1 Introduction

Gas analysis plays an important role in the field of environmental monitoring, to control the emission from vehicles, industrial safety, healthcare facility, food industry, petrochemical sector, technological processes and many more applications. To achieve the accuracy in monitoring results, there is requirement of calibration gas mixtures (CGMs). The gravimetrically prepared CGMs are the roots of metrological traceability chain in gas analysis and play an important role in developing the hierarchy of gas analysis that corroborates the gas analysis results are traceable to SI; mol. The key point of CGMs is their traceability to the international system of units (SI) [1,2,3]. The availability of CGMs with traceable values to the international system of units and its uncertainty estimation is required to achieve reliable result in gas measurements.

From past several years, environmental pollution has become serious concern to any developing country. There are several factors such as vehicular exhaust emission, industries which has been contributing in environmental pollution since several decades [4]. Vehicular emission produces large amount of harmful and trace gases such as carbon dioxide, carbon monoxide, hydrocarbons (HC), SOx, NOx, particulate matter and VOCs. Unburnt petrol and diesel released from vehicle’s exhaust are the main cause of hydrocarbon production in environment. Methane and non-methane hydrocarbons (NMHCs) have adverse effect on human directly and indirectly. Hydrocarbon (HC) has major participation in photochemical smog formation [5]. Propane does not affect environment directly but its participation in photochemical smog formation impact human health badly. Photochemical smog is intense air pollution which has adverse effect on human body such as eye and respiratory irritation [6]. Accurate and precise determination of hydrocarbon concentration in ambient air has been given great attention for developing the model of environmental trace gases. The instrumental techniques like gas chromatography equipped with thermal conductivity detector, flame ionization detector and mass spectrometer, etc., exhaust gas analyzers and non-methane hydrocarbon analyzers are used for the measurement of propane in ambient air [7]. The analytical instruments used for monitoring HC at ambient and emission levels requires highly accurate and stable CGMs so that we can get the reliable and traceable results. In pollution centers, Pollution under control (PUC) machines is used to detect various pollutants such as CO, NOx, SOx and HCs in vehicles. There is a requirement of traceable calibration gas mixtures (CGMs) with stated uncertainty for the calibration of these PUC machines/ instruments. Propane in nitrogen gas/ air is used for the calibration of PUC machines or environmental monitoring instruments.

Calibration gas mixtures can be prepared by many methods [8]. There are many challenges faced during past years for the development of calibration gas mixtures [9,10,11]. Calibration gas preparation methods can be divided into static and dynamic method and both of them have their own limitations and challenges [12]. Static method is more useful for the preparation of gas mixtures because it is simple and less expensive. Static method can be further divided into gravimetric preparation, volumetric method and barometric method. Among these methods, gravimetric method provides the accurate calibration gas mixtures with least uncertainty and their directly linkage to SI through mass. ISO 6142 is the standard document which is followed by all National Metrology Institutes (NMIs) including NPL India (Internationally known as NPLI) for gravimetric preparation of gas mixtures with highly accurate results [13, 14]. Gas mixtures composition expressed as mole fraction that is calculated on the basis of mass of gas filled in the cylinder, and the molar mass of the component gas and the purity of the mixing gases. The mass of each component gas can be calculated as the differences between the mass of the cylinder before and after filling the component gas. Accuracy and precision of prepared standard gas mixtures based on the type of weighing technique used, the purity of components and the quality of the cylinders. The gravimetric preparation of gas mixture leads to very small uncertainties and the values are traceable to SI.

In this paper, three CGMs of propane in nitrogen of amount-of-substance fraction in range of 5000–12,000 µmol/mol were prepared using pure propane gas according to ISO 6142 [15, 16]. Verification of prepared gas mixture was done using standard ISO 6143 [17]. Gas chromatography equipped with thermal conductivity detector and flame ionization detector is used for the verification of composition of propane in nitrogen gas mixtures [18, 19]. Uncertainty estimation of prepared binary component gas mixture was done using ISO-GUM approach [20]. According to ISO-GUM guidelines for estimation of uncertainty in any method consists of four main parts such as specification of measurand, identification and quantification of all possible sources of uncertainty and combination of all individual sources of uncertainty.

