J. Phys. Chem. 1993,97,2307-2313
2307
Vibrational Frequency Shifts of Fluid Nitrogen up to Ultrahigh Temperatures and Pressures George S. Devendorf and Dor Ben-Amotz' Department of Chemistry, Purdue University, West Lqfayette, Indiana 47907- 1393 Received: November 30, I992
Vibrational frequency shift measurements in fluid nitrogen (over a temperature range of 77 K < T < 3521 K and a pressure range of 1 bar < P < 158 kbar) are used to test the predictions of a generalized perturbed hard sphere fluid theory. The experimental data include new Raman scattering measurements, up to near the freezing density of N2 at 295 K, as well as previousstatic and shocked high-pressure and temperature measurements. These are compared with the predictions of a theory which independently treats the effects of short range repulsive, long range attractive, and centrifugal forces. Repulsive solvation forces are modeled using the cavity distribution function of a reference hard sphere fluid. The temperature and density dependence of the reference fluid diameter is determined by fitting N2 equation of state data to an expression of CarnahanStarling-van der Waals (CS-vdW) form. Attractive force perturbations are incorporated using a single temperature and density independent mean field parameter, and centrifugal forces are treated by interpolating tow and high collision frequency expressions. The results are found to accurately reproduce both equation of state and frequency shift measurements over the entire temperature and pressure range of interest.
1. Introduction
Pure fluid nitrogen is one of the simplest systems in which solvation-induced perturbations of molecular potential energy surfaces can be explored. The internal vibratfonal potential of each nitrogen molecule serves as a direct probe of the mean force potential imposed by neighboring molecules. In spite of the ubiquity and fundamental importance of nitrogen, surprisingly few pressure and temperature dependent measurements of its vibrationalRequency have been reported, and among these, there are significant discontinuities and discrepancies. We have thereforeundertakento measurethe Raman frequency of nitrogen over the entire fluid density range at 295 K, as well as to connect our results with previous lower and higher pressure and temperature measurements performed both in our owni and in other24 laboratories. The combined results are used to test and extend a recently developed perturbed hard sphere fluid model for mean force induced perturbations of molecular potential energy surfaces.7J In this model, patterned after the work of Chandler and co-workers?-I I repulsive solvation forces are calculated using a 'hard fluid" model representing the properties of an appropriate reference fluid consisting of hard spherical solvent molecules and a hard 'dumbbell" solute molecule composed of two interpenetrating hard spheres(or 'cavities"). Attractive forces are treated using a mean field approximation, which predicts a linearly density dependent attractive force induced frequency shift.9 The slope of this attractive shift is described by a single temperature and density independent parameter. Such a perturbed hard sphere fluid model has previously been found to reasonably represent gas to liquid and high pressurefrquency shift results in a number of diatomic and polyatomic liquid and supercritical fluid systems.I.7.8.lZ New featuresof the present ultrahigh temperatureand pressure model include the incorporation of a centrifugal potential that interpolates between expressions appropriate at low and very high collision frequencies. Furthermore, the effective hard sphere diameter of the reference system is taken to be a function of both density and temperature, in order to properly account for the softness of the intermolecular potential of N2.Io As in previous work,1s7JJthe reference system diameter is evaluated by fitting equationof state data to an expression of the CarnahanStarlingvan der Waals (CS-vdW) form.l3 The resulting model is found to accurately reproduce both equation of state and frequency 0022-3654/93/2097-2307S04.00/0
shift data, up to densities over twice that of the normal liquid (at 77 K and 1 bar), and absolute temperatures over 10 times room temperature. 2. Experimental Techuiques and Procedures
Raman frequency measurements were made using the 5017.16-Alineofa Coherent Inova70-5 argon-ionlaserat -100 mW. An Olympus 50 mm,f/l.2 camera lens both focused the laser onto the sample and collected the backscattered light which was dispersed with a Spex 1400 double monochromator (f/8). Diffraction gratings of 1200 grooves/mm and entrance and exit slit widths of 300 pm give an effective spectral width of 4 A and allow peak positions to bedetermined within 0.2 cm-I. Thesignal was detected using an EM1 4558 water-cooled photomultiplier tube, and after amplification and discrimination, pulses were counted for 1 s at each wavelength point using a Macintosh IIcx computer with Labview I1 software. A t ical spectrumconsisted of two to six repeated scans over a 40- wavelength range with 0 . 