METHOD FOR CALIBRATING A GAS PYCNOMETER AND A PYCNOMETER CONFIGURED TO IMPLEMENT SUCH A METHOD

Abstract

The invention relates to a calibration method (100) of a gas pycnometer, said gas pycnometer comprising at least two containers, a reference container having a reference volumetric capacity, referred to as the first capacity, and a receiving container configured to hold a sample whose volume is to be determined, said receiving container having a test volumetric capacity, referred to as the second capacity, said gas pycnometer being configured to determine the volume v of a sample housed in the receiving container, characterized in that said gas pycnometer is characterized by a calibration function which depends on at least two modeling parameters: a sensitivity coefficient which is a predetermined value and a pycnometer-to-empty ratio which is a measurement value of said pycnometer when empty, the relative measurement uncertainty(s) of said pycnometer being determined as a function of the sensitivity coefficient and the pycnometer-to-empty ratio.Figure to be published with the abbreviation: none.

Description

Title of the invention: METHOD FOR CALIBRATING A GAS PYCNOMETER AND A PYCNOMETER CONFIGURED TO IMPLEMENT SUCH A METHOD

[0001] The present invention relates to the general field of density metrology for solids, as well as to the field of gas pycnometers for measuring the density of solids. More particularly, the invention relates to a calibration method for gas pycnometers.

[0002] Indeed, the density of certain solids cannot be determined by a hydrostatic weighing method according to Archimedes' principle, for example, due to the porosity of these solids or because it is impossible to immerse them. For such solids, it is known to use a gas pycnometer to determine the volume of the solid in question, the mass of this solid being determined separately by weighing, and then to deduce the density.

[0003] It should be noted that below the term "volume" characterizes the volume occupied by a solid (and which is the quantity that we seek to determine), while the term "capacity" or "volumetric capacity" designates the capacity (to contain) of a container or receptacle.

[0004] Several pycnometers currently exist for carrying out these volume measurements. Known as gas pycnometers, their principle is generally based on Boyle-Mariotte's law and generally implement static pressure measurements between a first volumetric capacitance in which a predefined gas is placed, at a fixed pressure (for example atmospheric pressure), and a second volumetric capacitance in which are placed, on the one hand, the solid whose volume is sought, and, on the other hand, the aforementioned gas, at a pressure greater than said fixed pressure.

[0005] An expansion of the gas is thus carried out from the second volumetric capacity to the first volumetric capacity, and the volume of the solid is deduced, on the one hand, from the knowledge of the volumes of the first and second volumetric capacities, and, on the other hand, from a measurement of the pressure in the second volumetric capacity before and after expansion.

[0006] Whether the pycnometers implement a simple expansion from the second volumetric capacitance to the first volumetric capacitance or whether they implement an intermediate volumetric capacitance with variable volume, the pycnometers known at present exhibit measurement uncertainties on the order of a few fractions of percent, typically in the range of 0.15 to 0.2% in relative value, particularly when a substitution method is not applied or is applied incorrectly.

[0007] Furthermore, the size of solid samples that can be introduced into such pycnometers remains small, on the order of a few tens of cubic centimeters. In the case of solids exhibiting lattice structures or containing numerous small channels, such a limitation can introduce additional uncertainty regarding the representativeness of the small-volume sample used for volume measurement.

[0008] It should also be noted that there are now two types of gas pycnometers, constant-volume pycnometers, which are known and commercially available, and constant-flow pycnometers as defined in patent application FR 2 103 866 Al. It can also be mentioned that the name "constant-volume pycnometer" is a terminology adopted for the first time in the article entitled "Optimum design of the constant-volume gas pycnometer for determining the volume of solid particles" in the scientific journal "Measurement Science and Technology" of February 2004.

[0009] One of the problems associated with these gas pycnometers is the relative uncertainty on the volume(s), without even considering the contribution of uncertainty to the sample. Indeed, under optimal conditions, the relative uncertainty is on the order of 0.02%, but this requires the user to calibrate the pycnometer with a suitable standard sphere according to the volume of the sample being tested, a short time between calibration of the pycnometer with a standard sphere and measurement of the unknown volume, a similarity between the standard volume and the calibrated volume values, the highest possible volume / capacity ratio (for example 30%), and thermal stability of the pycnometer containers (commercial pycnometers generally provide temperature control by circulating temperature-regulated water around the containers).

