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

Abstract

The invention relates to a method (100) for calibrating a gas pycnometer, said gas pycnometer comprising at least two containers, a reference container having a reference volume capacity, called the first capacity, and a receiving container configured to house a sample whose volume is to be determined, said receiving container having a test volume capacity, called 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 an empty pycnometric ratio which is an empty measurement value of said pycnometer, the relative measurement uncertainty(s) of said pycnometer being determined as a function of the sensitivity coefficient and the empty pycnometric ratio.Figure to be published with the abstract: none.

Description

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

[0001] The present invention relates to the general field of metrology of the density of a solid, as well as to the field of gas pycnometers for measuring the density of a solid. The invention relates more particularly 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 because of the porosity of these solids or because it is impossible to immerse them. For such solids, it is known to use a gas pycnometer in order to determine the volume of the solid considered, the mass of this solid being determined elsewhere by weighing, then to deduce the density therefrom.

[0003] It will be noted that hereinafter 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 the Boyle-Mariotte law and generally implement static pressure measurements between a first volume capacity in which a predefined gas is placed, at a fixed pressure (for example atmospheric pressure), and a second volume capacity 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 higher than said fixed pressure.

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

[0006] Whether the pycnometers implement a simple expansion of the second volume capacity towards the first volume capacity or whether they implement an intermediate volume capacity with variable volume, the pycnometers currently known have measurement uncertainties of the order of a few fractions of a percent, typically of the order of 0.15 to 0.2% in relative value, in particular when a substitution method is not applied or is applied incorrectly.

[0007] Furthermore, the size of the samples of solids that can be introduced into such pycnometers remains small, of the order of a few tens of cubic centimeters. In the case of solids having lattice structures (or "lattice structure" in English) or containing numerous small channels, such a limitation can induce additional uncertainty resulting from the representativeness of the small volume sample used for the volume measurement.

[0008] It should also be noted that there are now two types of gas pycnometers, constant-volume pycnometers, known and commercially available, and constant-flow pycnometers as defined in patent application FR 2 103 866 A1. It may also be mentioned that the term "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 uncertainty contribution to the sample. Indeed, under optimal conditions, the relative uncertainty is of the order of 0.02%, but this requires the user to calibrate the pycnometer with a standard sphere suitably chosen according to the volume of the sample tested, a limited time between calibration of the pycnometer with a standard sphere and measurement of the unknown volume, a similarity of the standard volume-calibrated volume values, a volume / capacity ratio as high as possible (for example 30%), thermal stability of the pycnometer containers (commercial pycnometers generally provide temperature control by circulation of temperature-regulated water around the containers).

[0010] This calibration method is often referred to as a so-called "Borda" substitution calibration method. However, measurements with the smallest uncertainties are performed only over a short time after calibration of the pycnometer using standard volumes (e.g. standard spheres). In addition, the uncertainty performance (i.e. their minimization) with the substitution calibration method is mainly related to the repeatability of the measurements.

[0011] This method of calibration 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 user manuals for said pycnometers define an uncertainty valid for optimal conditions which are difficult to grasp and most often to respect. It is then necessary, if the user wants an estimate of the uncertainty of the pycnometer in the conditions of use, to carry out a metrological characterization, which requires the user to have solid expertise in metrology, this estimation of uncertainty being moreover 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, independently of the latter's metrology skills.

[0014] The present invention thus proposes to remedy at least one of the aforementioned drawbacks by proposing a new type of method for calibrating a gas pycnometer, said gas pycnometer comprising at least two containers, a reference container having a reference volume capacity, called the first (volume) capacity, and a receiving container configured to house a sample whose volume v is to be determined, said receiving container having a test volume capacity, called the second (volume) 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 f which depends on at least two modeling parameters: - a sensitivity coefficient S which is a predetermined value; - an empty pycnometric ratioR0 which is an empty measurement value of said pycnometer; the relative uncertainty(s) of measurement of said pycnometer being determined as a function of the sensitivity coefficient S and the empty pycnometric ratio Ro.

[0015] The calibration method according to the invention is thus advantageously based on: - a sensitivity coefficient S, generally expressed in cm 3, which is a predetermined value, for example determined by calibration or by the manufacturer of the pycnometer in 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 carried out by means of said pycnometer; - a pycnometric ratio to vacuum 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 to vacuum Ro making it possible to group together all the experimental corrections to be taken into account in the determination of the volume v of a sample by the pycnometer.

