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Virial Expansion and Its Application to Oxygen Spectroscopic Measurements at 1270 nm Band.

Byline: Muhammad Ahmad AL-Jalali, Issam Fawaz Aljghami and Yahia Mohamed Mahzia

Summary: A virial equation of state was applied to oxygen spectroscopic measurements at 1270 nm band. Second virial coefficients were evaluated from data analysis of compression factor as a power series in molar density and pressure. Similar virial expansion was applied to integrated absorption intensity (integrated absorbance), Voigt full width at half-maximum height (Voigt FWHM), also, as a power series in molar density and pressure. Some semi empirical formulae have been investigated, and some physical constants were calculated.

Keywords: Virial expansion, Virial coefficients, Compression factor, Voigt FWHM, Integrated absorption intensity.

Introduction

Experimental observations of oxygen spectral lines at 1270 nm band show two kinds of an oxygen molecule, O2 dimol (oxygen binary O2-O2) and O2 monomer, which took a lot of attention from researchers to study in greater depth [1-6].

The study will be devoted on near infrared (NIR) spectrum at1270 nm band, where the 1270 nm band extends from 1258 nm to 1273 nm wavelength.

O2 monomer has a discrete-line absorption band at 1268.4 0.2 nm wavelength with full width at half maximum height (FWHM) does not exceed 5 nm, and O2 dimol has a continuous absorption band at 1264.50.2 nm wavelength with FWHM between (10-30 nm). O2 monomer subject to this interaction:

(Equations)

Whereas, O2 dimol absorption process subject to this interaction:

(Equations)

Collision induced absorption (CIA) will prevail in pressure more than 2 bar, so O2 discrete spectrum, it may disappear, because it has a very narrowing bandwidth [7, 8]. Absorption intensities of ideal gas typically vary linearly with the numerical gas density, but may be as density squared, cubed etc. for real gas, because of CIA.

The aim of this paper is to connect the virial equating of state with oxygen absorption intensity and other factors, like, FWHM, integrated absorption intensity. Because these quantities may be given as a power series in a real gas density and conclude some semi empirical expressions which demonstrate some spectral facts.

Theoretical Background

Clausius 1870 [9] was the first one who had invented the virial theorem (the effective force of heat is proportional to the absolute temperature), which is a simple theory could be used in most of physics fields. It refers to a system of interacting particles with an effective potential energy of the form (Eq.) and effective force of the (Eq.).

The ideal gas is characterized by the equation of state:

(Equation)

Quantity (Z) is called the "compression factor", it is a useful measure of the deviation of a real gas (Eq.) from an ideal gas (Eq.) number of moles (mass), R = gas constant, T= absolute temperature, P = absolute pressure. The compression factor is defined by many methods, one of these as:

(Equations)

If Z less than 1, then attractive interactions dominate and the volume of the gas is less than that predicted by the ideal gas equation. If, however, Z greater than 1 then repulsive interactions dominate and the gas occupies a larger volume than expected. In general, real gases exhibit negative deviations from ideality at lower pressures and positive deviations at higher pressures.

In 1901 H. Kamerlingh-Omnes [10,11] was proposed a virial equation of state for real gases in Z as an infinite power series(The virial expansion) in terms of molar density (Eq.) for a pure gas:

(Equations)

The coefficient, B (T), is a function of temperature and is called the "second virial coefficient", while C (T) is the third virial coefficient, and rarely used in chemical thermodynamics [12-16].

The virial coefficients can be calculated from experimental and theoretical model of the intermolecular potential energy of the gas molecules, the B (T) may be given as follows [17, 18, 19]:

(Equations)

U (r) is the functional form of the pair- interaction between molecules and is only dependent on the inter-molecular distance r. Clearly, if U(r) =0, then B (T) =0, then ideal gas expression could be re-obtain.

A theoretically exact potential energy is not currently known or available [16], but some equations like, Lennard-Jones (LG 6-12), Morse and van der Waals potential energies [20-23] may be help to give some information about effective potential energy between two atoms or molecules.