[0010] This calibration method is often referred to as a Borda substitution calibration method. However, measurements with the smallest uncertainties are only obtained within a limited time after calibration of the pycnometer using standard volumes (e.g., standard spheres). Furthermore, the uncertainty performance (i.e., its minimization) with the substitution calibration method is primarily linked to the repeatability of the measurements.

[0011] This calibration method by substitution is tedious for the user of the pycnometer, because the pycnometer needs to be calibrated with standard volumes, judiciously chosen in relation to the volume of the object to be calibrated, this also requiring a certain know-how on the part of the user.

[0012] Furthermore, in the case of constant-volume gas pycnometers, the operating manuals for said pycnometers define an uncertainty valid for optimal conditions that are difficult to grasp and most often to maintain. Therefore, if the user wants an estimate of the pycnometer's uncertainty under operating conditions, a metrological characterization must be performed, which requires the user to possess solid expertise in metrology, as this uncertainty estimate is otherwise only empirical.

[0013] It is therefore necessary to find a solution to facilitate the calculation and reduction of measurement uncertainties when the user uses a gas pycnometer, regardless of the user's metrological skills.

[0014] The present invention thus proposes to remedy at least one of the aforementioned drawbacks by proposing a new type of calibration method for a gas pycnometer, said gas pycnometer comprising at least two containers, a reference container having a reference volumetric capacity, referred to as the first (volumetric) capacity, and a receiving container configured to hold a sample whose volume v is to be determined, said receiving container having a test volumetric capacity, referred to as the second (volumetric) capacity, said gas pycnometer being configured to determine the volume v of a sample housed in the receiving container,

[0015] characterized in that said gas pycnometer is characterized by a calibration function f which depends on at least two modeling parameters: - a sensitivity coefficient S which is a predetermined value; - a pycnometric ratio at no load Ro which is a measurement value at no load of said pycnometer; the relative measurement uncertainty(s) of said pycnometer being determined as a function of the sensitivity coefficient S and the pycnometric ratio at no load Ro.

[0016] The calibration method according to the invention is thus advantageously based on: - a sensitivity coefficient S, generally expressed in cm3, which is a predetermined value, for example determined by calibration or by the manufacturer of the pycnometer under the best possible experimental conditions, and which is given to the user of the pycnometer and / or integrated into the embedded software of said pycnometer, in order to determine the relative uncertainties of the measurements made using said pycnometer; - a pycnometric ratio at zero Ro, which is a dimensionless quantity, which corresponds to a measurement of zero, that is to say a measurement during which no sample having a volume v is introduced into the pycnometer, the pycnometric ratio at zero Ro allowing to group all the corrections experimental factors to be taken into account in determining the volume v of a sample by the pycnometer.

[0017] The use of such parameters makes it easy to determine the contribution of the pycnometer's operating environment to the measurement uncertainty, and also to integrate these parameters into a mathematical model within the pycnometer's embedded software so that the measurement uncertainties are calculated automatically. Furthermore, it is no longer necessary to have one or more calibration spheres to calibrate the pycnometer.

[0018] The invention therefore makes it possible to restrict the calibration operation to a zero measurement operation. The use of a gas pycnometer is thus simplified, as the user of the pycnometer no longer needs to perform complicated manipulations and / or have advanced expertise in metrology to determine the accuracy of the measurements made using said pycnometer.

[0019] According to one possible characteristic, the sensitivity coefficient S is a function of at least one of the volumetric capacities of said pycnometer. It is advantageous to have a sensitivity coefficient S that depends only on values ​​of one or more volumetric capacities of the pycnometer containers, because said sensitivity coefficient S can then be determined beforehand very precisely, for example by the manufacturer of the pycnometer and / or by calibration.

[0020] According to another possible characteristic, the pycnometric ratio at vacuum Ro is a function of a physical quantity relating to a pressure or a rate of change of pressure in the first and / or second containers during a vacuum measurement of said pycnometer.

[0021] According to another possible feature, there is determination of a calibration function f relating the sample volume to be determined with the sensitivity coefficient S, the pycnometric ratio at empty Ro, and a pycnometric ratio Rm which is a function of a physical quantity relating to the pressure or the rate of change of pressure in the first and / or second containers during a measurement of said pycnometer when a sample is housed in the receiving container. It should be noted that in the case of a constant volume pycnometer, the physical quantity is a pressure, while in the case of a constant flow rate pycnometer, the physical quantity is a rate of pressure change (or a pressure change with respect to time).

