Precise determination of the compressibility factor of a gas from refractive index measurements

We claim:

1. A method for the precise determination of the compressibility factor of a gas sample, wherein use is made of two grating interferometers coupled together with one interferometer defining a refractive index interferometer adapted to provide a signal of information related to the refractive index of said gas sample and the other interferometer defining a pressure interferometer adapted to provide another signal of information related to the pressure of said gas sample whereby to permit measurement of the refractive index as a function of pressure, each interferometer being capable of dividing a linearly polarized monochromatic laser beam into a measuring beam and a reference beam and causing said measuring beam and reference beam to travel along respective optical paths extending in close parallel relationship over predetermined optical path lengths, said refractive index interferometer and pressure interferometer comprising respectively two optical cells in tandem alignment and a single optical cell with each cell having elongated measuring and reference compartments arranged in close parallel relationship along the parallel optical paths of said measuring beam and reference beam to receive same therethrough, the measuring compartments of both cells of said refractive index interferometer being interconnected to permit gas expansion therebetween with the measuring compartment of one of said cells being connected to the measuring compartment of said single cell of said pressure interferometer via a pressure equilibrium chamber to provide said interferometer coupling, said method comprising the steps of:

(a) maintaining all said optical cells at a constant predetermined temperature value;

(b) filling the measuring compartments of said one cell and said single cell respectively with said gas sample and a selected gas at a same pressure above atmospheric pressure and allowing said gas sample and selected gas to reach thermodynamic equilibrium in said pressure equilibrium chamber;

(c) stepwise lowering the pressure of said gas sample and selected gas until substantially vacuum is attained in the measuring compartments of both said one cell and said single cell while maintaining the reference compartments thereof substantially under vacuum and recording at each step the signals of information provided by both interferometers once the thermodynamic equilibrium has been re-established in said pressure equilibrium chamber, thereby obtaining first and second sets of data;

(d) uncoupling said refractive index interferometer and pressure interferometer from one another;

(e) evacuating the measuring compartment of the other cell of said refractive index interferometer, re-filling the measuring compartment of said one cell with said gas sample at superatmospheric pressure, allowing said gas sample to reach thermodynamic equilibrium. recording the signal of information provided by said refractive index interferometer, causing said gas sample to expand into the measuring compartment of said other cell while maintaining the reference compartments of both said one cell and said other cell substantially under vacuum, and recording again the signal of information provided by said refractive index interferometer once the thermodynamic equilibrium of said gas sample has been re-established;

(f) evacuating the measuring compartments of both said one cell and said other cell, filling the measuring compartment of said other cell with said gas sample at a pressure equal to said surperatmospheric pressure of step (e), allowing said gas sample to reach thermodynamic equilibrium, recording the signal of information provided by said refractive index interferometer, causing said gas sample to expand into the measuring compartment of said one cell while still maintaining the reference compartments of both said one cell and said other cell substantially under vacuum, and recording again the signal of information provided by said refractive index interferometer once the thermodynamic equilibrium of said gas sample has been re-established;

(g) repeating steps (e) and (f) several times as paired steps after evacuation of the measuring compartment of said one cell, each of said paired steps being carried out at a different pressure above atmospheric pressure, thereby obtaining a third set of data; and

(h) processing said first, second and third sets of data obtained in steps (c) and (g) together with the temperature value of step (a) through circuitry means to obtain the compressibility factor of said gas sample.

2. A method as claimed in claim 1, wherein said selected gas is nitrogen of at least research grade purity.

3. A method as claimed in claim 1, wherein the temperature value in step (a) is selected from the range of about 0.degree. to about 100.degree. C.

4. A method as claimed in claim 1, wherein the pressure of said gas sample and selected gas in step (b) is in the range of about 100 to about 500 bars.

5. A method as claimed in claim 1, wherein the pressure of said gas sample in steps (e) and (f) is selected from the range of about 50 to about 500 bars and said steps (e) and (f) are repeated at pressures above said selected pressure.