2 1 4 intervals between each point. The peak Raman signal intensity under these conditions was on the order of 9000 counts/s above a background of 2000 counts/s. Neon and argon Penray penlamps provided simultaneously collected reference lines for absolute frequency calibration. Plasma lines emanating from the Ar ion laser were eliminated with a Pellin-Broca prism and stray light was reduced by prefocusing the Raman light through a 400 pm diameter spatial filter before refocusing into the monochromator entrance slit. A gasketed, Merrill-Basset diamond anvil cell was used for high pressure measurement^.'^ Number X750 stainless steel gaskets, 0.25 mm thick, and drilled with a 0.7-mm hole, were sandwiched between 0.15-carat type l a diamonds (Drukker standard 16 sided, 3 mm girdle, 1.2 mm culet) affxed directly to 3/8 in. thick sapphire window (Crystal Systems, Inc.) backing plates. This design withstood pressures well above the 24 kbar freezing pressure of nitrogenls and provided a wider angle of collection and optical access to the sample than conventional steel backing plate designs. Nitrogen was loaded in the diamond anvil cell by opening a cell with a preindented gasket under commercial grade, liquid N2.16 Expansion of the pressurizing screws while warming necessitated clamping and repeated tightening of the screws to keep the cell sealed as it warmed. Pressure measurements were determined by the ruby fluorescence technique." Over our
r
0 1993 American Chemical Society
Devendorf and Ben-Amotz
2308 The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 experimental pressure range the ruby R I peak fluorescence wavelength is linearly dependent on pressure with a slope of 0.365 A/kbar. Three chips were placed in the high-pressure cell and fluorescence measurements were made on each chip to verify pressure uniformity (hydrostaticconditions) of the sample. These were compared with measurements performed on ruby at atmospheric pressure, in order to determine the sample pressure. Laser power was reduced to 0.03mW in the ruby fluorescence measurements. Tests at higher laser power confirmed the absence of any significant laser induced heating of the sample in either the ruby fluorescence or Raman measurements. Pressure to density conversions were performed using the CSvdW equation of state (as described in section 3). The bard sphere diameter used in this equation was obtained through fits to IUPAC equation of statedatala and the higher pressure shocked data given in Table I1 (see section 3.1). Two to three Raman and ruby fluorescence measurements were made at each pressure. Absolute frequencies were determined from the difference between the spectral peak of interest and either the 5689.8-Aemission line from a neon pen lamp (for N2 Raman measurements) or the 6965.4-Aargon pen lamp line (for ruby fluorescence measurements). The pen lamp calibration peaks were recorded simultaneously (in the same scan) as the spectral peaks. Raman vibrational frequencies were obtained from the difference between the nitrogen Raman peak and the laser line. Frequency shifts relative to the dilute vapor were determined using the zero density fundamental pure vibrational frequency, io, Q-branch origin of 2329.91 cm-1 reported by Lav0re1.I~ Reported experimental vibrational frequency and pressure values are averages of at least three measurements at each pressure. Experimental errors are determined using an average of the standard deviations of measurements at each pressure and appropriate 1-values. Experimental errors of f0.5 cm-I for frequency shifts and i0.5 kbar for pressures reflect a confidence level of over 95%. 3. Perturbed ‘Hard Fluid” Model for Vibrational Frequency Shifts The total shift in Vibrational frequency of a solute diatomic, relative to its vapor phase Q-branch origin (pure vibrational transition), is treated as the sum of the repulsive, Pur,attractive, Aua, and the centrifugal, Avcrforce induced shifts. Aulota,= Aur + Aua
+ AuC
empirically or through a theoretical dispersion force expression (section 3.2). Centrifugal shifts are calculated from the bond length derivatives of the centrifugal potential of the rotating diatomic solution at finite temperatures and densities (section 3.3). 3.1. Repulsive Frequency Shift. Packing forces (Le. steric interactions) play a dominant role in the structure and thermodynamics of dense fluids.1°q2’ Such repulsive excluded-volume forces can be approximatedusing a fluid composed of hard spheres, and their influence on the vibrational potential of a diatomic solute can be related to the properties of an appropriate twocavity distribution function in the reference fluid? This distribution function describes the probability of finding two cavities (of diameters equal to those of the diatomic atom hard cores) at a separation equal to the diatomic bond length. The “hard fluid” model offers accurate analytical expressions for this distribution function and its derivatives with respect to solute bond length (see Appendi~).l,~**-12 The hard-sphere diameter of the solvent is the single most sensitive parameter in the hard fluid model. Variations of a few percent in this diameter can cause significant errors in the predicted frequency shifts. As pointed out in previous studies,13 an appropriate effective hard sphere diameter can often be obtained by fitting high pressure equation of state data to a perturbed hard sphere fluid equation of state of the CS-vdW form13