[0022] According to another possible feature, the pycnometric ratio at vacuum Ro is determined before and after the determination of the volume v of a sample. Advantageously, the variation of the pycnometric ratio at empty time Ro before and after the measurement of a volume v of a sample characterizes the stability of the parameters experimental, for example the temperature homogeneity in the pycnometer, and allows the calculation of the uncertainty contribution of the pycnometer in its environment, therefore independently of the uncertainty contribution of said sample.

[0023] According to another possible characteristic, the function f which relates the volume v to be determined of a sample to the sensitivity coefficient S and the pycnometric ratio at empty Ro is of the form: W-1 = -Sr where W is the pycnometric ratio Rm normalized by the pycnometric ratio at empty Ro.

[0024] According to another possible characteristic, there is determination of one or more of the components of an uncertainty relating to the measurement of the volume v of the sample: - determination of an uncertainty component of the sensitivity corresponding to the uncertainty of the model deduced from the calibration of the gas pycnometer, said uncertainty component of sensitivity being a function of (1 / s2 • ( 1- Il )) u(S ), where U(S) is the uncertainty of the sensitivity and the term (1 / S2. ( ] _ yy )) is the associated sensitivity coefficient; - determination of a repeatability uncertainty component that is a function of (1 / 151 ■ Rq ) U ( R ) , where (R ) is the repeatability uncertainty and the term (1 / 151 • Æq) the associated sensitivity coefficient - determination of an uncertainty component related to the repeatability and reproducibility of the determination of the Roqui no-load pycnometric ratio is a function of ( W / LSI • R^ u(Rq)' °where is the uncertainty of the pycnometric ratio at idle Roet the term (yy / |$| . u(Rq) 'c associated sensitivity coefficient

[0025] It should be noted that the different components of the uncertainty relating to the measurement performed by the pycnometer are determined by analytical calculation based on the law of propagation of variances, while assuming that the different uncertainty components are independent of each other. According to another possible characteristic, the gas pycnometer is a constant volume pycnometer or a constant flow rate pycnometer.

[0026] According to another possible characteristic, the sensitivity coefficient S is: - for a pycnometer with constant volume, a function of the first and second capacities; - for a constant flow pycnometer which includes an intermediate container having an intermediate capacity, called third capacity, a function of the second and third capacities.

[0027] The invention also relates to a gas pycnometer, characterized in that said pycnometer is configured to implement the calibration method as defined above.

[0028] Other features and advantages of the invention will become apparent from the following description on the one hand, and from several illustrative and non-limiting examples of embodiments given with reference to the accompanying schematic drawings on the other hand, in which: - Fig. 1 illustrates a very schematic view of a first type of gas pycnometer according to the invention; - Fig. 2 illustrates a very schematic view of a second type of gas pycnometer according to the invention; - [[Fig.3]] illustrates a flowchart of a calibration method for a pycnometer from [Fig.1] or [Fig.2].

[0029] Of course, the features, variants, and different embodiments of the invention can be combined in various ways, provided they are not incompatible or mutually exclusive. In particular, variants of the invention may be conceived comprising only a selection of features, described hereafter in isolation from the other described features, if this selection of features is sufficient to confer a technical advantage or to differentiate the invention from the prior art.

[0030] In particular, all the variants and embodiments described are combinable with each other provided there are no technical obstacles to such combination. Furthermore, in the various figures, the elements common to several figures retain the same reference numeral.

[0031] Fig. 1 is a very schematic and partial view of a first type of gas pycnometer 1 according to the invention, this first type of pycnometer being a constant volume gas pycnometer, and also referred to hereafter as a constant volume pycnometer.

[0032] The constant volume pycnometer 1 thus comprises at least two containers 3 and 5 fluidically connected to each other via a valve 6, such as a shut-off valve (or "on-off valve").

[0033] More specifically, containers 3 and 5 of the pycnometer 1 are: - a reference container 3 having a reference volumetric capacity CR, called the first volumetric capacity; - a receiving container 5 configured to house a solid sample 7 whose volume v is to be determined, said receiving container 5 having a volumetric test capacity CT, called second capacity.

