 --L 2N02 for two reasons. First, it absorbs in the visible wavenumber range due to the presence of N02, while it is normally assumed that N2O4 does not absorb in this range.7 Second, the dispersion of the electrooptical parameters (complex refractive index A(@ and the related complex mean polarizability &(u))of the neat components NO2 and N2O4 is known only for very few discrete O).596
* U.H. is author for correspondence of the experimental part, and G.M. is author for correspondence of the theoretical part. Permanent address: Department of Chemistry, Physical Chemistry Laboratory, GR-26110 Patras, Greece. Abstract published in Advance ACS Abstracrs, November 15, 1994. @
frequencies.* Also, from the theoretical point of view, obviously nothing is known about the polarizabilities and permanent electric moments of NO2 and NzO4. Therefore, in order to support a most efficient evaluation of the experimental refractive index data, the static dipole polarizability tensor as well as the quadrupole and hexadecapole moment tensors of N204 are calculated by one of us (G.M.). The paper is organized as follows. In section 2, the relation between the complex refractive index A(u) and the complex dipole polarizability &(a) is evaluated. In section 3.1, the experimental setup used is briefly described, whereas the performance and evaluation of the measurements are outlined in sections 3.2, 3.3, and 3.4. The experimental results are presented in section 4. In section 5 , the quantum-chemical ab initio calculations are presented in some detail, followed by the discussion section 6 at the end of this paper.
2. The Complex Refractive Index ri
In this section we will describe briefly the connection between the complex refractive index A(u) and the complex mean dipole polarizability &(a) of isotropic species, bearing in mind that we are always dealing with low electromagnetic field strengths and that we are completely neglecting field gradient effects. However, these assumptions are perfectly fulfilled in our experiments. The starting point is eq 1. By using a Lorentzian local field and the Maxwell relation A = JC&R for nonconducting materials? one can derive the complex Lorentz-Lorenz relation, also valid for absorbing materials.' ( Z R = $//EO is the complex relative dielectric permittivity, and PR = P/po is the complex relative magnetic permeability.) However, in the present case two further simplifications can be made in the visible wavenumber range. First, in the case Of the gaseous mixture N02/ NzO4 the magnetic susceptibility y, = @R - 1) is on the order of with a negligible imaginary part, leading to A = JCR.
0022-3654/94/2098-13123$04.50/0 0 1994 American Chemical Society
13124 J. Phys. Chem., Vol. 98, No. 50, I994
Goebel et al.
8'
Second, assuming further that there is also no influence of intermolecular interactions on the dipole polarizability, the Lorentz-Lorenz relation can finally be written in the following complex notation:'
where & is the complex mean linear dipole polarizability with the dimension of volume. (For conversion factors to other units see footnote of Table 2.) e is the molar density, and N A is Avogardro's constant. The complex polarizability &(a) = b(a,T,e=O>can now be written in terms of no = n(a)and n1 a(a)/(4na),eq 1, as
&(a)= a - ia'
(3)
a = ( I / c D > [ ( ~ ,+~ n1212+ n: - n12 - 21
(4)
a'
= 6non,/CD
(5)
Real and imaginary parts of fi(o) and &(a), respectively, are related by the Kramers-Kronig relations.* In principle, knowledge of either real or imaginary parts is sufficient for the calculation of the other one, provided that it is known over the entire frequency range 0 I w < -. However, this is normally not the case. If one is especially interested in the real part n no of the complex refractive index of a low-pressure gas, eq 2 may be simplified further. In the low-pressure case the molar density e may be approximated by the density 40 = p/RT of the corresponding ideal gas
B(T) is the second thermodynamic virial coefficient and in our case assumed to be on the order of -120 cm3 mol-'. (See the calculations made by Plekhotkin. 13) Additionally, using only the real parts of eqs 2 and 3, the Lorentz-Lorenz equation 2 can now be written as
Within our experimental conditions (see section 3), the relative deviation between the real parts of the left-hand sides of eqs 2 and 8 is less than 10-l0, and between the real parts of the leftand right-hand side of eq 8 less than 5 x which is absolutely negligible in our case.