6. A method as claimed in claim 1, wherein the pressure of said gas sample in steps (e) and (f) is selected from the range of about 50 to about 500 bars and said steps (e) and (f) are repeated at pressures below said selected pressure.

7. A method as claimed in claim 1, wherein after expansion in step (e) or (f) said gas sample has a density reduced substantially by half.

8. A method as claimed in claim 1, wherein the laser beam of each interferometer is a He-Ne laser beam polarized at 45.degree. and the reference beam issued from said laser beam is passed through a half-wavelength plate such that said measuring beam and reference beam have respective polarization planes which are perpendicular to one another.

9. A method as claimed in claim 8, wherein the measuring beam and reference beam of each interferometer after having travelled said predetermined optical path lengths are focussed on a grating to produce three beams corresponding to selected superposed diffraction orders including a zero order of diffraction, said three beams are rendered parallel and caused to impinge on a double refracting quartz plate with a quarter-wavelength plate being incorporated in the optical path of the beam of the zero order of diffraction to produce three pairs of phase modulated optical signals, each pair of optical signals being in-phase and anti-phase with one pair being in quadrature with the other two pairs, and said optical signals are converted into corresponding push-pull modulated electrical signals which are processed electronically to produce a d.c. compensated rotating electrical field associated with each interferometer, and wherein the thermodynamic equilibrium of said gas sample in steps (b), (c), (e) and (f) and of said selected gas in steps (b) and (c) is ascertained by observing the stability of the electrical fields associated respectively with said refractive index interferometer and said pressure interferometer.

10. An apparatus for the precise determination of the compressibility factor of a gas sample, comprising two grating interferometers coupled together with one interferometer defining a refractive index interferometer adapted to provide a signal of information related to the refractive index of said gas sample and the other interferometer defining a pressure interferometer adapted to provide another signal of information related to the pressure of said gas sample whereby to permit measurement of the refractive index as a function of pressure, each interferometer including means for dividing a linearly polarized monochromatic laser beam into a measuring beam and a reference beam and means for causing said measuring beam and reference beam to travel along respective optical paths extending in close parallel relationship over predetermined optical path lengths, said refractive index interferometer and pressure interferometer comprising respectively two optical cells in tandem alignment and a single optical cell with each cell having elongated measuring and reference compartments arranged in close parallel relationship along the parallel optical paths of said measuring beam and reference beam to receive same therethrough, said apparatus further including means for maintaining all said optical cells at a constant predetermined temperature value, first valved conduit means connecting the measuring compartment of one of said cells of said refractive index interferometer with the measuring compartment of said single cell of said pressure interferometer via a pressure equilibrium chamber to selectively couple or uncouple said interferometers, second valved conduit means interconnecting both cells of said refractive index interferometer to permit gas expansion therebetween when said interferometers are uncoupled and thereby enable said refractive index interferometer to provide a further signal of information, third valved conduit means adapted to connect a vacuum means to the reference compartments of all said cells for maintaining same substan tially under vacuum and to the measuring compartments of both cells of said refractive index interferometer for evacuating same after said gas expansion, fourth valved conduit means adapted to connect a source of said gas sample to the measuring compartments of both cells of said refractive index interferometer for filling the measuring compartment of either cell with said gas sample at superatmospheric pressure, fifth valved conduit means adapted to connect a source of a selected gas to the measuring compartment of said single cell of said pressure interferometer for filling same with said selected gas at superatmospheric pressure, said first and third valved conduit means being operative to stepwise lowering the pressure of said gas sample and selected gas until substantially vacuum is attained in the measuring compartments of both said one cell and said single cell when said interferometers are coupled, and circuitry means for processing the signals of information provided by both interferometers as a result of said pressure lowering and said gas expansion together with said temperature value to obtain the compressibility factor of said gas sample.