[0034] It should be noted that hereafter the term "volume" characterizes the volume v occupied by a solid, while the term "capacity" or "volumetric capacity" designates the capacity (to contain) of a container or receptacle. Volume and volumetric capacities have the dimensions of a length cubed.

[0035] It should also be noted that the reference container 3 and the receiving container 5 are advantageously made of the same material, for example stainless steel.

[0036] Said pycnometer 1 also includes an injection system not shown for injecting a predetermined quantity of gas into the reference container 3. The injected gas is preferably an inert or low-reactivity gas, such as helium, dinitrogen, etc.

[0037] By way of non-exclusive example, the gas chosen is advantageously nitrogen, which is inexpensive, has, for a given pressure and temperature, a compressibility coefficient close to that of an ideal gas, and has very low chemical reactivity with a wide variety of materials.

[0038] Said pycnometer 1 further includes a pressure measuring device 9, for example a capacitive type manometer, configured to measure the pressure of the gas injected into the reference container 3 and into the two containers 3 and 5, this after opening of the valve 6 and expansion of the gas injected into the two containers 3 and 5.

[0039] The pycnometer 1 may optionally include at least one thermometric probe 11 (or temperature probe) configured to determine a temperature relative to the reference container 3 and / or the test container 5. Preferably, the thermometric probe 11 is associated with the test container 5 or placed on the test container 5 in order to estimate the sample temperature. This advantageous configuration has the effect of associating a temperature with the measurement of the volume v in the event that temperature is an influencing factor for said measurement.

[0040] Thus, to determine the volume v of a solid sample placed in the receiving container 5, there is: - injection of a gas into the reference container 3, the gas then has a pressure Pi and a theoretical temperature 1), the valve 6 being closed (preventing the gas from spreading / expanding into the receiving container 5); - expansion of the gas in the reference container 5, after opening of the valve 6 and fluidic communication of said containers 3 and 5, the gas then having, after expansion, a pressure Pf and a theoretical temperature Tf.

[0041] Here, the chosen gas is considered to behave as an ideal gas, and Boyle's law therefore applies to this gas. Thus, applying the ideal gas law, after expansion of the gas, leads to the equation: P,CR Pf(CR+C rv) T, ~ Tf

[0042] The equation can be rewritten by using the experimental pycnometric ratio Rm which is the ratio of the initial pressures P; and final pressure Pf, thus "pi . — P f

[0043] In general, we can postulate the existence of other corrective terms due to the experiment as a whole and group them with AT ITf in a global relative correction term denoted a, term a relatively small compared to 1, and where AT = Tf-Ti.

[0044] We therefore have nfv \ . ( ~cT~ ~cZ '^ + a^ \ KK /

[0045] The above equation also applies for v = 0, and allows us to obtain a no-load pycnometric ratio Ro or a no-load pycnometric ratio of the form: „ CR+CT Note ~ ""c""" • ( 1 + a ) '

[0046] Furthermore, we let W be the ratio Rm normalized by Rq, and we obtain the function f: Il v, or (f): VF - 1 = - S v where S is a sensitivity coefficient associated with said pycnometer 1. The sensitivity coefficient S is generally expressed in cm 3, and is here equal to _ .,,.,,,1.,,.,.. Cfî+Cr

[0047] This function f thus corresponds to a mathematical calibration model for a constant volume pycnometer. The sensitivity coefficient S is therefore a function of the first and second capacities CR and CT respectively of the reference container 3 and receiving container 5, while the pycnometric ratio at empty Ro is a function of the pressure values ​​in the first and second containers 3, 5 during an empty measurement of said pycnometer 1.

[0048] Fig. 2, on the other hand, is a very schematic and partial view of a second type of gas pycnometer 1' according to the invention, this second type of pycnometer 1' being a constant gas flow pycnometer, and is also referred to below as a constant flow pycnometer.

[0049] Said constant flow pycnometer 1' also includes at least two containers 3 and 5 fluidically connected to each other through one or more valves, such as a two-way valve 6' or in an alternative embodiment not represented by the combination of several valves.

[0050] More specifically, containers 3 and 5 of the pycnometer 1' are: - a reference container 3 having a reference volumetric capacity CR, called the first volumetric capacity; - a receiving container 5 configured to house a solid sample 7 whose volume v is to be determined, said receiving container 5 having a volumetric test capacity CT, called second capacity.