At low to moderately high pressures (up to tens of kbar) the effectivehard sphere diameter, UHS, may be taken to be a function of temperature only, and the attractive mean field parameter, 7 , either constant or mildly dependent on temperature.13 The following two parameter functional forms for the temperature dependence of UHS (derived from exact expressionsfor the thermal turning point in a bead on collision between two Lennard-Jones spheres) and the mean field parameter 7 , have been found to reasonably represent the equation of state of atomic and polyatomic high pressure liquids and supercritical fluids.13
(1)
These frequency shift contributions are in turn related to the first and second bond length derivatives of the excess potential energy of the solute (relative to the isolated solute at 0 K). For a cubic anharmonic solute oscillator the resulting frequency shift contributions can be expressed as20
where Av represents each of the individual shift contributions in q 1, V is the corresponding excess potential energy, ve is the classical frequency, re is the equilibrium bond length, andfand g are the quadratic and cubic force constants, respectively, of the diatomic ~ o l u t e . ~ Repulsive contributions to the frequency shift are determined by relating F and G to the hard sphere reference fluid cavity distribution function (section 3.1 ) . ’ ~ 7 ~ 8 J 2Attractive contributions are assumed to contribute a linearly density dependent mean potential energy field, whose magnitude is determined either
7(T)
= 7,
+-T, T
(4b)
The values of the two parameters in this expression, as well as those describingthe temperature dependenceof 7, have previously been determined by fitting equation of state data for a large number of compounds.I3 The corresponding parameters for N2 (derived from data spanning the 220 K < T < 380 K and 1 kbar < P < 10 kbar temperature and pressure range)22 are given in Table I. These are found to accurately represent the equation of state over a 200 K < T < 600 K and 0.1 kbar < P < 10 kbar temperature and pressure rangel* (see Figure 1 and section 4.1 for discussion). At the ultrahigh pressures and temperatures of interest in this study, further refinements of the reference fluid model, allowing for both temperature and density dependence of the hard sphere diameter, must be introduced.lOJ In order to determine an appropriate functional form for the dependence of the diameter on density and temperature, we again fit equation of state data to the CS-vdW equation. The fit diameters are assumed to be representable by a generalized form of eq 4a, with additional temperature and density dependent flexibility. After trying
The Journal of Physical Chemistry, VO~. 97, NO. 10, 1993 2309
Vibrational Frequency Shifts of Fluid Nitrogen TABLE I: Nitrogen Parameters ref gas phase bond length classical vibration frquency gas phase transition frequency quadratic force constant cubic force constant ionization potential polarizability bond polarizability derivative in cq 4a (and eq 3) 7-
in cq 4b (and cq 3)
T, ut a1
in cq 5 (and cq 3)
a2 01 T
1.09768A 2358.57 cm-’ 2329.91 cm-I 22.95 X losd/cm -56.69 X 10” d/cm2 15.51 eV 1.76 1.75 A2 3.956 A 182 K 353.4 K 17.1 K 4.0572A 130.3K -3.5195 X 10-4 nm6 7.2536 X 10-6K-’ 388.1 K
31 31 19 38 38 39 9 9 13
a
This work.
various such expressions, we have found the following to be the simplest one capable of accurately reproducing equation of state data over the temperature and density range of interest.
ambient Alternatively, a direct theoretical estimate of C,, based upon a simple dispersion force model, may be used7
c, =,wa-Araa ar
(7)
uHS
where W is the ionization energy, a the polarizability, Ar the bond length change of the solute upon vibrational excitation, and aa/ar the bond polarizability derivative. Equation 7 is derived from eq 2 using standard dispersion force expressions, with the added assumptionsthat the higher order derivatives of a are equal to zero and the radial distribution function of the solvent about the solute is equal to one,’ In a previous study of high pressure frequency shifts for acetonitrile, CH3CN,dissolved in various solvents, an expression analogous to eq 7 has been found to underestimate empirical attractive frequencyshift coefficients. For such a polar polyatomic solute, however, the discrepancy may reasonably be attributed to the influenceof significant dipolar and multipolar interactions. In the case of Nlr on the other hand, a purely dispersive model for C, may be expected to be more accurate. This expectation is borne out by comparison with empirical attractive force shift coefficients (see section 4 and Figure 2). 3.3. Centrifugal Frequency Shifts. Centrifugal forces result from the rotational motion of molecules at finite temperatures and, as discussed below, these may also be expected to depend on density, particularly when intermolecularcollision frequencies approach the vibrational time scale. The standard treatment of centrifugal forces begins with the definition of the classical rotational energy of a diatomic