2 Experimental Work

2.1 Preparation of Binary Component Gas Mixtures

Two components (Comp-1; Propane, C3H8 and Comp-2; Nitrogen, N2) calibration gas mixtures were prepared using gravimetric method following the protocols of ISO 6142. This is the primary method, to convince the international linkages for preparation of CGMs. Final mass of component gas and diluents gas after the preparation can be determined by the method given in ISO 6142. For the preparation of CGMs, selected cylinder is evacuated up to 10–3 mbar with simultaneously heating of cylinder at (70 ± 5) °C. After the evacuation, purging of the cylinder is done with nitrogen gas. Target mass of component gas and balance gas is transferred in 10 L aluminum cylinder by using stainless steel tube. Target mass of component gas and balance gas can be calculated by Eq. (1)

$$m_{{\text{i}}} = \frac{{x_{{\text{i}}} \times P \times V \times M_{{\text{i}}} }}{R \times T \times Z}$$
(1)

where mi mass of the component i in the mixture, xi indented mole fraction of components in mol/mol, P final filling pressure in Pascal; 120 × 105 Pa, V volume of cylinder in cubic meters; 10 L, i.e., 0.01 m3, Mi molar mass of ith component, (MC3H8 = 44.08 g mol−1, MN2 = 28.0135 g mol−1), R gas constant; 8.314 J mol−1 K−1, T temperature in Kelvin; 298 K, Z compressibility factor of the mixture at T and P, i.e., 1.0.

The component and diluent gas were accurately weighed by using Equal arm gas balance (Raymer HCE-25G; 25 kg capacity with 1 mg sensitivity). The environmental conditions, temperature and relative humidity (RH) of the laboratory were maintained at (23 ± 2) °C and (45 ± 15) %, respectively. Reference cylinder is used in opposite to the sample cylinder for the weighing in equal arm gas balance. The difference in mass of the sample cylinder and reference cylinder gives the mass of the component gas filled in sample cylinder. Weighing is done in sequence of E-S-E-S-E-S-E (E, empty pan; S, with cylinders) to get three sets of mass readings. The repeatability in weighing is calculated from these three sets of mass data. The buoyancy effect generated by the cylinder could be diminished by reference weighing. The preparation scheme for binary component mixture is shown in Fig. 1.

Fig. 1
figure 1

Preparation scheme for binary component gas mixture

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For the preparation of calibration gas mixtures, starting material propane and nitrogen were used as a parent gas. Propane gas with > 99.7% purity and N2 gas of purity > 99.999% were purchased. Aluminum cylinders (Make; Luxfer, UK) with 10 L water capacity were used for gas mixture preparation. Vacuum system (Make; Zinke, Germany) was used for the evacuation of cylinder up to 10–3 mbar with simultaneously heating to remove trace gases present in the cylinder. Transfer of component gas (pure propane) and nitrogen was done using a gas filling system (Make: Zinke, Germany).

Weighing Process—An equal arm double pan gas balance (Raymer HCE-25G) with 25 kg maximum capacity and 1 mg sensitivity, calibrated at CSIR-NPL, is used for preparation of calibration gas mixtures (CGMs). This balance has equal arm type, equipped with magnetic dampers and a weighing chain of 1000 mg with digital display. Figure 2 shows the weighing using ‘equal arm double pan balance’ with sample cylinder (Right pan) and reference cylinder (Left pan): (a) digital display for showing added mass in mg on right pan for balancing through chain weight (b) digits showing after balancing both sides, i.e., null point (c) handle for weighing (Up at rest; Down at time of weighing) (d) calibrated mass pieces used as auxiliary weight. The balance is kept in a vibration free room and temperature and RH maintained at (23 ± 2) °C and (45 ± 15) % throughout the period of preparation of calibration gas mixtures.

Fig. 2
figure 2

Weighing using Equal arm Double Pan Balance with sample cylinder (Right pan) and reference cylinder (Left pan) a digital display for showing added mass in mg on right pan for balancing through chain weight b digits showing after balancing both sides, i.e., null point c handle for weighing (Up at rest; Down at time of weighing) d calibrated mass pieces used as auxiliary weight

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2.2 Calculation of Molar Composition

Based on the preparation scheme given in Fig. 1 final molar composition of CGMs can be calculated by Eq. (2)

$$x_{{\text{i}}} = \frac{{n_{{\text{i}}} }}{{\mathop \sum \nolimits_{{{\text{i}} = 1}}^{2} n_{{\text{i}}} }}$$
(2)

where xi is the mole fraction of component gas in mixture and ni is the number of mole of component ‘i’ present in binary mixture.

$$n_{{\text{i}}} = \frac{{m_{{\text{i}}} }}{{M_{{\text{i}}} }}$$
(3)

where mi and Mi are the mass and molecular mass of component ‘i’ present in binary mixture. Table 10 represents the mole fraction of CGMs that is calculated after preparation procedure.

