Uncertainty of factor Z in the gravimetric volume measurement


ACTA IMEKO 
ISSN: 2221-870X 
September 2021, Volume 10, Number 3, 198 - 201 

 

ACTA IMEKO | www.imeko.org September 2021 | Volume 10 | Number 3 | 198 

Uncertainty of factor Z in gravimetric volume measurement 

Mar Lar Win1 

1 National Institute of Metrology (Myanmar) - NIMM, Yangon, Myanmar 

 

 

Section: RESEARCH PAPER  

Keywords: Density of air; density of water; value of factor Z; uncertainty of factor Z; gravimetric volume measurement 

Citation: Mar Lar Win, Uncertainty of factor Z in the gravimetric volume measurement, Acta IMEKO, vol. 10, no. 3, article 27, September 2021, identifier: 
IMEKO-ACTA-10 (2021)-03-27 

Section Editor: Andy Knott, National Physical Laboratory, United Kingdom 

Received: May 7, 2021; In final form: August 13, 2021; Published: September 2021 

Copyright: This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, 
distribution, and reproduction in any medium, provided the original author and source are credited. 

Corresponding author: Mar Lar Win, e-mail: marlarwin99@gmail.com  

 

1. INTRODUCTION 

The ISO 4787 standard describes the general equation for the 
calculation of the volume at the reference temperature of 20 °C, 
𝑉20, from the apparent mass of the water, be it contained or 
delivered, as follows from [1], B.1: 

𝑉20 = (𝐼L − 𝐼E) (
1

𝜌W − 𝜌A
) (1 −

𝜌A
𝜌B

) [1 − 𝛾(𝑡 − 20)] , (1) 

where 𝐼L is the balance reading of the vessel containing water (g), 
𝐼E is the balance reading of empty vessel (g), 𝜌W is the density 
(g/mL) of the water at temperature 𝑡, 𝜌A is the density of air 
(g/mL), 𝜌B is the actual density of the reference weights 
(8 g/mL), 𝛾 is the coefficient of the cubic thermal expansion of 
the material (°C−1), and 𝑡 is the temperature of the water used in 
the testing (°C). 

Since this equation is somewhat complicated to work with, 
the factor 𝑍 is provided to simplify the volume calculation: 

𝑉20 = 𝑚 𝑍 [1 − 𝛾(𝑡 − 20)] , (2) 

where 𝑚 is the net weighed values of 𝐼L − 𝐼E and 𝑍 is the 
conversion factor of the mass of the water measured in terms of 
volume at the measurement temperature. 

2. DETERMINATION OF FACTOR Z 

The factor 𝑍 depends on the air density, water density, water 
temperature, and the density of the reference weights. The factor 
can be derived from (1) as follows: 

𝑍 = (
1

𝜌W − 𝜌A
) (1 −

𝜌A
𝜌B

) . (3) 

To determine the value of factor 𝑍, the following parameters 
must be taken into account: 

• density of air 𝜌A in relation to atmospheric pressure, 

temperature and a relative humidity of 40 % – 90 %; 

• density of water 𝜌W in relation to temperature; 

• density of the reference weights 𝜌B of the balance used. 

2.1. Determining the air density  

The air density 𝜌A in kg m
−3 can be determined with sufficient 

uncertainty from the following CIPM approximation formula for 
air density [2], E.3-1:  

𝜌A =  
0.348 48 𝑃 − 0.009 ℎ𝑟 × 𝑒0.061×𝑡

273.15 + 𝑡
  , (4) 

where 𝑃 is the atmospheric pressure (hPa), ℎ𝑟 is the relative 

humidity (%), and 𝑡 is the atmospheric temperature (°C).  