3. Experimental Section 3.1. Equipment. The dispersive Fourier spectrometer used is an evacuated Michelson twin interferometer, described in detail e l s e ~ h e r e . ~ItJ ~consists of a reference and measuring Fourier interferometer, which are motion-coupled and nearly identical in construction. A schematic sketch of the measuring interferometer is given in Figure 1. This interferometer is illuminated with a collimated white-light source WLS (100 W halogen lamp or 150 W Xe arc lamp). The light is split by the beam splitter cube BSC. The measuring beam A goes through the sample cell SC and is reflected on the fixed end mirror MA. The reference beam B traverses an evacuated reference cell RC and is reflected at the movable end mirror MB (planarity of
f
612 M0 I
\
d{ I PMT
sc
\
I
\
A
pJ Figure 1. Schematic drawing of the measuring Michelson interfer-
ometer (see text).
MA and MB 12/10 EZ: 63 nm). Both cells are made from quartz glass. They are approximately 1/2 = 50 cm in length and 8 mm in diameter. After superposition of A and B at the beam splitter cube BSC, the resulting interference pattem is recorded with a photomultiplier tube PMT with pinhole aperture, having nearly constant sensitivity between 300 and 700 nm. The optical path difference Aw(o) between the two arms of this interferometer is given by
Aw(@ = [n(@ - 111 - 6
+~ X U )
(9)
where n(a)is the real part of the refractive index of the probe, 1 is twice the geometrical length of the sample cell SC, 6/2 is the geometrical path difference between the two arms of the interferometer (both cells empty), and 6da) is a small term for correcting a possible small residual dispersion of the apparatus without gas. The path difference 6 is determined by counting the interference fringe shifts of the coupled reference interferometer (without cells), which is illuminated with a HeNe laser of wavelength A0 = 632.99 nm. The motions of MB and the corresponding end mirror of this reference interferometer are strongly coupled, allowing unique determination of changes of S(t) with time t. 3.2. Performance of the Measurements. Two kinds of measurements have been performed, which are described in full detail in refs 5 and 10. The first one is the separate recording of the two dispersive Fourier interferograms [Z(6) - I(-)] of the apparatus without gas (apparatus interferogram, Figure 2) and the apparatus plus gas (sample interferogram, Figure 3) by moving the end mirror MB, keeping temperature T and pressure p of the sample fixed (DFTS measurement). At present, the maximum path difference used is 6 = &lo0 pm, giving a spectral resolution of 100 cm-' (unapodized). However, the largest usable path difference in ow interferometer is 6 = k25 mm, yielding a resolution of 0.4 cm-'. The mirror velocity is about 100pm/min; the number of sampled data points is 60 000 per interferogram. The maximum useful spectral range is at present limited to 11 000-22 000 cm-'. The second kind of measurement is the isothermal absolute determination of the real part n(a0)of the refractive index of the gaseous sample for one reference wavenumber a0 = 15 798 cm-' (20 = 632.99 nm of a HeNe laser) in dependence on p and T, keeping 6 fixed. This measurement is simply performed by the isothermal
Complex Refractive Index A(a,T,e) of NO2/N2O4
J. Phys. Chem., Vol. 98, No. 50, I994 13125 h
1.55
h
10, Y
O
c
Y
‘I
KI
t
1
11000
-50
-1 00
=,6
100
50
0
&Pm
Figure 2. Apparatus interferogram [I(d) - I(=)].
I
I
I
1 t
1
I
1
I
-50
O
44,
1 50
100
w m
Figure 3. Sample interferogram [I(& - I(=)] determined atp = 3144 Pa and T = 309.99 K.
lowering of the pressure p of the sample down to p = 0 and counting the resulting interference fringe shift N = [n(ao,p,T) - l]aoZ. The sample N02/N204 is supplied by Linde AG, Germany, with a purity of 99.0% and used without further purification. 33. Evaluation of Measurements. The evaluation of the measurements is described in full detail in ref 5. Here, only a very brief outline is given. The recorded sample interferogram [I(S) - I(-)] is related to the complex refractive index spectrum A(@ via
4 6 ) - I(..) = J-:mA(a) e x p [ i k d ] da
(10)
with the abbreviation
A(@ = B(a)exp{ -ikaZ[A(a) - 11)
130OO
15000
17000 00
19000 oicm-’
Figure 4. Refractivity [n(a) - 13 and absorption coefficient a(a) of N O D 2 0 4 in the VIS determined by DFTS at T = 309.99 K and p = 3144 Pa, obtained from the interferograms in Figures 2 and 3.