3 Measurement Uncertainty Estimation

Uncertainty associated with gravimetric preparation procedure can be identified by establishment of cause-effect diagram/ Ishikawa diagram. A cause-effect diagram for the uncertainty estimation of gravimetric preparation of calibration gas mixture is shown in Fig. 3. Major uncertainty components affecting the final amount of gas fraction are weighing of components, purity of component gases, residual gas, expansion of cylinder, handling of cylinder and molecular mass of component gases, etc. The detailed description with quantification of each uncertainty component is explained.

Fig. 3
figure 3

Fish bone diagram for uncertainty components associated with amount-of-substance fraction of propane in nitrogen in gravimetric preparation

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3.1 Weighing of Components

In weighing procedure uncertainty parameters associated are balance, weighing process and auxiliary mass pieces uncertainty.

3.1.1 Balance

Balance is calibrated from Mass metrology section at NPLI. The calibration certificate of balance has reported expanded uncertainty ± 4 mg at coverage factor k = 2 which corresponds to a coverage probability 95% for a normal distribution. The standard uncertainty due to balance is calculated as Eq. (4)

$$u_{{{\text{balance}}}} = \frac{U}{k} = 2 {\text{mg}}$$
(4)

3.1.2 Mass Pieces Uncertainty

E-2 Class stainless steel mass pieces (1 mg–5 kg), calibrated at CSIR-NPL India were used for gravimetric preparation of gas mixtures. These calibrated mass pieces are used as auxiliary weights to compensate the difference between mass of the reference cylinder and sample cylinder. The used mass pieces uncertainty was taken from the calibration certificate as given in Tables 1, 4 and 7 for empty pan weighing, component-1 weighing and component-2 weighing, respectively.

3.1.3 Weighing Process and Associated Uncertainty

Weighing process has major part in the gravimetric preparation of gas mixtures to calculate the accurate mass of each gas component. It includes empty pan weighing, buoyancy effect, component-1 and component-2 weighing, which contributes uncertainty at every step, respectively. The mass of the cylinder having component was determined by comparing it with the closely equal weight reference cylinder. Sample cylinder and reference cylinders are placed on pan hanging using brackets on right and left pan, respectively, as shown in Fig. 2. The weighing process of cylinder was carried out with the sequence E0-S1-E1-S2-E2-S3-E3 (E = empty pan; S = sample cylinder) to get three sets of readings of component mass. Empty pan readings were taken before every weighing to cancel the impact of air dynamics on the balance pan. The standard deviation obtained from the repeated weighing sequence of empty pan with sample cylinder was used to calculate the standard uncertainty due to the weighing process.

Equation (5) is the equation used in ISO 6142:2001, for calculation of density of moist air by measuring the temperature, pressure and relative humidity in the balance room.

$$\rho = \frac{{3.484~88~P~ - ~\left( {8.083~7~ + ~737.4~ \times 10^{{ - 3}} ~t~ + ~975.25~ \times ~10^{{ - 6}} t^{3} } \right) \times h}}{{\left( {273.15~ + ~t} \right)~ \times ~10^{3} }}$$
(5)

where ρ is air density, p is the pressure in Pa, t is the temperature in °C and h is the relative humidity as % RH. A calibrated digital temperature, RH and pressure meter (Model; TR-73 U, S/N E00099, Make; T & D Corp, Japan) was used for measurement of T, RH and P in the balance room. As per CIPM formula (1981/91) [21, 22], an approximate formula used for the calculation of air density is given in Eq. (6)

$$\rho = \frac{{0.34848 p - 0.009\left( {hr} \right) \times {\text{exp}}\left( {0.062 t} \right)}}{273.15 + t}$$
(6)

where the density of air is obtained in kg m−3, the pressure p is given in hPa, the relative humidity, hr expressed as a %, and the temperature, t in °C.