[0016] The use of such parameters makes it possible to simply determine the contribution of the working environment of the pycnometer to the measurement uncertainty, but also to integrate these into a mathematical model in the embedded software of the pycnometer so that the measurement uncertainties are calculated automatically. elsewhere, it is no longer necessary to have one or more standard spheres to carry out the calibration of the pycnometer.

[0017] 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, the user of the pycnometer no longer needing to carry out complicated manipulations and / or to have advanced expertise in metrology to determine the accuracy of the measurements carried out by means of said pycnometer.

[0018] According to a possible characteristic, the sensitivity coefficient S is a function of at least one of the volume capacities of said pycnometer. It is advantageous to have a sensitivity coefficient S which depends only on values ​​of one or more volume capacities of the pycnometer containers, because said sensitivity coefficient S can then be determined in advance very precisely, for example by the manufacturer of the pycnometer and / or by calibration.

[0019] According to another possible characteristic, the vacuum pycnometric ratio Ro is a function of a physical quantity relating to a pressure or to a rate of pressure variation in the first and / or second containers during a vacuum measurement of said pycnometer.

[0020] According to another possible characteristic, there is determination of a calibration function f linking the sample volume to be determined with the sensitivity coefficient S, the vacuum pycnometric ratio Ro, and a pycnometric ratio Rm which is a function of a physical quantity relating to the pressure or the rate of pressure variation in the first and / or second containers during a measurement of said pycnometer when a sample is housed in the receiving container. Note 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 change of pressure (or a change of pressure with respect to time).

[0021] According to another possible characteristic, the vacuum pycnometric ratio Ro is determined before and after the determination of the volume v of a sample. Advantageously, the variation of the vacuum pycnometric ratio Ro before and after the measurement of a volume v of a sample characterizes the stability of the experimental parameters, for example the temperature homogeneity in the pycnometer, and allows the calculation of the uncertainty contribution of the pycnometer in its environment, this therefore independently of the uncertainty contribution of said sample.

[0022] According to another possible characteristic, the function f which links the volume v to be determined of a sample to the sensitivity coefficient S and to the empty pycnometric ratio Ro is of the form: W - 1 = - S Vwhere W is the pycnometric ratio Rm normalized by the pycno- ratio

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[0027] metric empty Ro. According to another possible characteristic, there is determination of one or more of the components of an uncertainty relative to the measurement of the volume v of the sample: - 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 (] / 52 . (1-W)^m(5), where u(S) is the sensitivity uncertainty and the term (1 / S2. ( 1 - IV is the associated sensitivity coefficient; - determination of a repeatability uncertainty component which is a function of ( Il |S| R u( R) ' U (R) the repeatability uncertainty and the term 1 / j 5 j • R j The associated sensitivity coefficient - determination of an uncertainty component linked to the repeatability and reproducibility of the determination of the empty pycnometric ratio Ro which is a function of ( IV / |S| • R u(Ro) ■> °where w(^o) is the uncertainty of the empty pycnometric ratio Ro and the term (yy / |5| ■ R ) tt(R(}) The associated sensitivity coefficient It should be noted that the different components of the uncertainty relative to the measurement carried out by the pycnometer are determined by analytical calculation on the basis of 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. According to another possible characteristic, the sensitivity coefficient S is: - for a constant volume pycnometer, function of the first and second capacities; - for a constant flow pycnometer which includes an intermediate container having an intermediate capacity, called the third capacity, which is a function of the second and third capacities. The invention also relates to a gas pycnometer, characterized in that said pycnometer is configured to implement the calibration method as defined above. Other characteristics and advantages of the invention will become apparent from the following description on the one hand, and from several examples of embodiment given for informational and non-limiting purposes with reference to the attached schematic drawings on the other hand, in which: - [Fig.l] 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 of [Fig.l] or [Fig.2].

[0028] Of course, the features, variants and different embodiments of the invention may be combined with each other, in various combinations, provided that they are not incompatible or mutually exclusive. In particular, variants of the invention may be imagined comprising only a selection of features described below in isolation from the other features described, if this selection of features is sufficient to confer a technical advantage or to differentiate the invention from the prior art.

[0029] In particular, all the variants and all the embodiments described can be combined with each other if nothing prevents this combination from a technical point of view. In addition, in the different figures, the elements common to several figures retain the same reference.

[0030] [Fig.l] 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 gas volume pycnometer, and also referred to hereinafter as a constant volume pycnometer.

[0031] The constant volume pycnometer 1 thus comprises at least two containers 3 and 5 fluidly connected to each other via a valve 6, such as a stop valve (or “on-off valve”).