The parameters UI, a l ,(12, and a3 in this expression, as well as the mean field constant T (taken to be temperature and density independent), were determined from a least-squares best fitZ3to N 2equation of state of data, over an extended temperature range of 200 K < T < 3521 K and a pressure range of 0.05 kbar < P < 158 kbar (see parameters in Table I).$3 The fit data include Schmidt’s reported densities, pressures and temperatures for shocked N2 (the highest pressure, 344 kbar, point was excluded where 1 is the angular momentum and I is the moment of inertia from thefit),andIUPACequationofstatedata.18 Thediameters of the diatomic. If, in addition, angular momentum is assumed obtained with eq 5 reproduce the experimental pressures with an to be conserved during a vibrational period, then the following average error of 1.1% (see section 4.1). expression is obtained for the centrifugal potential as a function of the diatomic bond length (expanded about the equilibrium The hard fluid model requires input of effective hard sphere bond length re).24,30 diameters for both the “solvent” N2 molecules, whose diameter is determined from eqs 4 or 5 above, and the two atoms of the “solute” N2 molecule whose frequency shift is to be determined. In previous work various routes to the estimation of atomic (or pseudoatomic) solute diameters have been p r o p o ~ e d . ~ *In ~-~-~~ Utilizing eq 2 (with F = -2keT/rC and C = 3k~T/r,2)the following this work, nitrogen atom diameters are determined by requiring expression for the average centrifugal frequency shift, Aut, is the total volume of the diatomic (hard “dumb-bell”) of bond obtained. length rc,2J to be equal to that of the effective hard sphere representingmolecular N2. This self-consistent definition of the AuC(cm-I) = -0.006044T (K) (10) diameter of each nitrogen atom constrains the atomic diameters This result is within 7% of that derived independently using to track the temperature and densitydependence of the molecular empirical rotational and rotation-vibration coupling mnstants3l diameter obtained from the equation of state data. Thus the in an expression for the shift of a motionally narrowed Q-branch diameters for both the “solvent” (spherical) and “solute” (diband.32 The next higher order temperature correction to this atomic) nitrogen are derived directly from equation of state data. centrifugal shift19 has also been evaluated and found not to These diameters are used without further adjustment in the contribute significantly, even at the extremely high temperatures repulsive frequency shift calculations. of interest in this study. For example, at 3500 K, the second3.2. Attractive Frequency Shift. Comparedto repulsive forces, order centrifugal shift contribution is only 0.5 cm-’, which is well attractive forces are typically longer range and more slowly within the 2 cm-I experimental measurement uncertainty for the varying, and thus can be assumed tocontribute a uniform, linearly high temperature data (see Table 11), and in any case only density dependent, background potential.9v26 The resulting representsa 2% correctionto the first-order centrifugalshift given attractive frequency shift is in turn predicted to be a linear function by eq 10. of d e n ~ i t y .The ~ corresponding constant of proportionality, C,, On the other hand, the assumption that angular momentum represents theinfluenceof long range dispersive (as well as dipolar is conserved during a vibrational period is expected to contribute and multipolar) attractive solvent-solute interaction^.^ more significant discrepanciesat the highest densities of interest. Under these extreme conditions the collision frequency may Aua = Cap (6) approach or even exceed the vibrational frequency of N2,and thus the solute’s angular momentum may be assumed to be more In previous vibrational frequency shift studies C, has often nearly in equilibrium with the solvent, rather than conserved, as been treated as an adjustable parameter whose magnitude is fixed the solute bond length changes. Belak has derived the following using empirical gas to liquid vibrational frequency shifts at
Devendorf and Ben-Amotz
2310 The Journal of Physical Chemistry, Vol. 97, No. 10, 1993
expression for the centrifugal potential in this high density limit.33,34 10
When this expression is combined with eq 2 (F= -2k~T/r, and G = kBT/r,2) the centrifugal shift is again predicted to be proportional to T, but in this case with a larger negative slope. Avc (cm-I) = -0.008399T (K)
(12)
A reasonable expression which interpolates between the above low and high density limiting results is one which reduces to eq 10 at low and eq 12 at high collision frequencies. For lack of a better model we assume an exponential interpolation between these two limits, as a function of the collision time divided by the vibrational period, Tc/Tvib.
*6WK
2
--
e 3 m ~ M2WK
0 0
5
10
15
20
25
Density (molecules/nm3)
Av, (cm-I) = -(0.006044e-rc~rvib + 0.008399( 1 - e-rc/rvib)JT (K) (1 3)
-
The vibrational period of N2 is, Tvib 1.42 X l e i 4s, and the collision time, T,, is taken to be that of the corresponding hard sphere reference fluid (with the CarnahanStarling hard sphere pressure equation) .21
The predictions of eq 13 only deviate significantly from those of eq 10 at the very high temperature and density shock wave conditions. Attempts to refine the aboveexpression,using e7rJrvib in place of e-rc/rvlb, did not significantly improve the theoretical fits to experimental frequency shifts (although slightly better fits were obtained with y 0.75).