i
-500
0.04
0*06 0.02
1
0.00
-500
I
0.08
(1 1)
B(a)is the complex refractive index spectrum of the apparatus and is determined by Fourier transformation of the apparatus interferogram. Hence, Fourier transformation of [I(& - I(-)] for both interferograms allows for a unique determination of the dispersion of the complex refractive index of the gaseous sample. Examples of both interferograms are shown in Figures 2 and 3, and the resulting Fourier transformation is shown in Figure 4.
3.4. Uncertainties. The evaluation of the uncertainties of the real refractive index n(a)and the mean dipole polarizability
a(a) is discussed in ref 9. Here we will only present the resulting uncertainties and some specific results concerning NO21 N204. The relative error of the real refractive index n(a) can be obtained by setting the maximum uncertainties in the recorded fringe shift N to AN/N = 4.2 x the length of the sample cell Al/Z = 1.7 x and the wavenumber of the laser used to A a h = 1.6 x The uncertainty in the refractivity results in A[n(a)- l]/[n(a) - 11 = 6.1 x loW4.This uncertainty holds for both the absolute measurement made at a0 and the DFTS measurement, because the uncertainty Aala does not make a significant contribution to this uncertainty.1° The uncertainty of the ideal gas density eo = p/RT is obtained using Ap/p = 5 x and ATIT = 1 x This leads to A&o = 5.1 x which, however, is a quite pessimistic estimate. The large uncertainty in the pressure measurement results mainly from the fact that time-dependent adsorption effects of the NO2 sample prevent a more precise pressure measurement. Therefore, using eq 8, the real part of the mean dipole polarizability has an uncertainty of A a ( a ) / a ( a )< 5.2 x Again, this result seems to be a pessimistic estimate and is not the typical uncertainty associated with our equipment, which is usually 5-10 times ~ m a l l e r . ~ *The ~ ’ absolute uncertainty of the absorption coefficient a(a) was estimated by recording an absorption spectrum of CHq, which, beside overtones of the C-H stretching mode, should show no absorption in the wavenumber range covered in our experiments.1° Nevertheless, an “apparent absorption coefficient” of la(a)l < 5 x cm-l was obtained for CHq by experiment, which is assumed to be the absolute uncertainty Aa(a) of the measured absorption coefficient a(a) of NO2 in the covered spectral range. (N2O4 does not absorb significantly?) Using eq 5 , the absolute uncertainty of the imaginary part of the mean dipole polarizability a’(@can now estimated to be Aa’(a) = 5.5 x m3.
4. Experimental Results 4.1. Refractive Index and Polarizability at 00 = 15798 cm-’. The refractivity [n(ao)- 11 of mixtures NOD204 obtained in our experiments at a0 = 15 798 cm-l in dependence on temperature T and ideal gas density eo is shown in Figure 5. The fit parameters of the smoothed curves are given in terms of the Lorentz-Lorenz function (eq 8) in Table 1.44 In order to obtain the real parts of A(a0) and &(GO) of the neat components, the mole fractions XN~&,T) = p ~ ~ o , /and p m@(p,T) =p ~ d are p calculated with the aid of the equilibrium constant K,(T) = ~ N % ~ / ~ given N ~by o ~Bodenstein and Bo&*
Goebel et al.
13126 J. Phys. Chem., Vol. 98, No. 50, 1994 TIK 289.42
m.57 319.01
328.99 339.91 349.28
2
"0
4
6
8
pdmoi m-3
Figure 5. Refractivity [n(oo)- 11 of N O f l ~ 0 4at uo= 15 798 cm-I as a function of eo = p/RT at different T. For the sake of clarity, only
the smoothed curves are shown (see text).
(p = PNO? -t- p ~ ~ 0Using ~ ) . the isotherms shown in Figure 5 , the refractivity [n(ao)- 11can be calculated for various densities eo in dependence on the mole fraction X N ~ .The result is shown in Figure 6. On the assumption that the equilibrium N204 * 2N02 forms an ideal gas mixture, the mixing rule aM