[0032] More particularly, the containers 3 and 5 of the pycnometer 1 are: - a reference container 3 having a reference volume capacity CR, called the first volume 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 test volume capacity CT, called second capacity.

[0033] It will be noted that hereinafter 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. The volume and the volume capacities are homogeneous to a length cubed.

[0034] It will also be noted that the reference container 3 and the receiving container 5 are advantageously made of the same material, for example stainless steel.

[0035] Said pycnometer 1 also comprises an injection system not shown. allowing a predetermined quantity of gas to be injected into the reference container 3. The injected gas is preferably an inert or slightly reactive gas, such as helium, nitrogen, etc.

[0036] As a non-exclusive example, the gas chosen is advantageously nitrogen, which is low cost, which has, for a given pressure and temperature, a compressibility coefficient close to that of an ideal gas, and which has very low chemical reactivity with a wide variety of materials.

[0037] Said pycnometer 1 further comprises a pressure measuring member 9, for example a capacitive type pressure gauge, 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 the valve 6 and expansion of the gas injected into the two containers 3 and 5.

[0038] The pycnometer 1 may optionally comprise at least one thermometric probe 11 (or temperature probe) configured to determine a temperature relative to the reference container 3 and / or to 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 temperature of the sample. This advantageous configuration has the effect of associating a temperature with the measurement of the volume v in the event that the temperature is an influencing quantity for said measurement.

[0039] Thus, to carry out the determination of the volume v of a solid sample placed in the receiving container 5, there are: - injection of a gas into the reference container 3, the gas then has a pressure P; 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 the valve 6 and placing said containers 3 and 5 in fluid communication, the gas then having, after expansion, a pressure Pf and a theoretical temperature Tf.

[0040] Here we consider that the chosen gas behaves like a so-called perfect gas, and that the Boyle-Mariotte law therefore applies to this gas. Thus, the application of the perfect gas law, after expansion of the gas, leads to the equation: P^Cr ( Cf^CrV ) T ~ Tf

[0041] The equation can be rewritten by using the experimental pycnometric ratio Rm which is the ratio of the initial pressures P; and final pressures Pf, thus n _ A . p

[0042] Generally speaking, we can postulate the existence of other corrective terms due to the experiment as a whole and group them with AT / Tf in a global relative correction term noted a, term a relatively small compared to 1, and where A T ^Tf-Ti.

[0043] We therefore have Rn = . (1 + ct).

[0044] The above equation also applies for v = 0, and makes it possible to obtain a pycnometric ratio to empty Ro or a pycnometric ratio to empty of the form: Æo=^.(l + a)-

[0045] Furthermore, we set W as the ratio Rm normalized by Æo, and we obtain the function f: [y = — = 1 - 5 - p , or again (f): W - 1 = - S ■ v ; where S is a sensitivity coefficient associated with said pycnometer 1. The sensitivity coefficient S is generally expressed in cm3, and is here equal to _ ........L.... CR+CT

[0046] This function f thus corresponds to a mathematical calibration model for a constant volume pycnometer. The sensitivity coefficient S is thus a function of the first and second capacities CR and CT respectively of the reference 3 and receiving 5 containers, while the vacuum pycnometric ratio Ro is a function of the pressure values ​​in the first and second containers 3, 5 during a vacuum measurement of said pycnometer 1.

[0047] [Fig.2], for its part, 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 pycnometer with constant gas flow rate, and is also referred to hereinafter as a constant flow rate pycnometer.

[0048] Said constant flow pycnometer 1' also comprises at least two containers 3 and 5 fluidically connected to each other via one or more valves, such as a two-way valve 6' or in an alternative embodiment not shown by the combination of several valves.

[0049] More particularly, the containers 3 and 5 of the pycnometer 1' are: - a reference container 3 having a reference volume capacity CR, called the first volume 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 test volume capacity CT, called second capacity.

[0050] Said pycnometer 1' further comprises an intermediate container 8, or buffer container, connected to the reference 3 and reception 5 containers, via the valve 6', said intermediate container 8 having an intermediate volume capacity Cb called the third capacity. It will be noted that the third capacity C, preferably has the smallest possible value and is small compared to the first and second capacities CR and CT.

[0051] Said valve 6' is thus configured to put the reference containers 3 and intermediate 8, this by fluidly isolating the receiving container 5, or else to put the receiving containers 5 and intermediate 8 into fluid communication, this by fluidly isolating the reference container 3.

[0052] Said pycnometer 1' also comprises an injection system 15 making it possible to inject 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 slightly reactive gas, such as helium, nitrogen, etc.