[0051] Said pycnometer 1' further comprises an intermediate container 8, or buffer container, connected to the reference container 3 and receiving container 5, via the valve 6', said intermediate container 8 having an intermediate volumetric capacity Ci, referred to as the third capacity. It should be noted that the third capacity Ci preferably has the smallest possible value and is small compared to the first and second capacities CR and CT.

[0052] Said valve 6' is thus configured to put the reference container 3 and intermediate container 8 into fluidic communication, this by fluidically isolating the receiving container 5, or else to put the receiving container 5 and intermediate container 8 into fluidic communication, this by fluidly isolating the reference container 3.

[0053] Said pycnometer 1' also includes an injection system 15 for injecting a gas at a constant flow rate q into the reference container 3 and the test container 5. The injected gas is preferably an inert or low-reactivity gas, such as helium, dinitrogen, etc.

[0054] Said injection system is further configured to inject a gas at a constant flow rate q into the reference container 3 or the receiving container 5, via the intermediate container 8 and the valve 6. It should be noted that the intermediate capacity Ci can represent the only volume of all the pipes and elements of the pycnometer 1' which are or which connect the reference container 3 and the receiving container 5 to each other and / or to the injection system 15 and flow control.

[0055] It should be noted that it is possible to consider that the constant-volume pycnometer 1 also has a third container with a third capacity corresponding to the pipes connecting the different elements of the pycnometer, but due to the structure of said pycnometer, these pipes and their associated capacities are implicitly included in the first and / or second capacity. It is therefore not necessary to consider these capacities related to the pipes separately.

[0056] Furthermore, said constant flow pycnometer 1' may include one or more thermometric probes (or temperature probes) 11', 12' configured to measure (directly or indirectly) the temperature of the gas located respectively in the reference container 3 and the test container 5. This advantageous configuration makes it possible to determine and verify that the temperature homogeneity remains stable over time during the use of the constant flow pycnometer 1'. However, in the context of the invention, it is not essential to experimentally consider the temperature of the gas.

[0057] Thus, to determine the volume v of a solid sample 7 placed in the receiving container 5, there is: - injection of a gas at a constant flow rate into the reference container 3 and the intermediate container 8, until the pressure (in said containers 3 and 8) changes from an initial value Pf' to a final value Pf', the variation of pressure as a function of time measured by the pressure measuring device 9 allowing to determine a rate of variation of the pressure PR (in containers 3 and 8); - injection of a gas at a constant flow rate q into the receiving container 5 and the intermediate container 8, until the pressure (in said containers 5 and 8) changes from the initial value Pf' to the final value Pf', the variation of pressure as a function of time measured by the pressure measuring device 9 allowing to determine a rate of variation of the pressure PT (in containers 5 and 8).

[0058] The application of the ideal gas law gives the expression for the molar gas flow rate q = dN / dt in the respective containers 3, 5 and 8 of the pycnometer 1', and allows us to obtain, assuming the temperature TR in the reference container 3 is constant: „ _ » ir .r1 \ 1 ; with R the molar gas constant. y - R + RTlt

[0059] Furthermore, when injecting gas at a constant flow rate into the receiving container 5, assuming the temperature Tt in the receiving container 5 is constant, the gas flow rate q can be formulated as follows j 1.

[0060] The equation can be rewritten by using the experimental pycnometric ratio R'm which is the ratio of the (successive) pressure variations in containers 3 and 5 (in association with the intermediate container 8), the experimental pycnometric ratio being of the form: pt. K m “ PR

[0061] In general, one can postulate the existence of other corrective terms due to the experimentation as a whole and group them with AT / T f in an overall relative correction term «', relatively small compared to 1, where AT -Tt-Tr .