An equivalent form of the virial expansion is an infinite series in powers of the pressure:

(Equations)

The new virial coefficients, (Eqs.) could be calculated from the original virial coefficients, B, C, these equations are easily solved to give B' and C' in terms of B, C, and R, and may be given as follows:

(Equations)

A similar virial expansion could obtain from absorption intensity, integrated absorption intensity, FWHM and other factors connected with them as a series power in molar density and pressure. Thus, general relation may be described by [24]:

(Equation)

Where (I) may be represented absorption intensity, integrated absorption intensity or FWHM. The coefficients in relation (9) referring to monomer A", and induced binary B" contributions. The intensity of absorption by non-interacting molecules therefore varies linearly with the density, Ideal (Eq.) while the intensity of collision-induced phenomena varies with the second or higher powers of the density:

(Equation)

At low pressures, the second and higher terms are insignificant, whereas at high pressures many-body interactions may be expected to dominate optical properties.

The relation between effective collision time and Lorentzian full width at half-maximum height ( (Eq.) = FWHM) was found to be given by [25, 26]:

(Equations)

where (P) is the pressure, (n) is the number density of the gas, (Eq.) is elastic or inelastic collision cross-section, (Eq.) is the average speed of reduced mass (Eq.).

The coefficient (Eq.) is well defined the pressure broadening coefficient, and clarify the changes in the collision-induced absorption states, so the change in line width is linear in pressure P (virial equilibrium),and at relatively low pressures of gas the pressure broadening coefficient was, in general, empirically observed to increase linearly as the gas pressure is increasing.

In addition, Doppler broadening full width (Gaussian width= (Eq.)) is given by:

(Equations)

The relationship between area version of Gaussian width and Gaussian FWHM is given by:

(Equation)

Whereas Lorentzian width (Eq.) equals to Lorentz FWHM.

The Area version of the Voigt FWHM is identical with Humlicek expression [20], where he developed a Voigt profile, and gave the equation of the Voigt spectral line width (Voigt FWHM) as follows:

(Equation)

Relation (14) will be used in all data analysis. When Doppler (Gaussian) component is constant and small, Lorentzian distribution will prevail.

Experimental

Spectra of pure oxygen gas (99.999%,) were measured in spectroscopy laboratory, in the NIR region of the electromagnet spectrum (1200-1300 nm), by using Cary5000 UV-VIS-NIR spectrophotometer, VARIAN company, equipped with multipath absorption gas cell with a fixed path length of 9.6 m and with 1.7L volume, at temperatures between 298 to 373 K and pressures between 1 to 25 bar. Fig. (1) shows some of many experimental plots between absorption intensity as a function of wavelength in the near infrared region, between 1200 -1300 nm wavelengths.

Data Analysis

All experimental data were analyzed by Origin Pro-Lab software 2015, where the values of compression factor, integrated absorption intensity, Voigt full width at half-maximum height, and others have been calculated. 3D Charts belong to P (Eqs.) data, V oigt FW HM P (Eqs.) data, and integrated abs.int. (Eq.) P (Eqs.) data have bee drawn below.

Fig. (2) shows experimental results between pressures as a function of temperature and molar density, which would be represented the viral equation of state.

Fig. (3) shows Voigt FWHM dependence of pressure and molar density, where The Voigt width deduced from the line profile analysis.

Fig. (4) shows the changes in total integrated absorption intensity of (O2+ O2-O2) gas as a function of pressure and temperature.

Integrated absorption intensity was calculated by the integral of the entire area under the absorption curve from 1200 to 1300 nm wavelength range by using the Gadgets in the Origin Lab program.

All data, which represented by Fig. s (2, 3, 4) will subject to Virial expansion in the results and discussion part.

Results and Discussion

Deconvolution of Voigt function of total absorption intensity will be done in another work, this paper will deal with additive behavior of full width at half maximum as a Voigt width, so, the small width at 1268 nm will merge with the wide width of 1264nm line, and all results will practically belong to Voigt width at 1264 nm wavelength.

Fig. (5) shows the compression factor, calculated from ratio of real gas pressure to ideal gas pressure, as a function of pressure (Fig, 5a) and molar density (fig.5b), which represent the virial equation of state, according to relations (5, 7).

Table-1 contains values of calculated virial coefficients, which appeared in relations (5,7), by using a linear least squares fitting routine, the polynomial relations of order three fitting (which written on the top of the table (1), was the best as a relationship between compression factors as a function of molar density and pressure.

(Equations)

Fig. 6 shows the temperature dependence of the second (Fig. 6 a) and third (Fig. 6 b) virial coefficients, where, mathematical relations exist inside Fig. 6 (a, b), their dependence is similar as in relation (15) in reference [25, 26]. Fig. 7 shows the temperature dependence of the pressure virial coefficient, where the Fig. 8 shows the relation between second virial coefficient and pressure virial coefficient.