[0053] Said injection system is furthermore 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 will be noted that the intermediate capacity Q can represent the sole volume of all the pipes and elements of the pycnometer 1' which are or which put the reference container 3 and the receiving container 5 into fluid communication with each other and / or with the injection system 15 and flow control.

[0054] It will be noted that it is possible to consider that the constant volume pycnometer 1 also has a third container with a third capacity which corresponds to the pipes connecting the different elements of the pycnometer, but due to the structure of said pycnometer these pipes and the associated capacities are implicitly included in the first and / or the second capacity. It is therefore not necessary to consider these capacities relating to the pipes separately.

[0055] Furthermore, said constant flow rate pycnometer 1' may comprise 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 3 and test 5 containers. 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 rate pycnometer 1'. However, in the context of the invention, it is not essential to experimentally consider the temperature of the gas.

[0056] Thus, to carry out the determination of the volume v of a solid sample 7 placed in the receiving container 5, there are: - 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) passes from an initial value P;' to a final value Pf', the variation of the pressure as a function of time measured by the pressure measuring device 9 making it possible to determine a rate of variation of the pressure (in the 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) drops from

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[0066] the initial value P;' to the final value Pf', the variation of the pressure as a function of time measured by the pressure measuring device 9 making it possible to determine a rate of variation of the pressure Pr (in the containers 5 and 8). The application of the ideal gas law gives the expression of the molar gas flow rate q = dN / dt in the respective containers 3, 5 and 8 of the pycnometer 1', and makes it possible to obtain, assuming the temperature TR in the reference container 3 to be constant: q _ p^(J + Cfj ' with the molar constant of the gases. Furthermore, when injecting gas at a constant flow rate into the receiving container 5, assuming the temperature Tt in the receiving container 5 to be constant, the gas flow rate q can be formulated as follows: q _ p^C + ' The equation can be rewritten by using the experimental pycnometric ratio R'm which is the ratio of the (successive) variations of the pressures in containers 3 and 5 (in association with the intermediate container 8), the experimental pycnometric ratio being of the form: n< „ . m ““ / > LR Generally speaking, one can postulate the existence of other corrective terms due to the experiment as a whole and group them with AT / T f into an overall relative correction term a', relatively small compared to 1, where AT = Tt- T r . We therefore have an experimental pycnometric ratio: R' - ( __E— 1 . ( 1 + a' ) \ Cr+Cr Cr+Ci J ' 1 u ' While the pycnometric ratio of an empty measurement or empty calibration coefficient Rq, therefore for v = 0, is written as follows: ^'o- •(! + «') We set IV' the ratio R'm normalized by R'o, therefore ' - ^2 - J _ 5. v, or even the ^0 function (f'): IV' - 1 = - 5' • v ; where S' is a sensitivity coefficient associated with said pycnometer 1'. The sensitivity coefficient S' is generally expressed in cm3, and is here equal to __JL_. This equation f' also corresponds to the mathematical model for calibrating 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, whatever 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 completed 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 carried out using said pycnometer; - a pycnometric ratio to vacuum 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 to vacuum Ro grouping together all the experimental corrections to be taken into account when determining the volume v of a sample by the pycnometer 1, 1'.

[0067] The sensitivity coefficient S for the pycnometer 1 at constant volume is therefore a function of the first CR and second capacities 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 capacities Ci of said pycnometer 1' at constant flow rate.

[0068] It will be noted that the volume 1 or constant flow rate 1' pycnometers advantageously comprise electronic entities, such as electronic cards, microprocessors, memories, etc. configured to be connected to the various elements making it possible to measure physical quantities, such as the pressure measuring member 9, at least one thermometric probe 11, etc. to memorize 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 empty pycnometric ratio Ro, R'o.

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

[0070] [Fig.3] is therefore a flowchart of the different stages of the calibration method 100 of a pycnometer 1.1' which thus comprises a: - empty measurement Si of the pycnometer 1.1' to determine the calibration coefficient R o, 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 empty pycnometric ratio Ro, R'o-

[0071] More particularly, the step S4 of determining one or more of the components of an uncertainty relating to the measurement comprises one or more of the following sub-steps: - 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 (] / 52 . (1-W)^m(5), where u(S) is the sensitivity uncertainty and the term (1 / S2. ( 1 - IV is the associated sensitivity coefficient; - determination of a repeatability uncertainty component which is a function of l / |S | • R ) h (R) ■> °ù M ( ) is the repeatability uncertainty and the term [ 1 / jS| • R ) 'c associated sensitivity coefficient; - determination of an uncertainty component linked to the repeatability and reproducibility of the determination of the empty pycnometric ratio Ro which is a function of ( IV / 151 R ) u(R ) °where w(^o) is the uncertainty of the empty pycnometric ratio Ro and the term (jy / . R ju(R0) 'c associated sensitivity coefficient.