[0062] We therefore have an experimental pycnometric ratio: = +

[0063] Whereas the pycnometric ratio of an open-circuit measurement or open-circuit calibration coefficient Rq, therefore for v = 0, is written as follows:

[0064]

[0065]

[0066]

[0067] Let W be the ratio R'm normalized by , therefore jy » _ r . r, or the function (f') : W' -1 = - S'- v ; where S' is a sensitivity coefficient associated with the pycnometer 1'. The sensitivity coefficient S' is generally expressed in cm³, and is here equal to .......l....... c r +c7 This equation f' also corresponds to the mathematical calibration model of a gas pycnometer, here at constant flow rate, and is similar in form to that of the constant volume pycnometer. It is thus possible to determine a mathematical calibration function f or f' having the same form, regardless of the type of gas pycnometer 1 or 1', the function f, f' being a function of at least two modeling parameters: - the sensitivity coefficient S, S' which is a predetermined value, for example determined by the manufacturer of the pycnometer 1,1' under the best possible experimental conditions, and which is given to the user of the pycnometer and / or integrated into the embedded software of said pycnometer in order to determine the relative uncertainties of the measurements made using said pycnometer; - a pycnometric ratio at zero Ro, R'o which is a dimensionless quantity, which corresponds to a measurement of zero, that is to say a measurement during which no solid sample having a volume v is introduced into the pycnometer 1, 1', the pycnometric ratio at zero Ro grouping all the experimental corrections to be taken into account when determining the volume v of a sample by the pycnometer 1, 1'. The sensitivity coefficient S for the pycnometer 1 at constant volume is therefore a function of the first CR and second capacitances CT of said pycnometer 1, while the sensitivity coefficient S' for the pycnometer 1' at constant flow rate is a function of the second CT and third capacitances C of said pycnometer 1' at constant flow rate.

[0068] It will be noted that volume 1 or constant flow 1' pycnometers advantageously include electronic entities, such as electronic boards, microprocessors, memories, etc., configured to be connected to the various elements enabling the measurement of physical quantities, such as the pressure measuring element 9, at least one thermometric probe 11, etc., to store the values ​​of the physical quantities measured by said pycnometer 1, 1', and to calculate the relative measurement uncertainty(s) of the pycnometer 1, 1' as a function of the sensitivity coefficient S, S' and the pycnometric ratio at no load Ro, R'o.

[0069] More particularly, the gas pycnometer 1, 1' is configured to implement a calibration method 100 according to the invention, prior to determining a volume v of a sample housed in the receiving container 5. Thus, said pycnometer 1, 1' includes embedded software in which is integrated the mathematical modeling function f, f', that is to say an equation of the form: W - 1 = - S v.

[0070] Figure 3 is therefore a flowchart of the different steps of the calibration method 100 of a pycnometer 1,1' which thus includes: - no-load measurement Si of the pycnometer 1.1' to determine the calibration coefficient Ro , R'o; - introduction S2 of the solid sample 7 of volume v (to be determined) into the receiving container 5; - measurement S3 of the experimental pycnometric ratio Rm, R'm with sample 7 housed in the receiving container 5; - determination S4 of one or more of the components of an uncertainty relating to the measurement of the volume v of the sample as a function of the sensitivity coefficient S, S' and / or the pycnometric ratio at empty Ro, R'o.

[0071] More specifically, the step S4 of determining one or more components of a measurement uncertainty comprises one or more of the following substeps: - determination of a sensitivity uncertainty component corresponding to the model uncertainty deduced from the calibration of the gas pycnometer, said sensitivity uncertainty component being a function of (1 / 52. ( 1- VF)) u(S) ■> °where is the sensitivity uncertainty and the term (1 / S2 • ( | -W)) is the associated sensitivity coefficient; - determination of a repeatability uncertainty component that is a function of ( 1 / | SI • ) U( R ) , where u ( R ) is the repeatability uncertainty and the term (1 / ISi • ^o) the associated sensitivity coefficient; - determination of an uncertainty component related to the repeatability and reproducibility of the determination of the Roqui no-load pycnometric ratio is a function of ( VF / 151 • Rq) u(R0) °where is the uncertainty of the pycnometric ratio at no load Roet the term (jy / \S\ ■ Ro) u( Rq) 'c associated sensitivity coefficient.

[0072] Thus U(S) is an uncertainty related to the determination of the sensitivity coefficient S, ll(R) is an uncertainty related to the determination of the measurement pycnometric ratio Rm, and u( Rq) is an uncertainty related to the determination of the no-load pycnometric ratio Ro.

[0073] It should be noted that determining an uncertainty component related to the repeatability and reproducibility of the determination of the pycnometric ratio at idle Ro requires an additional subsequent step, which is a new measurement at idle of the ratio pycnometric Ro after the measurement of the pycnometric ratio of measurement Rm and therefore of the withdrawal of the sample 7 from the receiving container 5.