ABSTRACT 
In the gravimetric volume measurement method, the factor Z is generally used to facilitate an easy conversion from the apparent mass 
obtained using a balance to the liquid volume. The uncertainty of the measurement used for the liquid volume can be divided into two 
specific contributions: one from the components related to the mass measurements and one from those related to the mass-to-volume 
conversion. However, some ISO standards and calibration guides have suggested that the uncertainty due to the factor Z is generally 
neglected in the uncertainty calculation pertaining to gravimetric volume measurement. This paper describes the combined effects of 
the density of the water, the density of the reference weights, and the air buoyancy on the uncertainty of factor Z in terms of how they 
subsequently affect the uncertainty of the measurement results. 

mailto:marlarwin99@gmail.com


 

ACTA IMEKO | www.imeko.org September 2021 | Volume 10 | Number 3 | 199 

Table 1 presents the air density to temperature relationship 
at 1,013.25 hPa and a 50 % relative humidity. 

2.2 Determining the water density 

The density of air-free water 𝜌W′ at a pressure of 1,013.25 hPa 
and a temperature 𝑡 between 0 °C and 40 °C expressed in terms 
of the ITS-90 is given by Tanaka et al. [3] as follows: 

𝜌W′ =  𝑎5 [1 −
(𝑡 + 𝑎1)

2(𝑡 + 𝑎2)

𝑎3(𝑡 + 𝑎4)
] , (5) 

where  

𝑎1 = (− 3.983 035 ± 0.000 67) C  

𝑎2 = 301.797 C  

𝑎3 = 522 528.9 C
2  

𝑎4 = 69.348 81 C  

𝑎5 = (0.999 974 950 ±  0.000 000 84) g/mL  

𝑡 =  water temperature in C .  

Since the pure water that is used in the laboratory is generally 
air-saturated, the density must be corrected. To adjust the air-free 
water density as described in (5) to air-saturated water between 
0 °C and 25 °C (the standard laboratory condition), we can use 
the following [3], 5.2: 

∆𝜌 = 𝑠0 + 𝑠1 × 𝑡 , (6) 

where 

𝑠0
kg m−3

= − 4.612 × 10−3  

𝑠1
kg m−3 °C−1

= 0.106 × 10−3 .  

Moreover, since water is slightly compressible, a small 
correction may be required under typical laboratory conditions. 

The compressibility factor 𝐹c [3], 5.3, for the water density 
used at different pressures is as follows: 

𝐹c = 1 + [(𝑘0 + 𝑘1 𝑡 + 𝑘2 𝑡
2) ∆𝑃] , (7) 

where 

∆𝑃 = 𝑃 − (101 325 Pa)  
 

𝑘0 = 50.74 × 10
−11 Pa−1  

𝑘1 = −0.326 × 10
−11 Pa−1 C−1  

𝑘2 = 0.004 16 × 10
−11 Pa−1 C−2  

𝜌W = (𝜌W´ × 𝐹𝑐 + ∆𝜌)  

with 𝜌W = the density of air-saturated water and 

𝜌W´= the density of air-free water. 

(8) 

 

 

Table 2 presents examples of water density as a function of 
water temperature for distilled water at 1,013.25 hPa after 
correction for any dissolved gasses in the water. 

Meanwhile, Table 3 presents examples of calculated air-
saturated water density as a function of water temperature and 
pressure after applying corrections for the dissolved gasses in the 
water. 

2.3 Density of the reference weights 

The density of the reference weights/mass pieces 𝜌B that are 
used for the balance calibration is normally presented in the 
calibration certificate of the weights. Alternatively, the 
uncertainties corresponding to the used weight class according 
to OIML R 111-1 [2] can be used. If the density of the reference 
weights are not known, 8.0 g/mL is generally used. 

2.4 Determining the factor Z values 

The values of the conversion factor 𝑍 (mL/g) as a function 
of temperature and pressure as given in the existing literature 
mainly pertain to distilled water. When liquids other than distilled 
water are used, the correction factors (factor 𝑍) for the specific 
liquid must be determined. 

Table 4 presents the factor Z values in relation to 
temperature and pressure. The calculations were based on (3) and 
(8). 

3. THE UNCERTAINTY OF FACTOR Z 

3.1 Sources of uncertainty in factor Z  

Having identified the input quantities of factor Z using (3), it 
is possible to determine the sources of uncertainty pertaining to 
the different input quantities, which are: 

Table 1. Air density values as a function of temperature at 1,013.25 hPa and 
a 50 % relative humidity. 