-
4. Results and Discussion
The accuracy of vibrationalfrequency shift predictionsobtained by using the perturbed hard sphere fluid model described above depends critically on the availability of accurate hard sphere diameters. As described in section 3.1, these are obtained by fitting equation of state data to the CS-vdW equation, in such a way as to produce a self-consistent molecular hard core volume for N2. The quality of the equation of state data fits is a good preliminary indication of the accuracy and predictive utility of the corresponding perturbed hard sphere fluid model. Comparisons of experimental equation of state data for N2 with the predictions of two different CS-vdW based models are illustrated in Figure 1. The points in this figure represent experimental data5918 and the solid curves are CS-vdW (eq 3) predictions with a T and p dependent hard sphere diameter (eq 5 ) and a constant T value. The calculated isotherms lie within about 1% in both pressure and density for all the points in this figure. The dashed lines in Figure 1 represent CS-vdW predictions using previously reported temperature-dependent u and T parameters for N2.13 Clearly either model is capable of accurately reproducing equation of state datal* over the 200 K < T < 600 K and 0.1 kbar < P < 10 kbar temperature and pressure range represented in the main part of this figure. The quality of the CS-vdW fit to equation of state data at much higher temperatures and pressure is illustrated in the insert to Figure 1. In this case the results are displayed as a percent deviation between the experimental (shock wave)5 and theoretically calculated pressures, at higher temperatures and pressures. The calculated pressures are determined using the same UHS(p,nand T parameters used to generate the lower temperature (solid curve) predictions. Other data, not included in the equation of state fits, is that by Mills and co-worker~,~~ spanning a temperature and pressure
Figure 1. Equation of state data for N2 is compared with theoretical predictions. The solid curves represent predictions of the CS-vdW equation (eq 3) with a density and temperature dependent hard sphere diameter, U H S (eq 5) and a constant mean field parameter, z. The coefficients defining these parameters were fit to the 200-3521 K temperature data shown in this figure. The dashed curves represent CS-vdW predictions using previously reportedI3temperature dependent diameter and mean field parameters (see eq 4 and Table I). The insert figure shows the percent deviation between experimental5and predicted pressure, at very high temperatures and pressures,using thesame equation of state model represented by the solid lower temperature curves.
range of 247 K < T < 320 K and 3 kbar < P < 22 kbar. Attempts to include these data produced a significant deterioration in fits to both the lower and higher temperature and density data points shown in Figure 1. Furthermore, Mills' densities appear to increase more rapidly with pressure than the IUPAC and more recent measurement^,'^ although these only extend to 10 kbar. Since no other data are available above 10 kbar, it is not possible to discern whether it is Mills' data or the CS-vdW equation of state model, which is responsible for the observed disparities. In any case, we have chosen to leave Mills' data out of our analysis, as the net result is a better fit to both the remaining equation of state and the vibrational frequency shift data in Table I1 (as discussed below). Table 11 contains both our frequency shift measurements at 295 K and pressures up to near the freezing density of N2,1J7as well as previously reported measurements in fluid N2 at higher temperatures and pressures. (Note that at the highest temperatures the fluid phase density of N2 extends up to more than twice that of liquid nitrogen at its normal boiling point, 17.4 nm"). The first set of literature data was obtained with a resistively heated diamond anvil cell bySchiferl6J'Jwhile the second set was obtained in shock compressed N2 by Schmidt et a1.5 Densities were not determined in the diamond anvil cell studies and were thus derived from the reported pressures and temperatures using the CS-vdW equation of state (eq 3, with U H S ( ~ , ~ ' ) from eq 5 and parameters from Table I). Figure 2 displays our experimental frequency shift data along with other room temperature data found in the literature.24 The solid and dashed curves in this figure represent frequency shifts predicted by the perturbed hard-sphere fluid theory, using two different hard sphere diameter and centrifugal force models. The dashed curve is calculated using a temperature-dependent diameter defined by eq 4a (with coefficients in Table I) and a centrifugal shift defined by eq 10. The solid curve is calculated using a temperature and density dependent diameter defined by eq 5 (with coefficients in Table I) and a centrifugal shift defined by eq 13. In both cases the attractive force coefficient, C,, was obtained from a linear least-squared fit to the frequency shift data (solid symbol points). The dot-dash curve in Figure 2 is calculated with no adjustable parameters, using a diameter from
The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2311
Vibrational Frequency Shifts of Fluid Nitrogen
. 4
TABLE II: Experimental and Calculated Raman Vibrational Frearwacies
f'
~~
P (kbar)
T(K)
2.21 4.27 7.44 8.38 11.61 18.55
295
p (nm-3)
Y
295 K DAC 16.10 19.22 22.20 22.55 24.10 26.43
44 50 61 78 87 100.6 116.2 119.4
946 927 878 828 758 645 654 619
High T DAC6." 27.98 29.01 30.72 32.80 33.75 34.95 36.12 36.33
0.03 93 102 156 144 158
83 2015 2196 3521 2271 2480
.oo
1.04 1.16 1.25 1.27 1.37 1.38 0.001
T.'
(calcd)
2326.7 2326.8 2326.6 2326.9 2326.5 2326.8 2326.7 2326.7 2326.5 2326.7 2326.8
2338 2340 2342 2347 2349 2351 2356 2356
2334.9 2336.6 2339.7 2344.5 2347.1 2350.9 2355.0 2355.6
Shock CompressedS 16.17 2328.1 32.90 2343.5 33.54 2344.7 36.34 2351.3 37.20 2356.0 38.06 2356.4
-g >'
? a
.--eq.4
, ;'
eq. 10 eq. 13 eq. 13
4.2179p 4.2104p
- eq.5 eq. 5
4.2242 p
h
0
-4k
b(
0
;
d I
-
I
5
~
"
10
~
~
~ 15
"
" " 20
'
~ " 25
"
~
+
77-83K
: :
-
./'
r
#' .I'
L
"
30
Density (molecules/nm? Flgure 2. Vibrational frequency shift of N2 at 295 K, relative to the vapor phase Q-branch origin (pure vibrational transition frequency of isolated N2). The solid circles represent the results of new Raman measurementsin a diamond anvil cell. The solid squares are our previous lower pressure results.' The remaining symbols represent other previous room temperature measurements by LeDuff (X): Wang and Wright (A): and Kroon et al. (O).' The three curves are theoretical predictions. The solid and long-dashed curve are obtained using different hard sphere diameter and centrifugal force models, and a best fit attractive force coefficient, C,. The dot-dashed curve is obtained with no independently adjustable parameters (see text for details).