11. An apparatus as claimed in claim 10, wherein the measuring compartments of both cells of said refractive index interferometer have substantially equal volumes, and the measuring compartment of said single cell of said pressure interferometer has a volume substantially twice the volume of either cell of said refractive index inter- ferometer.

12. An apparatus as claimed in claim 11, wherein the measuring compartments of both cells of said refractive index interferometer have substantially equal lengths.

13. An apparatus as claimed in claim 10, wherein the measuring compartment of each cell is provided with gas inlet and outlet means in gas flow communication with gas permeable lining means extending longitudinally of the measuring compartment over the whole length thereof, said gas permeable lining means allowing uniform gas distribution or evacuation over substantially the whole compartment length.

14. An apparatus as claimed in claim 13, wherein the measuring compartment of each cell has a cylindrical cross-section which is constant from end to end and wherein said gas permeable lining means comprise a plurality of tubular lining elements of similar wall thickness arranged coaxially in abutting engagement with one another to define an unsealed joint between two adjacent lining elements whereby to permit said uniform gas distribution or evacuation.

15. An apparatus as claimed in claim 10, wherein each interferometer includes a He-Ne laser source means adapted to generate a monochromatic laser beam linearly polarized at 45.degree. and a half-wavelength plate arranged in the optical path of the reference beam issued from said laser team such that said measuring beam and reference beam have respective polarization planes which are perpendicular to one another.

16. An apparatus as claimed in claim 15, wherein each interferometer further includes means for focussing said measuring beam and reference beam after having travelled said predetermined optical path lengths on a grating to produce three beams corresponding to selected superposed diffraction orders including a zero order of diffraction; means for rendering said three beams parallel with one another; a double refracting quartz plate arranged in the optical paths of said three parallel beams with a quarter-wavelength plate being incorporated in the optical path of the beam of the zero order of diffraction to produce three pairs of phase-modulated optical signals, each pair of optical signals being in-phase and anti-phase with one pair being in quadrature with the other two pairs; and means for converting said optical signals into corresponding push-pull modulated electrical signals.

17. An apparatus as claimed in claim 16, wherein the signal converting means of each interferometer has first, second and third outputs associated respectively with said three pairs of push-pull modulated electrical signals, said second output being associated with the pair of signals which is in quadrature with the other two pairs, and wherein said circuitry means comprise first and second differential amplifiers associated with the signal converting means of each interferometer, said first differential amplifier having two inputs and an output and said second differential amplifier having an input and an output, the inputs of said first differential amplifier being connected to the first and third outputs of said signal converting means whereby to produce a d.c. compensated output signal and the input of said second differential amplifier being connected to the second output of said signal converting means; a phase detector having two inputs and an output with the inputs being connected to the outputs of said first and second differential amplifiers whereby to produce a d.c. compensated rotating electrical field at the output of said phase detector; a counter having an input and an output with the input being connected to the output of said phase detector whereby to produce at the output of said counter a fringe count associated with each interferometer; and a micro-processor connected to the counter of each interferometer for processing the fringe count associated therewith together with said predetermined temperature value to obtain the compressibility factor of said gas sample.

18. An optical cell for use in a grating interferometer in which a laser beam is divided into a measuring beam and a reference beam travelling along respective optical paths extending in close parallel relationship over predetermined optical path lengths, said cell comprising a body formed with two elongated bores extending through said body in close parallel relation with one another to define elongated measuring and reference compartments for receiving respectively said measuring beam and reference beam therethrough, each compartment being provided with gas inlet and outlet means in gas flow communication with gas permeable lining means extending longitudinally of the compartment over the whole length thereof, said gas permeable lining means allowing uniform gas distribution or evacuation over substantially the whole compartment length.

19. An optical cell as claimed in claim 18, wherein each compartment has a cylindrical cross-section which is constant from end to end and wherein said gas permeable lining means comprise a plurality of tubular lining elements of similar wall thickness arranged coaxially in abutting engagement with one another to define an unsealed joint between two adjacent lining elements whereby to permit said uniform gas distribution or evacuation.