Temperature in °C Density of air in g/mL 

15 0.001 225 

20 0.001 204 

25 0.001 184 

Table 2. Water density values after correction for air-saturated distilled 
water. 

Water 
temperature 

(°C) 

Air-free water 
density (g/mL) 

Correction for air 
saturation 

(g/mL) 

Air-saturated 
water density 

(g/mL) 

15 0.999 102 57 - 0.000 003 02 0.999 099 55 

20 0.998 206 75 - 0.000 002 49 0.998 204 25 

25 0.997 047 02 - 0.000 001 96 0.997 045 06 

Table 3. Air-saturated water density as a function of water temperature and 
pressure. 

Water 
temperature 

(°C) 

Air-saturated water density (g/mL) at 
atmospheric pressure of 

900 hPa 1,013.25 hPa 1,050 hPa 

15 0.999 094 26 0.999 099 55 0.999 101 27 

20 0.998 199 07 0.998 204 25 0.998 205 94 

25 0.997 039 96 0.997 045 06 0.997 046 72 

Table 4. Factor Z values as a function of water temperature and pressure. 

Water 
temperature 

(°C) 

Value of factor Z (mL/g) at 
atmospheric pressure of 

900 hPa 1,013.25 hPa 1,050 hPa 

15 1.001 845 1.001 958 1.001 995 

20 1.002 745 1.002 858 1. 002 895 

25 1.003 912 1.004 026 1.004 062 

27 1.004 448 1.004 562 1.004 599 



 

ACTA IMEKO | www.imeko.org September 2021 | Volume 10 | Number 3 | 200 

• water density 

• air density 

• density of the reference weights. 

The uncertainty of factor Z can be determined using the 
GUM law of propagation of uncertainty [4] as follows: 

𝑢(𝑍) = √(
𝜕𝑍

𝜕𝜌A
𝑢(𝜌A))

2

+ (
𝜕𝑍

𝜕𝜌W
𝑢(𝜌W))

2

+ (
𝜕𝑍

𝜕𝜌B
𝑢(𝜌B))

2

 (9) 

where 𝑢(𝜌A) is the standard uncertainty of the air density (g/mL), 
𝑢(𝜌W) is the standard uncertainty of the water density (g/mL), 
and 𝑢(𝜌B) is the standard uncertainty of the reference weights 
(g/mL). 

3.2 Uncertainty of air density 

At a relative humidity of hr = 0.5 (50 %), a temperature of 
20 °C and a pressure of 1,013.25 hPa, the uncertainty of the air 
density can be estimated according to OIML R111 [2], C.6.3-3, 
as follows: 

𝑢(𝜌A)

𝜌A
= √

(1 × 10−4)2 + (−3.4 × 10−3 × 𝑢(𝑡))
2

 

+(1 × 10−3 ×  𝑢(𝑃))2 + (−10−2  × 𝑢(ℎ𝑟))2
 , (10) 

where 𝑢(𝑡) is the uncertainty of the air temperature 𝑡 (°C), 𝑢(𝑃) 
is the uncertainty of the air pressure 𝑃 (hPa), and 𝑢(ℎ𝑟) is the 
uncertainty of the relative humidity ℎ𝑟 (%). 

For the uncertainty values (k = 2) from the calibration of the 
pressure, temperature and humidity sensors, the uncertainty due 
to the temperature of the air is 0.1 °C, the uncertainty due to the 
barometric pressure is 0.1 hPa, and the uncertainty due to the 
relative humidity is 1 %. This yields the following: 

𝑢(𝜌A) = 0.0012 g mL⁄

× √
(1 × 10−4)2 + (−3.4 × 10−3  ×

0.1

2
)

2

 

+ (1 × 10−3 ×
0.1

2
)

2

+ (−1 × 10−2  ×
0.01

2
)

2 

 = 2.52 × 10−7 g mL⁄  . 