Buoyancy effect can be calculated by Eq. (7).

$$m_{{{\text{buoyancy}}}} = \rho \times \left( {V_{{\text{S}}} - V_{{\text{R}}} } \right)$$
(7)

where ρ is air density, VS is the volume of sample cylinder and VR is the volume of reference cylinder. As the volume of both cylinders is same, so the mass due the air buoyancy is due to the volume of auxiliary mass pieces used on pan for balancing the cylinders. Hence, the mass due to buoyancy correction is calculated according to Eq. (8) which is used in calculation of components masses and given in Tables 3, 6 and 9. It majorly affects the weighing of component-2, i.e., diluents gas, as the auxiliary mass pieces used in weighing of diluents gas is approx 1.4 kg which has large volume hence, large buoyancy effects.

$$m_{{{\text{buoyancy}}}} = \rho \times V_{{\text{mass pieces}}}$$
(8)

Standard uncertainty due to weighing process, i.e., uncertainty in resultant masses m0, m1 and m2 can be obtained by combining uncertainty of repeatability in weighing, mass (includes both digits and mass pieces uncertainty) and balance as given in Eq. (9).

$$u_{{\text{resultant mass}}} = \sqrt {u_{{{\text{rep}}}}^{2} + u_{{\text{mass }}}^{2} + u_{{{\text{balance}}}}^{2} }$$
(9)

3.1.4 Uncertainty due to Empty Cylinder Weighing

Sample cylinder position = Right pan.

Reference Cylinder position = Left Pan.

Chain wt position = Right panside always (added in form of digits up to 1000 mg).

Mass of Reference cylinder (mref cyl) − Mass of Sample cylinder (msam cyl) = 20 g.

Auxiliary mass piece position = Right pan (as the mref cyl > msam cyl).

Tables 1, 2 and 3 represent the data related to mass pieces, empty pan weighing and resultant weighing with associated uncertainty, respectively.

Table 1 Mass, volume and uncertainty data for mass pieces used in empty pan weighing
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Table 2 Weighing cycle data of empty pan weighing
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Table 3 Resultant weighing data and associated uncertainty in empty pan weighing
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3.1.5 Weighing of Component-1

Difference in cylinder mass after transfer of comp-1 (propane) = 8.05 g.

mref cyl − msam cyl = 8.05 g.

Auxiliary mass piece position = Right pan (as the mref cyl. > msam cyl).

Tables 4, 5 and 6 represent the data related to auxiliary mass pieces put on right pan, comp-1 weighing and resultant component weighing with associated uncertainty, respectively.

Table 4 Mass, volume and uncertainty data for mass pieces used in Comp-1 weighing
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Table 5 Weighing cycle data of component-1 weighing
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Table 6 Resultant weighing data and associated uncertainty in component-1 weighing
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Mass of Comp 1 (m1–m0) (propane) = 11.728 g.

Corrected mass of propane = 11.693 g (for 99.7% purity of propane).

3.1.6 Weighing of Component-2, Diluent Gas

Difference in cylinder mass after transfer of comp-2 (Nitrogen) = 1359.75 g.

mref cyl − msam cyl = 1359.75 g.

Auxiliary mass piece position = Left pan (mref cyl < msam cyl).

Data related to auxiliary mass pieces put on left pan, comp-2 weighing and resultant component weighing data with associated uncertainty are given in Tables 7, 8 and 9.

Table 7 Mass, volume and uncertainty data for mass pieces used in component 2 weighing
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Table 8 Weighing cycle data of component-2 weighing
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Table 9 Resultant weighing data and associated uncertainty in component-2 weighing
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Mass of comp-2 (m2 + m1) = 1368.488 g (as mass piece in m1 are on other pan, hence added to the resultant weight of comp-2).

Corrected mass of comp-2, i.e., nitrogen = 1368.474 g (for 99.999% purity of nitrogen).

Number of moles, ni and mole fraction of components, xi, are calculated as per Eq. (2) and (3) using masses of component 1 and 2, respectively. And data for all the prepared three calibration gas mixtures is given in Table 11.