20. An optical cell as claimed in claim 19, wherein said lining elements are removably inserted inside each compartment and are held in place by window means at the ends of each compartment, said window means sealingly engaging outermost lining elements and being transparent to said measuring beam and reference beam for allowing passage of same therethrough, releasable retaining means being provided for releasably retaining said window means in sealing engagement with said outermost lining elements.

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BACKGROUND OF THE INVENTION

The present invention relates to a method and apparatus for the interferometric determination of the compressibility factor of a gas. More particularly, the invention is directed toward the precise determination of the compressibility factor from refractive index measurements.

The compressibility factor measures the deviation from the ideal gas law which describes the behavior of a perfect gas. A perfect gas assumes that there are no interactions between molecules. Although a perfect gas does not exist, most gases at low densities resemble closely the perfect gas. A perfect gas follows an ideal gas law given by

PV=nRT (1)

where P is the pressure, V is the volume, R is the universal gas constant, T is the absolute temperature and n is the number of moles. An ideal gas, however, is totally inadequate to describe the behavior of high-pressure gases. The ideal gas equation (1) can be modified to handle real gases by inserting the compressibility factor Z. Thus, the gas law can now be written as

PV=ZnRT (2)

The compressibility factor which must be determined from experiments is a function of temperature, pressure and gas composition. The precision in the measurement of the compressibility factor is important both from the point of view of fundamental as well as applied science. In molecular physics the compressibility factor is a direct measure of the importance of molecular interactions. In gas industry the compressibility factor is necessary to calculate the cost of natural gas. The cost of gas which depends on the heat content is calculated on the basis of heat per unit mass. The mass m of natural gas is derived from the compressibility factor by using the formula

m=MPV/ZRT (3)

where M is the molecular weight.

Until now, the most commonly used methods for the determination of the compressibility factor have been the Burnett expansion technique and constant or variable volume methods. In one commercial application of the Burnett mcthod, for example, the gas under test is contained at a measured pressure P.sub.1 above atmospheric pressure in one chambcr of volume V.sub.1 of a double chamber vessel which is in a constant temperature bath. The second chamber has a volume V.sub.2 usually at atmospheric pressure P.sub.2. The test gas is expanded to fill both chambers and the pressure P.sub.3 of the gas in the resultant volume V.sub.1 +V.sub.2 is measured. The compressibility factor Z.sub.1 is given by: ##EQU1## where K is the ratio V.sub.1 /V.sub.2, Z.sub.2 and Z.sub.3 are experimental values determined by iteration and represent respectively the compressibility factors at P.sub.2 and P.sub.3. Thus, the volume ratio K as well as the values of Z.sub.2 and Z.sub.3 must be determined experimentally. Although such a method enables one to determine the compressibility factor with a precision of about 0.01%, it is in general very time consuming particularly when use is made of a dead weight gauge to measure the pressure, in which case it may take several days to obtain a certain number of experimental values. Thus, only a limited number of experimental values can be obtained per unit of time with the Burnett method. The same applies with respcct to the constant or variable volume methods.

SUMMARY OF THE INVENTION

It is therefore an object of the present invention to overcome the above drawback and to provide a method and apparatus enabling the determination of the compressibility factor of a gas in a more rapid manner than the prior art while still obtaining a precision of about 0.01%.

In accordance with one aspect of the invention, there is provided a method for the precise determination of the compressibility factor of a gas sample, wherein use is made of two grating interferometers coupled together with one interferometer defining a refractive index interferometer adapted to provide a signal of information related to the refractive index of the gas sample and the other interferometer defining a pressure interferometer adapted to provide another signal of information related to the pressure of the gas sample whereby to permit measurement of the refractive index as a function of pressure. Each interferometer is capable of dividing a linearly polarized monochromatic laser beam into a measuring beam and a reference beam and causing the measuring beam and reference beam to travel along respective optical paths extending in close parallel relationship over predetermined optical path lengths. The refractive index interferometer and pressure interferometer comprise respectively two optical cells in tandem alignment and a single optical cell with each cell having elongated measuring and reference compartments arranged in close parallel relationship along the parallel optical paths of the measuring beam and reference beam to receive same therethrough, the measuring compartments of both cells of the refractive index interferometer being interconnected to permit gas expansion therebetween with the measuring compartment of one of the cells being connected to the measuring compartment of the single cell of the pressure interferometer via a pressure equilibrium chamber to provide the interferometer coupling. The method according to the invention comprises the steps of:

(a) maintaining all the optical cells at a constant predetermined temperature value;

(b) filling the measuring compartments of the one cell and the single cell respectively with the gas sample and a selected gas at a same pressure above atmospheric pressure and allowing the gas sample and selected gas to reach thermodynamic equilibrium in the pressure equilibrium chamber;

(c) stepwise lowering the pressure of the gas sample and selected gas until substantially vacuum is attained in the measuring compartments of both the one cell and the single cell while maintaining the reference compartments thereof substantially under vacuum and recording at each step the signals of information provided by both interferometers once the thermodynamic equilibrium has been re-established in the pressure equilibrium chamber, thereby obtaining first and second sets of data;

(d) uncoupling the refractive index interferometer and pressure interferometer from one another;

(e) evacuating the measuring compartment of the other cell of the refractive index interferometer, re-filling the measuring compartment of the one cell with the gas sample at superatmospheric pressure, allowing the gas sample to reach thermodynamic equilibrium, recording the signal of information provided by the refractive index interferometer, causing the gas sample to expand into the measuring compartment of the other cell while maintaining the reference compartments of both the one cell and the other cell substantially under vacuum, and recording again the signal of information provided by the refractive index interferometer once the thermodynamic equilibrium of the gas sample has been re-established;

(f) evacuating the measuring compartments of both the one cell and the other cell, filling the measuring compartment of the other cell with the gas sample at a pressure equal to the superatmospheric pressure of step (e), allowing the gas sample to reach thermodynamic equilibrium, recording the signal of information provided by the refractive index interferometer, causing the gas sample to expand into the measuring compartment of the one cell while still maintaining the reference compartments of both the one cell and the other cell substantially under vacuum, and recording again the signal of information provided by the refractive index interferometer once the thermodynamic equilibrium of the gas sample has been re-established;

(g) repeating steps (e) and (f) several times as paired steps after evacuation of the measuring compartment of the one cell, each of the paired steps being carried out at a different pressure above atmospheric pressure, thereby obtaining a third set of data; and

(h) processing the first, second and third sets of data obtained in steps (c) and (g) together with the temperature value of step (a) through circuitry means to obtain the compressibility factor of the gas sample.

The gas which is used in the pressure interferometer is selected for its well known or established thermodynamic properties. The selected gas is preferably an inert gas whose second pressure virial coefficient is low, for example nitrogen or argon.

The constant temperature at which all the optical cells are maintained is generally in the range of about 0.degree. to about 100.degree. C. To this end, the cells can be immersed in a thermostatic bath which allows passage of the measuring beam and reference beam without contacting the thermostatic liquid contained in the bath.

The pressure of the gas sample and selected gas in step (b) is in general selected from the range of about 100 to about 500 bars depending on the nature of the gas sample. The pressure of the gas sample in steps (e) and (f) is preferably selected from the range of about 50 to about 500 bars depending again on the nature of the gas sample, and steps (e) and (f) can be repeated at pressures either above or below the selected pressure.