[0072] Thus / . / (5) is an uncertainty linked to the determination of the sensitivity coefficient S, U ( R ) is an uncertainty linked to the determination of the measurement pycnometric ratio Rm, and u( Rq) is an uncertainty linked to the determination of the empty pycnometric ratio Ro.

[0073] It will be noted that the determination of an uncertainty component linked to the repeatability and reproducibility of the determination of the empty pycnometric ratio Ro requires an additional subsequent step which is a new empty measurement of the pycnometric ratio Ro after the measurement of the measurement pycnometric ratio Rm and therefore the removal of the sample 7 from the receiving container 5.

[0074] Of course, the invention is not limited to the examples which have just been described and numerous adjustments can be made to these examples without departing from the scope of the invention. In particular, the different characteristics, forms, variants and embodiments of the invention can be associated with each other in various combinations insofar as they are not incompatible or mutually exclusive. In particular, all the variants and embodiments described above can be combined with each other.

Claims

Claims

1. Method for calibrating (100) a gas pycnometer (1; 1'), said gas pycnometer (1; 1') comprising at least two containers (3, 5), a reference container (3) having a reference volume capacity (CR), called first capacity, and a receiving container (5) configured to house a sample (7) whose volume v is to be determined, said receiving container (5) having a test volume capacity (CT), called second capacity, said gas pycnometer (1; 1') being configured to determine the volume v of a sample (7) housed in the receiving container (5), characterized in that said gas pycnometer (1, 1') is characterized by a calibration function ( / ; / ') which depends on at least two modeling parameters: - a sensitivity coefficient (S; S') which is a predetermined value; - an empty pycnometric ratio (Ro; R'o) which is an empty measurement value of said pycnometer (1; 1');the relative uncertainty(s) of measurement of said pycnometer (1; 1') being determined as a function of the sensitivity coefficient (S; S') and the empty pycnometric ratio (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 volume capacities (CT, CR) of said pycnometer (1; 1').

3. Calibration method (100) according to any one of the preceding claims, characterized in that the vacuum pycnometric ratio (Ro; R'o) 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 vacuum measurement of said pycnometer (1; 1').

4. Calibration method (100) according to any one of the preceding claims, characterized in that the vacuum pycnometric ratio (Ro; R'o) is determined before and after determining the volume v of a sample (7).

5. Calibration method (100) according to any one of the preceding claims, characterized in that there is determination of a calibration function ( / ; / ') linking the volume v of sample (7) to be determined with the sensitivity coefficient (S ; S'), the vacuum pycnometric ratio (Ro ; R'o), and a pycnometric ratio (Rm ; R'm) which is a function of a physical quantity relating to a pressure or to 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).

6. Calibration method (100) according to the preceding claim, characterized in that the calibration function ( / ; / ') which links the volume v to be determined of a sample (7) to the sensitivity coefficient (S ; S') and the empty pycnometric ratio (Ro ; R'o) is of the form: W - 1 = - S V where W is the pycnometric ratio (Rm ; R'm) normalized by the empty pycnometric ratio (Ro ; R'o).

7. 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 a component of uncertainty of the sensitivity corresponding to the uncertainty of the model deduced from the calibration of the gas pycnometer, said component of uncertainty of sensitivity being a function of ( [ / 5 2. (1-W))m(5),°where is the uncertainty of the sensitivity and the term (1 / S2 • (] - [y)j is the associated sensitivity coefficient; - determination of a component of uncertainty of repeatability which is a function of 1 / |$| . ju(R)' °where u(R) is the uncertainty of repeatability and the term iy j 'c associated sensitivity coefficient - determination of an uncertainty component linked to the repeatability and reproducibility of the determination of the empty pycnometric ratio Ro which is a function of (jy y |$| .ju[Rq) ' °ù ^(Rq) is the uncertainty of the empty pycnometric ratio Ro and the associated sensitivity efficient term (yyy ■ R ^ u^Rq) 'c co'.

8. A calibration method (100) according to any preceding claim, characterized in that the gas pycnometer (1; 1') is a constant volume gas pycnometer (1) or a constant flow rate gas pycnometer (1').

9. 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); - function, for a constant flow pycnometer (1') which comprises an intermediate container (8) having an intermediate capacity (CO, called third capacity, second and third capacities (CT, CO.

10. 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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