3.2 Purity of Component Gases

Total impurities in the parent gases determine the purity of gas and can be the major contributor to the uncertainty of final mixture composition. In most of the cases, the uncertainty of diluent gas is of more important. A pure propane gas with > 99.7% purity mean it can have at most 0.003 mol/mol or 3000 ppmv impurities. Similarly, > 99.999% pure nitrogen means it can have at most 10 µmol/mol or 10 ppmv impurities. The mole fraction of pure gas can be determined conventionally by Eq. (10).

$$x_{{{\text{pure}}}} = 1 - \mathop \sum \limits_{{{\text{i}} = 1}}^{{\text{N}}} x_{{\text{i}}}$$
(10)

where xpure is mole fraction of pure gas, xi mole fraction of impurity i, determined by analysis or taken from certificate and N is the number of impurities present in the final mixture.

3.3 Residual Gas

As per the procedure given in Fig. 1, the cylinder is purged with nitrogen before evacuation. When the cylinder is evacuated up to 10–3 mbar, the actual pressure inside the cylinder is higher so assuming that the pressure inside the evacuated cylinder is 1 mbar, the residual gas at 1 mbar for 10 L cylinder volume is calculated as per Eq. (1) and found to be 11.30 mg considering nitrogen gas at 1 bar pressure at same condition which comes out 11.30 g. So, the standard uncertainty was calculated by applying rectangular distribution as shown in Eq. (11).

$$u_{{{\text{res}}}} = \frac{11.30}{{1000 \times \sqrt 3 }} = 0.0065 {\text{g}}$$
(11)

3.4 Expansion of Cylinder

Aluminum alloy cylinders (Al-alloy 6061, Luxfer, UK) with nominal capacity of 10 L are used for the preparation of calibration gas mixtures. In ISO 6142 it is clearly mention that with increase in the pressure of 15 MPa there is change in volume cylinder about 20 cm3. Experimentally also it is proven that there is a linear pattern changes in volume of cylinder with filling pressure of a gas cylinder from 0.1 to 12 MPa [23]. We fill the gas cylinder at 12 MPa, i.e., 120 bar so the change in volume at high pressure is expected to 20 cm3. So considering two extreme effect of air density, uncertainty due to expansion of cylinder for 10 L capacity is calculated 23.8 mg and the standard uncertainty can be calculated by applying rectangular distribution as given in Eq. (12).

$$u_{{{\text{exp}}}} = \frac{23.8}{{1000 \times \sqrt 3 }} = 0.014 {\text{g}}$$
(12)

3.5 Handling of Gas Cylinder

Uncertainty due to gas cylinder are mainly caused by adsorption or desorption effects on the external surface of cylinder, diffusion of gases into the cylinder due to vacuum, dust particle on cylinder and valve, loss of material or paints from external surface of cylinder or valve during the component gas transfer. The maximum standard uncertainty due to handling of gas cylinder is assumed as 0.01 g. So, applying rectangular probability distribution, the relative standard uncertainty due to handling of gas cylinder was calculated according to Eq. (13).

$$u_{{{\text{handing}}}} = \frac{0.01}{{\sqrt 3 }} = 0.0058\;{\text{g}}$$
(13)

3.6 Uncertainty due to Molecular Mass of Component Gas

Although uncertainty contribution due to molar mass of component gas is negligible but the gases having high molecular mass may be taken as one parameter and uncertainty contribution can be quantified using Eq. (14). IUPAC technical report provides the use of standard atomic weights and their related uncertainty [24].

$$u^{2} \left( {M_{{\text{r}}} } \right) = \mathop \sum \limits_{{\text{E}}} \upsilon_{{\text{E}}}^{2} u^{2} \left( {A_{{\text{r}}} \left( {\text{E}} \right)} \right)$$
(14)

where Ar(E) denotes the atomic weight of element E and νE the atomic composition coefficient of element E in the molecular formula (e.g., C = 3 and H = 8 in case of C3H8). So the uncertainty in molecular mass of propane is calculated as

$$u^{2} \left( {M_{{\text{r}}} ({\text{C}}_{3} {\text{H}}_{8} } \right)) = 3^{2} \times u^{2} \left( {A_{{\text{r}}} \left( {\text{C}} \right)} \right) + 8^{2} \times u^{2} \left( {A_{{{\text{r}} }} \left( {\text{H}} \right)} \right)$$
(15)