According to a particularly preferred embodiment of the invention, the laser beam of each interferometer is a He-Ne laser beam polarized at 45.degree. and the reference beam issued from such a laser beam is passed through a half-wavelength plate such that the measuring beam and reference beam have respective polarization planes which are perpendicular to one another. The measuring beam and reference beam of each interferometer after having travelled the predetermined optical path lengths are then focussed on a grating to produce three beams corresponding to selected superimposed diffraction orders including a zero order of diffraction, the three beams are rendered parallel and caused to impinge on a double refracting quartz plate with a quarter-wavelength plate being incorporated in the optical path of the beam of the zero order of diffraction to produce three pairs of phase modulated optical signals, each pair of optical signals being in-phase and anti-phase with one pair being in quadrature with the other two pairs, and the optical signals are converted into corresponding push-pull modulated electrical signals which are processed electronically to produce a d.c. compensated rotating electrical field associated with each interferometer. Thus, the thermodynamic equilibrium of the gas sample in steps (b), (c), (e) and (f) and of the selected gas in steps (b) and (c) can be conveniently ascertained by observing the stability of the electrical fields associated respectively with the refractive index interferometer and the pressure interferometer. Such a thermodynamic equilibrium is necessary in order to avoid the Joule-Thompson effect.

The present invention also provides, in a further aspect thereof, an apparatus for carrying out a method as defined above. The apparatus according to the invention comprises two grating interferometers coupled together with one interferometer defining a refractive index interferometer adapted to provide a signal of information related to the refractive index of the gas sample and the other interferometer defining a pressure interferometer adapted to provide another signal of information related to the pressure of the gas sample whereby to permit measurement of the refractive index as a function of pressure, each interferometer including means for dividing a linearly polarized monochromatic laser beam into a measuring beam and a reference beam and means for causing the measuring beam and reference beam to travel along respective optical paths extending in close parallel relationship over predetermined optical path lengths. The refractive index interferometer and pressure interferometer comprise respectively two optical cells in tandem alignment and a single optical cell with each cell having elongated measuring and reference compartments arranged in close parallel relationship along the parallel optical paths of the measuring beam and reference beam to receive same therethrough.

The apparatus of the invention further includes means for maintaining all the optical cells at a constant predetermined temperature value, first valved conduit means connecting the measuring compartment of one of the cells of the refractive index interferometer with the measuring compartment of the single cell of the pressure interferometer via a pressure equilibrium chamber to selectively couple or uncouple the interferometers, second valved conduit means interconnecting both cells of the refractive index interferometer to permit gas expansion therebetween when the interferometers are uncoupled and thereby enable the refractive index interferometer to provide a further signal of information, third valved conduit means adapted to connect a vacuum means to the reference compartments of all the cells for maintaining same substantially under vacuum and to the measuring compartments of both cells of the refractive index interferometer for evacuating same after the gas expansion, fourth valved conduit means adapted to connect a source of the gas sample to the measuring compartments of both cells of the refractive index interferometer for filling the measuring compartment of either cell with the gas sample at superatmospheric pressure, and fifth valved conduit means adapted to connect a source of a selected gas to the measuring compartment of the single cell of the pressure interferometer for filling same with the selected gas at superatmospheric pressure. The first and third valved conduit means are operative to stepwise lowering the pressure of the gas sample and selected gas until substantially vacuum is attained in the measuring compartments of both the one cell and the single cell when the interferometers are coupled. Circuitry means are also provided for processing the signals of information provided by both interferometers as a result of the pressure lowering and the gas expansion together with the temperature value to obtain the compressibility factor of the gas sample.

In order to prepare the output signal of each interferometer so as to be readily processed by the circuitry means, each interferometer preferably includes a He-Ne laser source means adapted to generate a monochromatic laser beam linearly polarized at 45.degree. and a half-wavelength plate arranged in the optical paths of the reference beam issued from such a laser beam so that the measuring beam and reference beam have respective polarization planes which are perpendicular to one another. Each interferometer further includes means for focussing the measuring beam and reference beam after having travelled the predetermined optical path lengths on a grating to produce three beams corresponding to selected superposed diffraction orders including a zero order of diffraction; means for rendering the three beams parallel with one another; a double refracting quartz plate arranged in the optical paths of the three parallel beams with a quarter-wavelength plate being incorporated in the optical path of the beam of the zero order of diffraction to produce three pairs of phase-modulated optical signals, each pair of optical signals being in-phase and anti-phase with one pair being in quadrature with the other two pairs; and means for converting these optical signals into corresponding push-pull modulated electrical signals. The signal converting means of each interferometer has first, second and third outputs associated respectively with the three pairs of push-pull modulated electrical signals, the second output being associated with the pair of signals which is in quadrature with the other two pairs.