10
20
I
, ,
1 30
40
Density (molecules/nm3) Figure 3. Comparison of experimental (Table 11) and theoretical frequency shifts of Nz,A ~ t o t a l ,after subtraction of the centrifugal shift contribution, AuC(see eq 1). The curves in part a are calculated using the perturbed fluid model with an effective hard sphere diameter given by eq 5, a centrifugal shift given by eq 13, and an attractive mean field shift given by eq 5, with C, = -0.2104 cm-1 nm3 determined from a least-squares fit to 295 K frequency shift data (see Figure 2). Verticle error bars reflect experimental frequency uncertainty and the horizontal bars on the solid square points represent sample densities calculated from a Monte-Carlo simulation equation of state3' (see section 4). The corresponding deviations between the experimental and calculated frequency shifts are displayed in part b, along with experimental results at 645-758 K and Monte-Carlosimulation measurement^,^' not included in part a. The good agreement between experimental and theoretical shifts is clearly evident, as are the systematically high Monte-Carlo simulation frequency shifts.
A v,
Av.
828.946K 0 e45.758~ X 285K
30-
0
Theoretical models
Av,
:
3521K
A 2015.2480K
'''_
2326.9 2344.4 2345.8 2351.3 2355.2 2357.2
" " 1 " " 1 ' " ' 1 " ' " " " ~ "
___
B 20-
a Densities derived from experimental temperatures and pressures using eq 3 and eq 5.
4
.... 3520K 22WK ..... WOK 295~ ... BOK
40-
2326.7 2327.3 2328.4 2328.6 2329.5 233 1.6
2326.6 2326.8 2326.5 2326.7 2326.5 2326.6 2326.6 2326.6 2326.5 2326.7 2327.7
Exprrinwnt
Theory
>-
77
1
295
Y
2326.9 2327.5 2328.4 2328.6 2329.4 2331.9
295 K SCF' 10.5 10.9 11.8 12.3 12.5 13.0 13.4 13.5 13.8 13.8 17.4
0.69 0.15 0.84
(exptl)
"
eq 4a, a centrifugal shift from eq 13, and an attractive force coefficient from eq 7. Comparison of the three curves clearly indicates the self-consistencyof the frequency shift model as well as its relative insensitivity to the density dependence of the hard spherediameter and the high density correction to the centrifugal force frequency shift, at this temperature and density range. Comparison of experimental (solid symbols) and theoretical (curves) frequency shifts at other temperatures and densities is displayed in Figure 3a. The corresponding differences between the calculated and experimental frequencies are plotted in Figure 3b. No further parameter adjustments were performed in calculating the theoretical frequency shifts. The hard sphere diameter and centrifugal shift parameters are taken from eqs 5 and 13 (with coefficients listed in Table I) and the attractive force coefficient, C, = -0.2104 cm-' nm3, is taken to be the one ~ obtained from a best fit to the 295 K frequency shift data in Figure 2. For most of the experimental data the agreement with theoretical predictions are well within the f 2 cm-l measurement errors reported in the high temperature and density studies,5,6 Slightly larger discrepanciesfound with the heated diamond anvil cell data may be the results of the sample density uncertainty. In particular, differences between sample densities calculated using the CS-vdW equation (square points in Figure 3a) and those calculated using another equation of state proposed by Belak33 to model Monte-Carlo simulation data for nitrogen (indicated by horizontal bars on the square points in Figure 3a) are sufficient to account for the observed deviations between
Devendorf and Ben-Amotz
2312 The Journal of Physical Chemistry, Vol. 97, No. 10, 1993
experimentaland theoretical frequency shift predictions. Similar agreement between experimental and theoretical results is obtained using the attractive force coefficient predicted by eq 7, Ce= -0.2242 cm-1 nm3. On the other hand, significantly poorer predicted frequency shifts are obtained using either the low (eq 10) or high (eq 12) collision frequency centrifugal force expressions, in place of the interpolated expression (eq 13). Attempts to further refine the frequency shift predictions by adding additional parameters, either by allowing a quadratic density dependent component to the attractive frequency shift (eq 6 ) or by adding a fit parameter, y (see section 3.3), to the centrifugal force density dependence, did not significantly improve the agreement between experimentaland theoretical frequency shifts. The open symbols in Figure 3b represent Monte-Carlo simulation measurements of Belak et al.,33evaluated at the same temperature and pressure conditions as the experimental data, using an interpolation function.33 These simulation results are qualitative consistent with the experimental and calculated frequency shifts, although they deviate systematically to higher frequency. The excessive stiffness of the simulated nitrogen vibration was also noted by Belak et al.33 The clearly better agreement between experimental results and perturbed hard sphere fluid predictions underscores the potential benefits of using analyticalstatistical mechanicsas opposed to molecular simulation techniques, in modeling condensed phase chemistry.