From IUPAC technical report, u(Ar(C)) = 0.00058 g and u(Ar(H)) = 0.00008 g, so we can calculate uncertainty in molecular mass of propane using Eq. (15). Similarly the uncertainty in molecular mass of nitrogen can also be calculated.

$$u(M_{{\text{r}}} \left( {{\text{C}}_{3} {\text{H}}_{8} } \right) = 0.00184 {\text{g}}$$
$$u\left( {M_{{\text{r}}} \left( {{\text{N}}_{2} } \right)} \right) = 0.0005 {\text{g}}$$

4 Combining of Standard Uncertainty Components and Value Assigned

The combined standard uncertainty of mass of component-1, uc(m1) and that of component-2, uc(m2) is estimated by square root of sum of square of all contributing parameters as per Eqs. (16) and (17).

$$u_{{\text{c}}} \left( {m_{1} } \right) = \sqrt {u_{{{\text{m}}_{1} }}^{2} + u_{{{\text{res}}}}^{2} + u_{{{\text{handling}}}}^{2} + u_{{{\text{M}}_{{\text{r}}} \left( {{\text{C}}_{3} {\text{H}}_{8} } \right)}}^{2} }$$
(16)
$$u_{{\text{c}}} \left( {m_{1} } \right) = \sqrt {\left( {0.0116} \right)^{2} + \left( {0.0065} \right)^{2} + \left( {0.0058} \right)^{2} + \left( {0.0018} \right)^{2} }$$
$$u_{{\text{c}}} \left( {m_{1} } \right) = 0.0146\;{\text{g}}$$

Similarly,

$$u_{{\text{c}}} \left( {m_{2} } \right) = \sqrt {u_{{m_{2} }}^{2} + u_{{{\text{res}}}}^{2} + u_{{{\text{exp}}}}^{2} + u_{{{\text{handling}}}}^{2} + u_{{M_{{\text{r}}} \left( {{\text{N}}_{2} } \right)}}^{2} }$$
(17)

where um1, um2, ures, uexp, uhandling, uMr(C3H8), uMr(N2) are the standard uncertainty of corresponding parameters and estimated in earlier sections of this paper.

\(u_{{\text{c}}} \left( {m_{2} } \right) = 0.0216\) g.

Now according to Eq. (2), model for the mole fraction of propane (xP) in nitrogen is

$$x_{{\text{P}}} = \frac{{\frac{{m_{{\text{P}}} }}{{M_{{\text{P}}} }}}}{{\frac{{m_{{\text{P}}} }}{{M_{{\text{P}}} }} + \frac{{m_{{\text{N}}} }}{{M_{{\text{N}}} }}}}$$
(18)

where mass of propane, mP = m1 and mass of nitrogen, mN = m2.

Combined standard uncertainty can be calculated according to Eq. (19), when the variables are not dependant on each other and there is no correlation between the input quantities.

$$u_{{\text{c}}}^{2} \left( {y\left( {x_{1} ,x_{2} \ldots . } \right)} \right) = \mathop \sum \limits_{{{\text{i}} = 1,{\text{n}}}} c_{{\text{i}}}^{2} u(x_{{\text{i}}} )^{2} = \mathop \sum \limits_{{{\text{i}} = 1,{\text{n}}}} u(y,x_{{\text{i}}} )^{2}$$
(19)

where u(xi) is the uncertainty of all independent parameters associated with value y and ci is the sensitivity coefficient for the corresponding parameter [25].

The uncertainty of the amount fraction computed from gravimetry is calculated by the application of law of propagation of uncertainty to formula (19), molecular mass of components being constants are not taken into account.

$$u^{2} \left( {x_{{{\text{P}},{\text{ grav}}}} } \right) = \left[ {\frac{{\partial x_{{\text{P}}} }}{{\partial m_{{\text{P}}} }}} \right]^{2} \times u^{2} \left( {m_{{\text{p}}} } \right) + \left[ {\frac{{\partial x_{{\text{P}}} }}{{\partial m_{{\text{N}}} }}} \right]^{2} \times u^{2} \left( {m_{{\text{N}}} } \right)$$
(20)

where (∂xp/∂mp) and (∂xp/∂mN) are the sensitive coefficients associated with mp and mN and given as