3.3 Uncertainty of water density 

The uncertainty of the water density can be evaluated from 
the formulation given by EURAMET cg 19 [5], 7.3.3, as follows: 

𝑢(𝜌W) = √𝑢
2(𝜌W,F) + 𝑢

2(𝜌W,𝑡) + 𝑢
2(𝛿𝜌W) , (11) 

where 

𝑢(𝜌W,𝑡) = 𝑢(𝑡W)  ×  𝛽 ×  𝜌W(𝑡W) 

𝑢(𝑡W) =  [(
 𝑈(ther)

𝑘
)

2

+ 𝑢(res)2]

0.5

 

𝑢(𝑡W) =  [(
0.01 °C

2
)

2

+ (
0.01 °C

2√3
)

2

]

0.5

 

𝑢(𝑡W) = 0.006 °C . 

The expansion coefficient can be estimated as follows: 

𝛽 = (−0.117 6 𝑡2 + 15.846 𝑡 − 62.677) × 10−6 ℃−1 

𝛽 = (−0.117 6 ×  202 + 15.846 × 20 − 62.677) × 10−6 ℃−1 

𝛽 = 0.000 21 °C−1 

𝜌W(𝑡W) = 0.998 204 25 g/mL (from Table 3) 

𝑢(𝜌W,𝑡) = 0.006 ×  0.000 21 ×  0.998 204 25 

𝑢(𝜌W,𝑡) = 1.26 × 10
−6g/mL~ 1 × 10−6g/mL .  

The standard uncertainty related to 𝑢(𝛿𝜌W) could range from 
a few ppm for highly pure water (measured by means of a high-
resolution density meter) to 10 ppm for distilled or de-ionised 
water, provided that the conductivity is less than 5 µS/cm, 
meaning it is assumed to be 5 × 10−6 g/mL. 

Meanwhile, 𝑢(𝜌W,F) is the estimated standard uncertainty of 

the formulation (4.5 × 10−7 g/mL), 𝑢(𝜌W,𝑡) is the uncertainty 
due to the temperature of the water (1 × 10−6 g/mL), and 
𝑢(𝛿𝜌W) is the uncertainty due to the purity of the water 
(5 × 10−6 g/mL) [5], 8.2.3. This yields the following: 

𝑢(𝜌W) = √(4.5 × 10
−7)2 + (1 × 10−6)2 + (5 × 10−6)2 g/mL 

 =  5.12 × 10−6 g/mL . 
(12) 

3.4 Uncertainty of the reference weights 

The uncertainty of the density of the weights/mass pieces is 
presented in the calibration certificate of the weights, or 
alternatively, according to OIML R111-1, the uncertainty of 
stainless-steel weights Class E2 is 140 kg/m3 (k = 2). 

The uncertainty of the density of the reference weights is 
normally presented in the calibration certificate of the weights. 
For example, the value presented in the calibration certificate of 
the set of weights used is 0.06 g/mL (k = 2): 

𝑢(𝜌B) =
𝜌r
2

=
0.06

2
 g mL⁄ = 0.03 g mL⁄  . (13) 

Alternatively, the uncertainty of the density of the weights, as 
presented in OIML R111-1, can also be used. 

3.5 Calculating the uncertainty of factor Z 

Table 5 presents example data for the calculation of the 
uncertainty of factor Z for air-saturated distilled water at a 
temperature of 20 °C and a barometric pressure of 1,013.25 hPa. 

Sensitivity Coefficient of Air Density 

𝜕𝑍

𝜕𝜌A
= (

1

𝜌W − 𝜌A
) (−

1

𝜌B
) + (1 −

𝜌A
𝜌B

) (
1

(𝜌W − 𝜌A)
2

) 

 = 0.880 500 mL²/g2 

Sensitivity Coefficient of Water Density 

𝜕𝑍

𝜕𝜌W
= (1 −

𝜌A
𝜌B

) (
−1

(𝜌W − 𝜌A)
2

) = −1.005 876 mL²/g2 

Sensitivity Coefficient of Density of Reference Weights 

𝜕𝑍

𝜕𝜌B
= (

1

𝜌W − 𝜌A
) (

𝜌A
𝜌B

2
) = 1.887 6 × 10−5 mL²/g2 

 

Table 5. Example data. 