[0074] Of course, the invention is not limited to the examples just described, and many modifications can be made to these examples without departing from the scope of the invention. In particular, the various features, forms, variants, and embodiments of the invention can be combined with one another in various ways, provided they are not incompatible or mutually exclusive. Specifically, all the variants and embodiments described above are combinable.

Claims

Demands

1. Calibration method (100) of a gas pycnometer (1; 1'), said gas pycnometer (1; 1') comprising at least two containers (3, 5), a reference container (3) having a reference volumetric capacity (RC), referred to as the first capacity, and a receiving container (5) configured to hold a sample (7) whose volume v is to be determined, said receiving container (5) having a test volumetric capacity (TC), referred to as the second capacity, said gas pycnometer (1; 1') being configured to determine the volume v of a sample (7) housed in the receiving container (5), said gas pycnometer (1, 1') being characterized by a calibration function (f; s) which depends on at least two modeling parameters: - a sensitivity coefficient (S; S') which is a predetermined value; - a pycnometric ratio at idle (RO; R'O) which is a measurement value at idle of said pycnometer (1; 1'); the relative measurement uncertainty(s) of said pycnometer (1; 1') being determined as a function of the sensitivity coefficient (S; S') and the pycnometric ratio at no load (RO; R'O); characterized in that there is determination of a calibration function ( / ; / ') relating the volume v of sample (7) to be determined with the sensitivity coefficient (S ; S'), the pycnometric ratio at empty (RO; R'O), and a pycnometric ratio (Rm ; R'm) which is a function of a physical quantity relating to a pressure or a rate of pressure variation in the first and / or second containers (3, 5) during a measurement of said pycnometer (1 ; 1') when a sample (7) is housed in the receiving container (5); and in that the calibration function ( / ; / ') which relates the volume v to be determined of a sample (7) to the sensitivity coefficient (S ; S') and the pycnometric ratio at empty (RO ; R'O) is of the form: W -1 = -5 v where W is the pycnometric ratio (Rm ; R'm) normalized by the pycnometric ratio at empty (RO; R'O).

2. Calibration method (100) according to the preceding claim, characterized in that the sensitivity coefficient (S; S') is a function of at least one of the volumetric capacities (CT, CR) of said pycnometer (1;1').

3. Calibration method (100) according to any one of the preceding claims, characterized in that the empty pycnometer ratio (RO; R'O) is a function of a physical quantity relating to a pressure or a rate of pressure change in the first and / or second containers (3, 5) during an empty measurement of said pycnometer (1; 1').

4. Calibration method (100) according to any one of the preceding claims, characterized in that the pycnometric ratio at empty (RO; R'O) is determined before and after the determination of the volume v of a sample (7).

5. Calibration method (100) according to the preceding claim, characterized in that there is determination of one or more of the components of an uncertainty relating to the measurement of the volume v of the sample (7): - determination of an uncertainty component of the sensitivity corresponding to the uncertainty of the model deduced from the calibration of the gas pycnometer, said uncertainty component of sensitivity being a function of (1 / 52.(lW))w(S),°ù i^S) is the uncertainty of the sensitivity and the term (1 / 52 • ( 1 - ) ) is the associated sensitivity coefficient; - determination of a repeatability uncertainty component which is a function of ( 1 / 851 'Æq) u(R) , where li(R) is the repeatability uncertainty and the term ( 1 / 151 • Rq) the associated sensitivity coefficient - determination of an uncertainty component related to the repeatability and reproducibility of the determination of the pycnometric ratio at no load R0 which is a function of (yy / | $ | , where « ( Æo ) is the uncertainty of the pycnometric ratio at no load R0 and the term ( TF / ISÎ • Rq) u(Rq) the associated sensitivity coefficient.

6. Calibration method (100) according to any one of the preceding claims, characterized in that the gas pycnometer (1; 1') is a constant volume gas pycnometer (1) or a constant flow gas pycnometer (1').

7. Calibration method (100) according to the preceding claim, characterized in that the sensitivity coefficient (S; S') is: - a function, for a constant volume pycnometer (1), of the first and second capacities (CR, CT);

8. - function, for a constant flow pycnometer (1') which includes an intermediate container (8) having an intermediate capacity (CI), called third capacity, of the second and third capacities (CT, CI). Gas pycnometer (1; 1'), characterized in that it is configured to implement the calibration method (100) according to any one of the preceding claims.

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