The circuitry means, on the other hand, preferably comprise first and second differential amplifiers associated with the signal converting means of each interferometer, the first differential amplifier having two inputs and an output and the second differential amplifier having an input and an output, the inputs of the first differential amplifier being connected to the first and third outputs of the signal converting means whereby to produce a d.c. compensated output signal and the input of the second differential amplifier being connected to the second output of the signal converting means. A phase detector having two inputs and an output has its inputs connected to the outputs of the first and second differential amplifiers whereby to produce a d.c. compensated rotating electrical field at the output of the phase detector. A counter having an input and an output has its input connected to the output of the phase detector whereby to produce at the output of the counter a fringe count associated with each interferometer. A micro-processor is connected to the counter of each interferometer for processing the fringe count associated therewith together with the aforesaid predetermined temperature value to obtain the compressibility factor of the gas sample.

In a preferred embodiment, the measuring compartments of both cells of the refractive index interferometer have substantially equal volumes and lengths in order to facilitate the measurements and the interpretation of the results. On the other hand, the measuring compartment of the single cell of the pressure interferometer preferably has a volume substantially twice the volume of either cell of the refractive index interferometer, for increased accuracy.

In another preferred embodiment of the invention, the measuring compartment of each cell is provided with gas inlet and outlet means in gas flow communication with gas permeable lining means extending longitudinally of the measuring compartment over the whole length thereof, the gas permeable lining means allowing uniform gas distribution or evacuation over substantially the whole compartment length. This avoids local turbulence which would otherwise be caused by gas entering the measuring compartment at a single point, and thus ensures stability of the measuring beam passing through the measuring compartment. The internal volume of the measuring compartment is also reduced by the provision of such gas permeable lining means so that on one hand a lesser quantity of gas sample is necessitated which may be important in the case of expensive gases and, on the other hand, the thermodynamic equilibrium is reached more rapidly. Preferably, both the measuring compartment and the reference compartment are provided with such gas permeable lining means for purpose of symmetry.

Accordingly, the present invention further provides an optical cell for use in a grating interferometer in which a laser beam is divided into a measuring beam and a reference beam travelling along respective optical paths extending in close parallel relationship over predetermined optical path lengths, which cell comprises a body formed with two elongated bores extending through the body in close parallel relation with one another to define elongated measuring and reference compartments for receiving respectively the measuring beam and reference beam therethrough, each compartment being provided with gas inlet and outlet means in gas flow communication with gas permeable lining means extending longitudinally of the compartment over the whole length thereof, the gas permeable lining means allowing uniform gas distribution or evacuation over substantially the whole compartment length.

Preferably, each compartment has a cylindrical cross-section which is constant from end to end and the gas permeable lining means comprise a plurality of tubular lining elements of similar wall thickness arranged coaxially in abutting engagement with one another to define an unsealed joint between two adjacent lining elements whereby to permit the aforesaid uniform gas distribution or evacuation. These lining elements may be removably inserted inside each compartment and held in place by window means at the ends of each compartment, the window means sealingly engaging outermost lining elements and being transparent to the measuring beam and reference beam for allowing passage of same therethrough, releasable retaining means being provided for releasably retaining the window means in sealing engagement with the outermost lining elements.