$$c_{1} = \frac{{\partial x_{{\text{P}}} }}{{\partial_{{m_{{\text{P}}} }} }} = \frac{{M_{{\text{P}}} m_{{\text{N}}} M_{{\text{N}}} }}{{\left( {m_{{\text{p}}}^{2} M_{{\text{N}}}^{2} + m_{{\text{N}}}^{2} M_{{\text{P}}}^{2} + 2 \times m_{{\text{P}}} M_{{\text{P}}} \times m_{{\text{N}}} M_{{\text{N}}} } \right)}} = 4.59 \times 10^{ - 4} \;{\text{mol mol}}^{{ - {1}}} {\text{g}}^{{ - {1}}}$$
$$c_{2} = \frac{{\partial x_{{\text{P}}} }}{{\partial_{{m_{{\text{N}}} }} }} = - \frac{{M_{{\text{P}}} m_{{\text{P}}} M_{{\text{N}}} }}{{\left( {m_{{\text{p}}}^{2} M_{{\text{N}}}^{2} + m_{{\text{N}}}^{2} M_{{\text{P}}}^{2} + 2 \times m_{{\text{P}}} M_{{\text{P}}} \times m_{{\text{N}}} M_{{\text{N}}} } \right)}} = 3.93 \times 10^{ - 6} \;{\text{mol mol}}^{{ - {1}}} {\text{g}}^{{ - {1}}}$$

\(u^{2} \left( {x_{{{\text{P}},{\text{ grav}}}} } \right) = 6.71 \times 10^{ - 6}\) mol2/mol2 = 6.71 µmol2/mol2

The amount-of-substance fraction of a particular component in mixture is expressed as Eq. (21) and the uncertainty associated with gravimetric preparation and that with the purity is represented as Eq. (22).

$$x_{{{\text{i}},{\text{ prep}}}} = x_{{{\text{i}},{\text{ grav}}}} + \Delta x_{{{\text{i}},{\text{ purity}}}}$$
(21)
$$u^{2} (x_{{{\text{i}},{\text{ prep}}}} ) = u^{2} (x_{{{\text{i}},{\text{ grav}}}} ) + u^{2} \left( {\Delta x_{{{\text{i}},{\text{ purity}}}} } \right)$$
(22)

Assuming contribution of uncertainty due to purity is 0.1% relative, the uncertainty due to preparation is calculated using Eq. (22) is.

\(u(x_{{{\text{i}},{\text{ prep}}}} ) = 8.61\) µmol/mol.

The effective degrees of freedom can be calculated using Welch-Satterthwaite approximation formula [23] given in Eq. (23)

$$\nu_{{{\text{eff}}}} = \frac{{u_{{\text{c}}}^{4} \left( y \right)}}{{\mathop \sum \nolimits_{{{\text{i}} = 1}}^{{\text{N}}} \frac{{u_{{\text{i}}}^{4} \left( {x_{{\text{i}}} } \right)}}{{\nu_{{\text{i}}} }}}}$$
(23)

where uc(y) is the combined standard uncertainty, ui(xi) is standard uncertainty of ith parameter) and νi is the degrees of freedom for corresponding ith parameter.

As νeff = ∞, coverage factor k = 2 at 95% confidence level. So the Expanded uncertainty is expressed as.

U (at k = 2) = 2 × u(xi, prep) = 2 × 8.61 = 17.23 µmol/mol.

5 Verification of Calibration Gas Mixture and Associated Uncertainty

A Gas Chromatograph (GC) model 6890 from Agilent, USA with gas sampling valve equipped with both TCD and FID was used for determination of propane in nitrogen gas mixtures. The linearity of independently prepared three CGMs was checked by analysis using gas chromatography FID. For introduction of the gas sample into gas sampling valve of GC, a mass flow controller MFC (Alicat Scientific, USA) was used to ensure consistent flow. A Porapak Q packed SS column, 1/8″ inch OD, 10 feet length was used with helium as carrier gas (25 ml/min flow rate) at 100 °C oven temperature and 250 °C detector temperature (H2 flow-40 mL/min; Air Flow-400 mL/min). The concentration of prepared gas mixture (µmol/mol) was plotted versus the instrument response (µV) from GC FID and linear fit found to follow the linear equation with linear regression coefficient r2 as 0.9993. The independently prepared gas mixtures can be checked with respect to each other following ISO 6143. The uncertainty due to analytical measurement of CGM by GC FID was found to be 0.3% relative. The uncertainty parameters responsible for the analytical measurement uncertainty are repeatability, reproducibility, GC response and uncertainty of calibration gas mixture used. The verification equation used for the CGM analytical verification is given in Eq. (24)