5. Conclusions Simple extensions of previous perturbed hard sphere fluid expressions are used to generate a quantitatively accurate theoretical model for solvent mean force perturbations of the vibrational potential of fluid N2, up to temperatures of 3520 K and pressures of 158 kbar. The effects of the centrifugal force experienced by the diatomic at finite temperatures and densities are included by interpolating between accurate high and low collision frequency limiting expressions. The softness of the intermolecular interaction potential is modeled using a density and temperature dependent reference hard sphere diameter, derived from the analysis of equation of state data. A mean field approximation is used to model long range attractive force interactions. Results of this work include analytical expressions for the effectivehard sphere diameter, equation of state, solvation mean force potential, and vibrational frequency shift of N2 up to ultrahigh temperatures and pressures. Acknowledgment. This work was support of the Exxon Education Foundation, a Presidential Young Investigator Award (for D.B.-A.) from the National Science Foundation CHE915735, and a National Needs Fellowship (for G.S.D.) from the Department of Education. We hope that this work reflects the inspiration we have derived from Dudley Herschbach’senthusiasm for seeing big pictures in simple ways. Appendix. Hard Fluid Results Repulsive solvation force contributions to the vibrational frequency shift of a homonuclear diatomic are calculated from eq 2. The corresponding repulsive mean force potential, V, is related in a simple way to the hard sphere two-cavity distribution function, y(r;p,UHS,UO),abbreviated as y(r)[BH93],where r is the N2 bond distance (cavity separation), p is the number density of N2 molecules, and U H S and uo are the effective hard sphere diameters Of the “so~vent”Nz molecules and ”Solute” N atoms, respectively.
V(r) = -kBT In y ( r ) I- k , a ~+ Br +
(AI) The coefficients A, B, and C are determined by the equation of state of the reference hard sphere fluid12 A = In y ( 0 )
(A2)
1 C = - # n y(uo) - In y ( 0 ) - Bug)
(444)
00
where y(u0) is the value of y(r) at a separation (bond distance) equal to the contact distance between the two solute atoms, while y ( 0 ) and [a In y(r)/dr],,orepresent the value and slope of y ( r ) at zero separation, respectively. Thesearedetermined fromexact thermodynamic relations to the equation of state of the corresponding mixed hard sphere fluidEJ2 lny(0) = dq X (-3d2 3d + 3)v2 (6d2 - 9d- 6 ) -~d2 + 6 d + 3 + (1 - d 3
c
+
+
1
(2d3- 3d2 + 1) lnl?]
1-d
(A5)
=
[ y r ) ] r=O
-311
[l
+ 2d + d2 + (-2 - d + d2)q + (1 - d)$] (1 - d 3
2%
1+
Y(.o)
=
6
) (1 -6d 3
d-29+
(A61
d2-5d+l$ (A7)
which depend only on the solvent-solute diameter ratio, d = UO/ U H S ,and packing fraction, Q (seeeq 2). The “solvent”and “solute” hard sphere diameters are determined directly from the equation of state of fluid N2, as described in section 3.1. The hard fluid model thus predicts the following linear and quadratic repulsive solvation force constants, which are in turn used to predict the repulsive frequency shift, Aur (in eqs 1 and 2) *
F = -kBT(B + 3Cre2)
(A8)
G = -k,T(3Cre) (A9) The above mean field approximation to G has been used, and compared with other approximations,in several recent vibrational frequency shift studies.1*7.8The resulting G values for pure fluid N1 contribute no more than 3% to the total repulsive frequency shift, over the entire temperature and pressure range of interest (see Table 11). References and Notes (1) Ben-Amotz, D.; LaPlant, F.; Shea, D.; Gardecki, J.; List, D. Supercrirical Fluid Technology: Theoretical and Applied approaches in Analyticul Chemistry; ACS Symposium Series 488; Bright, F. V., McNally. M. P., Eds.; American Chemical Society: Washington, DC, 1992; p 18. (2) Le Duff, Y. J . Chem. Phys. 1973,59, 1984. (3) Kroon, R.; Baggen, M.; Lagendijk, A. J . Chem. Phys. 1989,91.74. (4) Wang, C. H.; Wright, R. B. J . Chem. Phys. 1973,59, 1706. ( 5 ) Schmidt, S. C.; Moore, D. S.; Shaw, M. S.Phys. Rev. E 1987, 35, 493.