Turning to the mathematical development which has led to the present invention, the compressibility factor defined in equation (2) may be written as

Z=P/pRT (5)

where .rho. is the molar density. The invention is based on determining .rho. from the Lorentz-Lorenz equation ##EQU2## where n is the refractive index and A.sub.n, B.sub.n and C.sub.n are respectively the first, the second and the third refractivity virial coefficients. Equation (6) can also be written as ##EQU3## The first approximation for the density is given by

.rho..sub.1 =(L.sub.n /A.sub.n) (8)

With this walue of density one gets the first approximation for the compressibility factor

Z.sub.1 =(P/RT) (A.sub.n /L.sub.n)

The second approximation for .rho. is given by

.rho..sub.2 =(L.sub.n n/A.sub.n)-(B.sub.n /A.sub.n) (L.sub.n /A.sub.n).sup.2 ( 10)

Replacing equation (10) in equation (5) one gets for the compressibility factor

Z.sub.2 =(P/RT) [(A.sub.n /L.sub.n)+(B.sub.n /A.sub.n)] (11)

The third approximation for density given by ##EQU4## where Z.sub.3 is the compressibility factor incorporating the first three coefficients, A.sub.n, B.sub.n and C.sub.n, of the Lorentz-Lorenz equation.

Absolute refractive indices are calculated from laboratory measurements using the refractive index interferometer coupled to the pressure interferometer, by means of the formula:

n=(K.lambda./l)+1 (14)

where K is the total fringe count with reference to vacuum for a given pressure, .lambda. is the vacuum wavelength of light and l is the length of the measuring compartment of the optical cell of the refractive index interferometer, which is connected to the measuring compartment of the single cell of the pressure interferometer via the pressure equilibrium chamber.

The first refractivity virial coefficient A.sub.n is obtained by making an absolute measurement of the refractive index n as a function of pressure P. Expressing .rho. in terms of P as in equation (6), one gets

[(n.sup.2 -1)/(n.sup.2 +2)][RT/P]=A.sub.n +(B.sub.n -A.sub.n B.sub.p) (P/RT) (15)

where B.sub.p, the second pressure virial coefficient, is given by the expansion

(P/RT)=.rho.+B.sub.p .rho..sup.2 +. . . (16)

When one plots the left side of equation (15) against P/RT, one can determine A.sub.n from the intercept. Although B.sub.n appears in the second term, this absolute method cannot be used for a precision determination of B.sub.n Since B.sub.n is a small fraction of the second term in equation (15), an error as small as 2% in (B.sub.n -A.sub.n B.sub.p) could lead to an error as large as 100% in B.sub.n. In the present invention, B.sub.n is determined directly by using an expansion technique whereby the gas sample contained in the measuring compartment of one of the optical cells of the refractive index interferometer is expanded into the measuring compartment of the other cell of the refractive index interferometer.

The expansion technique is based on measuring the sum of optical path lengths of two similar compartments where one of them is filled with the gas sample at density p and the other is evacuated. After expansion, the density is nearly halved and one measures again the optical path lengths. Because the linear term in density remains the same before and after the expansion and only the quadratic and higher orders change, one can determine B.sub.n and C.sub.n from the change of the optical path lengths.

The development of (n-1) p.sup.-1 as a function of density is given by

(n-1).rho..sup.-1 =A.sub.m +B.sub.m .rho.+C.sub.m .rho..sup.2 +. . . . (17)

The coefficients of equation (17) are related to those of equation (6) through the following relations:

A.sub.n =(2/3)A.sub.m ( 18)

B.sub.n =(2/3)B.sub.m -(1/9)A.sub.m.sup.2 ( 19)

C.sub.n =(2/3)C.sub.m -(2/9)A.sub.m B.sub.m -(4/27)A.sub.m.sup.3 ( 20)

Considering the measuring compartments of both cells of the refractive index interferometer as having respectively volumes A and B given by V.sub.A =V(1+.delta.) and V.sub.B =V(1-.delta.) where .delta. is small and the optical path length of each volume as being given by l.sub.A =l(1+.DELTA.) and l.sub.B =l(1-.DELTA.), where .DELTA. is also small, if A initially contains a gas of refractive index n.sub.A, at a density .rho..sub.A. while B is evacuated, one observes a change (D.sub.A) of refractive index on e