$$\left| {x_{{{\text{grav}}}} - x_{{{\text{anal}}}} } \right| \le 2 \sqrt {u^{2} \left( {x_{{{\text{grav}}}} } \right) + u^{2} \left( {x_{{{\text{anal}}}} } \right)}$$
(24)

where xgrav and xanal are the gravimetric assigned amount-of-substance fraction and analytical determined amount-of-substance fraction.

The total uncertainty of the assigned value of the certified calibration gas mixture is given by Eq. (25)

$$u\left( {x_{{{\text{i}},{\text{ Assigned}}}} } \right) = \sqrt {u^{2} \left( {x_{{{\text{i}},{\text{ prep}}}} } \right) + u^{2} \left( {x_{{{\text{i}},{\text{ anal}}}} } \right)}$$
(25)

where ugrav and uanal are the uncertainty due to gravimetric preparation and analytical verification measurement of the prepared calibration gas mixture.

\(u\left( {x_{{{\text{Assigned}}}} } \right) = ~\sqrt {\left( {0.16} \right)^{2} + ~\left( {0.30} \right)^{2} } = 0.{\text{34}}\% {\text{ rel }}({\text{at}}\;k = {\text{1}})\) .

Table 11 represent the uncertainty associated with the gravimetrically prepared calibration gas mixture of propane in nitrogen (binary gas mixture) which include both preparation and analytical uncertainty.

The contribution of the uncertainty components contributing toward the weighing of component 1 and 2 is shown in Fig. 4a and b, respectively, as relative contribution to their mass weighing. Purity of component gas and weighing of component gas are the major contributing factor in the gravimetric preparation of binary component gas mixture as the component gas is always small in amount so its contribution to the uncertainty budget is large as compared to that of diluents gas. Tables 10 and 11 represent the amount-of-substance fraction calculated using Eqs. (2) and (3) for the gravimetrically prepared three CGMs of propane in nitrogen along with their expanded uncertainties for both preparative and analytical verification as explained in this paper for CGM-1. It shows the expanded uncertainty for all the three gas mixtures is 0.65% relative to the assigned value in which the contribution of preparative uncertainty is 0.3% relative.

Fig. 4
figure 4

Relative uncertainty contribution in weighing of components in binary mixture a for weighing of component-1; propane) b for weighing of component-2; nitrogen

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Table 10 Amount-of-substance fraction data of CGMs of propane in nitrogen
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Table 11 Uncertainties associated with certified values of CGMs propane in nitrogen
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6 Conclusion

Propane in nitrogen is taken as a case study for estimation of measurement uncertainty for gravimetric preparation of binary component gas mixture. The gravimetric values for the amount-of-substance fraction for propane in nitrogen CGMs are (5401 ± 17) µmol/mol, (6838 ± 18) µmol/mol and (11,413 ± 27) µmol/mol. The Expanded uncertainty in preparation of CGM of propane in nitrogen is found to be 0.34% relative (k = 2 at 95% confidence interval). The major contributing factors affecting uncertainty of the prepared calibration gas mixture are purity of component gas and its weighing. In weighing of component-1 (propane in this case), the uncertainty contribution mainly comes from the mass of component gas along with other factors like residual gas and handling of cylinder. Molecular mass of component gas has the least uncertainty contribution in weighing. In weighing of component-2, the major contributing factors are mass of component-2 and expansion of cylinder at high pressure filling. The use of equal arm double pan balance with reference cylinder in weighing process decreases the probability of buoyancy effect which could be major if single pan balance had been used. The buoyancy correction due to auxiliary mass pieces used for balancing of both cylinders is taken in to account in weighing process itself. In standard gas mixtures preparation, gravimetric method is the primary method and can achieve least uncertainty and it depends on the purity of gases used for the preparation of calibration gas mixtures. The uncertainty associated with the certified value of these gravimetrically prepared CGMs includes both preparative and analysis contribution of uncertainty. These primary standards can be used for assurance of traceability in gas measurements both for ambient air quality as well as vehicular emissions, with better precision and accuracy.