Parameter Value Uncertainty (k = 1) 

Air density 0.001 204 g/mL 2.52 × 10−7 g/mL 

Water density 0.998 204 g/mL 5.12 × 10−6 g/mL 

Mass density 8.0 g/mL 0.03 g/mL 

Factor Z 1.002 858 mL/g (Refer to the conditions in Table 4) 



 

ACTA IMEKO | www.imeko.org September 2021 | Volume 10 | Number 3 | 201 

Uncertainty of Factor Z (k = 1) 

𝑢(𝑍) = √(
𝜕𝑍

𝜕𝜌A
𝑢(𝜌A))

2

+ (
𝜕𝑍

𝜕𝜌W
𝑢(𝜌W))

2

+ (
𝜕𝑍

𝜕𝜌B
𝑢(𝜌B))

2

 

= √

(0.880 500 × 2.52 × 10−7)2

+(−1.005 876 × 5.12 × 10−6)2

+(1.887 6 × 10−5 × 0.03)2
 mL/g 

= 5.2 × 10−6 mL/g . 

4. DISCUSSION 

As shown in Table 4, the factor Z variation in distilled water 
due to water temperature is approximately 2.0 × 10−3 mL/g or 
2.0 × 10-4 mL/(g °C) between 15 °C and 25 °C. This implies that 
when the uncertainty of the temperature measurement of 
distilled water is 0.1 °C, this will present an uncertainty of 
approximately 0.002 % following the conversion to volume. 
Meanwhile, the influence of barometric pressure on the factor Z 
is approximately 1 × 10−4 mL/g or 0.01 % per pressure change 
of 100 hPa between 900 hPa and 1,050 hPa. The uncertainty of 
the pressure measurement in the laboratory is typically 5 hPa. 
This will cause the uncertainty to the conversion of volume by 
less than 0.000 5 %. 

5. CONCLUSIONS 

The factor Z does not only depend on the density of the liquid 
compensated for water temperature and pressure but also on the 
air density and the density of the weights used for the calibration 
of the balance. However, the contribution to the evaluation of 
uncertainty resulting from the factor Z for gravimetric volume 
measurement is extremely small compared to that resulting from 
the operator process (i.e., meniscus reading) [5], 8.2.7, and the 
mass of water determination, provided that the measuring 

instruments (balance, barometer, thermometer, etc.) are used in 
accordance with the specifications given in the standard 
operating procedure (SOP). Thus, it is good practice to neglect 
the uncertainty of the factor Z in the evaluation of uncertainty 
and to give only the components related to the mass 
measurement process [6]. 

ACKNOWLEDGEMENT 

This paper would not have been possible without the 
exceptional support of Bunjob Suktat, my former expert from 
Physikalisch-Technische Bundesanstalt (PTB) on the ‘PTB-
NIMM Strengthening Myanmar Quality Infrastructure Project’. 
His enthusiasm, knowledge, and minute attention to detail were 
an inspiration and ensured my work remained on track. As my 
teacher and mentor, he has taught me more than I could ever 
give him credit for here. 

REFERENCES 

[1] ISO 4787 (2010) - Laboratory glassware - Volumetric instruments 
- Methods for testing of capacity and for use. 

[2] OIML R111- Weights of classes E1, E2, F1, F2, M1, M2, M3 
Metrological and Technical Requirement, 2004. 

[3] M. Tanaka, G. Girard, R. Davis, A. Peuto, N. Bignell, 
Recommended table for the density of water between 0 °C and 
40 °C based on recent experimental reports, Metrologia 38 (2001) 
pp. 301-309.  
DOI: 10.1088/0026-1394/38/4/3  

[4] JCGM 100:2008 (GUM), Evaluation of measurement data – guide 
to the expression of uncertainty in measurement. 

[5] EURAMET cg-19, Version 3.0 (09/2018), Guidelines of 
determination of uncertainty in gravimetric volume calibration.  

[6] ISO 8655-6:2002(E), International Standard, Piston-operated 
volumetric apparatus, Part 6: Gravimetric methods for the 
determination of measurement error. 

 

 

https://doi.org/10.1088/0026-1394/38/4/3