(6) Etters, R. D.; Belak, J.; LeSar, R.Phys. Rev. E 1986,34,4221. Zinn, A. s.; Schiferl. D.;Nicol, M.F. J . Chem. Phys. 1987, 87, 1267. (7) Ben-Amotz, D.; Lee, M. R.; Cho, S . Y . ; List, D. J. J . Chem. Phys. 1992, 96, 8781.
( 8 ) Ben-Amotz, D.; Herschbach, D. R. J . Phys. Chem., this issue. (9) Schweizer, K. S.;Chandler, D. J . Chem. Phys. 1982, 76, 2296. (IO) Chandler, D.; Weeks, J. D.; Anderson, H. C. Science 1983,220.78 I . Anderson, H. C.; Chandler, D.; Weeks, J. D. Ado. Chem. Phys. 1976,34,105. ( I 1) Pratt, L. R.; Chandler, D. J . Chem. Phys. 1980,72,4045. Chandler, D.; Pratt, L. R. J . Chem. Phys. 1976,65,2925. Pratt, L. R.; Chandler, D. J . Chem. Phys. 1977,67, 3683. (12) Ben-Amotz. D. J . Phvs. Chem.. this issue. (13) Ben-Amotz, D.; Herkhbach, D: R. J . Chem. Phys. 1990,94, 1038. (14) Merrill, W.; Bassett, H. J. Rev. Sei. Insrrum. 1974, 45, 290. (15) Voss, W. L.; Schouten, J. A. J . Chem. Phys. 1989, 91, 6302. (16) Schiferl, D.; Cromer, D. T.; Mills, R. L. High Temp.-High Pressure 1978, IO,493.
Vibrational Frequency Shifts of Fluid Nitrogen (17) Barnett, J. D.; Block, S.; Piermarini, G. Rev. Sci. Insrrum. 1975.44, 1. (18) Angus, S.; deReuck, K. M.; Armstrong, B. International Thermodynamic Tables of the Liquid State 6-Nitrogen; Pergamon Press: Elmsford, NY, 1977. (19) Lavorel, B.; Chaux, R.; Saint-Loup, R.; Berger, H. Opr. Commun.
1987, 62, 25. (20) Buckingham, A. D. Proc. R. Soc. London Ser. A 1958, 248, 169; 1960, 255, 32. Trans. Faraday SOC.1960,56,753. (21) Hansen, J. P.; McDonald, 1. R. Theory of Simple Liquids, 2nd ed.; Academic Press: New York, 1986, and work cited therein. (22) Landolt-Bornstein Zahlenwerre und Funktionen; Springer: Berlin, 1971; Vol. 2, Part 1. (23) Press, W. H.; Flannery, B. F.; Teuolsky, S. A.; Vetterrling, W. T. NumericalRecipes. The Arr of Scienrijic Computins Cambridge University Press: Cambridge, 1986; pp 523-528. (24) Meyers, A. B.; Markel, F. Chem. Phys. 1990, 149, 21. (25) Bondi, A. J . Phys. Chem. 1964, 68, 441. (26) Longuet-Higgins, H. C.; Widom, B. Mol. Phys. 1964, 8, 549. (27) Zakin, M. R.; Herschbach, D. R. J . Chem. Phys. 1986, 85, 2376. (28) Zakin, M.R.; Herschbach, D. R. J . Chem. Phys. 1988, 89, 2380. (29) Ben-Amotz, D.; Zakin, M. R.; King, H. E., Jr.; Herschbach, D. R. J . Phys. Chem. 1988, 92, 1392.
The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2313 (30) LeSar, R. J . Chem. Phys. 1987,86,4138. (31) Huber. K.; Herzberg, G. Consrants of Polyatomic Molecules; Van Nostrand Reinhold: New York, 1979. (32) Brueck, S. R. J. Chem. Phys. f a r . 1977,50, 516. (33) Belak, J.; Etters, R. D.;LeSar, R . J. Chem. Phys. 1988, 89, 1625. (34) Belak, J. PhD thesis, Colorado State University, 1988. (35) Mills, R. L.; Liedenberg, D. H.; Bronson, J. C.J. Chem. Phys. 1975, 63, 1198. (36) Kortbeek, P. J.; Trappenier, N. J.; Biswas. N. S. fnr. J . Chem. Thermodyn. 1988, 9, 103. (37) Experimental frequency shifts at densities below -IO molecules/
nm3were excluded from theoretical analysis, since under these conditions the experimental Raman band of N2 was markedly broadened and asymmetric, leading to poor Gaussian peak fits. The band shape change undoubtedly results from the appearance free rotational structure in the Q-branch of N2 at low densities, although instrumental broadening in our apparatus prevented thisfrombeingfullyresolved. Thedensity belowwhich thepeakshapechange occurs is roughly consistent with that at which the collision frequency (given by q 14) becomes q u a l to the average rotational frequency of N2. (38) Herschbach, D. R.; Laurie, V. W . J. Chem. Phys. 1961, 35, 458. (39) Watanabe, K.; Nakayama, T.; Mottl, J. J . Quanr. Specrrosc. Radiat. Transfer 1962, 2, 369.