Table-2 contains the coefficients values of a similar virial expansion belong to integrated absorption intensity as a function of molar density, and integrated absorption as a function of pressure, and relationship between them are in the top of the Table-2, and according to the relation (9).

Table-1: Values of virial coefficients and pressure virial coefficients, according to relations in the top of table.

###TK###Z= 1 + B(T)p + C(T)p 2 + D(T)p 3###Z= 1 + Bp(T)P + Cp(T)P 2 + Dp(T)P 3

###B(T)###C(T)###D(T)###Bp(T)###Cp(T)###Dp(T)

###298###- 0.02388###4.38797E-4###9.14187E-5###- 9.63668E-4###1.65518E-6###2.28075E-9

###308###- 0.02206###5.23523E-4###8.97011E-5###- 8.61665E-4###1.54959E-6###2.06123E-9

###323###- 0.01956###6.29724E-4###8.60458E-5###- 7.28331E-4###1.40919E-6###1.7818E-9

###338###- 0.01728###7.1531E-4###8.16215E-5###- 6.14747E-4###1.28703E-6###1.55068E-9

###348###- 0.01586###7.62921E-4###7.84443E-5###- 5.48244E-4###1.21421E-6###1.41839E-9

###358###- 0.01453###8.04124E-4###7.51798E-5###- 4.88108E-4###1.14739E-6###1.30073E-9

###373###- 0.01266###8.55675E-4###7.02414E-5###- 4.08275E-4###1.05705E-6###1.14746E-9

In Table-2 P0 and P0 are constants, but physically equal zero. The coefficients (Eq.) and (Eq.) belong to very narrow discrete line of O 2 , which are not stable and very noisy, because CIA is dominant at all mechanisms. The induced binary (Eq.) and (Eq.) contributions belong to the intensity of collision- induced phenomena, and have a wide continuous line of O 2 -O 2 spectrum.

Fig. (9) represents variations of similar second virial coefficients values from table (2) as a function of temperature. Where linear and nonlinear mathematical relations between them exist at the Fig. (9).

From Fig. 3 and its data, an appropriate fitting between Voigt FWHM as a function of pressure and molar density were done, where results gave a nonlinear relation between Voigt FWHM and pressure, and, also, molar density, Where all coefficients during data analysis exist in table (3) according to relations (11, 14), and relation in the top of Table-3. Fig. 10 represents the relation between pressure-broadening coefficients, and molar density-broadening coefficients as a function of temperature, where the coefficients increase with temperature increasing and the Doppler effect, implicitly representative in this dependency.

Analogous with above similar virial expansion relations, a relation between absorption coefficient, absorption intensity, and molar density under the effect of pressure and temperature could be achieved.

Table-2: Values of similar virial coefficients, according to relations in the top of table.

###Ip###P0 A''' P + B''' P 2 + ...###Ip###p0 A'' p + B'' p 2 + ...###Tk

###B'''###A'''###P0###B''###A''###P0

###0.01215###0.00674###0.0431###8.01848###- 0.05763###0.06155###298

###0.01106###0.01697###0.00358###7.74416###0.24374###0.01898###308

###0.00942###0.03232###-0.0557###7.25971###0.64606###-0.02677###323

###0.00828###0.01425###0.00987###6.85161###0.29197###0.01781###338

###0.00753###0.0022###0.05358###6.55607###- 0.02284###0.05956###348

###0.00614###0.02245###-0.00102###5.64972###0.60287###0.00345###358

###0.00407###0.05283###-0.08295###4.05651###.59946###-0.08029###373

Table-3: Values of broadening coefficients comparing with relations (11, 14).

###Voigt FW HM###dp (T,s)py###Voigt FWHM bp (T,s)Pa###TK

###0.25637###6.52224###0.26806###11.1734###298

###0.23335###26.77478###0.25363###11.78468###308

###0.2196###27.44051###0.23307###12.71971###323

###0.22096###28.25997###0.22931###13.0743###338

###0.22186###28.80907###0.22693###13.30612###348

###0.21243###28.9818###0.22048###13.62952###358

###0.19859###29.20771###0.2111###14.11549###373

Conclusion

Virial theorem is useful to handle all the difficulties facing the spectrum changings, where these changings subject to change of pressure, temperature, and density of O2 real gas.

Virial expansion and similar Virial expansion were applied on compression factor, integrated absorption intensity, and FWHM as a function of molar density and pressure, and could be applied in a similar ways on all variables belong to O2 real gas spectrum. A good agreement between experimental and theoretical results has been investigated.

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