THE USE OF Z FACTORS IN GRAVIMETRIC VOLUME
MEASUREMENTS
Author: T. Mautjana
NMISA
Private Bag X 34
Lynwood Ridge; 0040
Email: [email protected]
Phone: 012 841 4798; Fax: 012 841 2131
Abstract
Using gravimetric method to determine volume is based on the determination of mass with a
balance of water delivered or contained. This method requires monitoring of the environmental
conditions. There is a need to convert mass of the liquid to volume which must take into account
the density of the liquid and evaporation during the cycle time. The Z factor must take into
account the air density and density of the mass pieces used to calibrate the balance and the
density of water adjusted to local temperature and pressure. This paper discusses the equation for
calculating volume conversion factors (Z) and gives typical examples for different liquids.
1. Introduction
Volume measurements are important in industrial and analytical measurements operations. The
instruments are common in field like chemistry, pharmacy, biology (biochemistry,
microbiology), etc. It is important that these instruments give reliable and accurate results during
operation to give the end user confidence in measurement results. The instrument would
normally get calibrated using gravimetric method which links them directly with the SI unit of
mass [1]. It is therefore vital that possible errors be eliminated through the use of correct method
and it is important to understand formulae (their conditions) and any applicable correction
factors which might be necessary to give accurate results.
2. Gravimetric method
The standard method used by NMI’s and accredited laboratory to determine volume of high
accurate instrument is to weigh the liquid. The method requires the weighing of empty and full
volumetric measure or the content of the measure/instrument can be emptied into a weighing
vessel following a specified procedure [1].
This method should be carried out in a laboratory with stable environmental conditions.
Temperature control is recommended as temperatures of liquid and the measure/instrument must
be stabilized, and monitored.
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The method requires liquid of high purity to be used as density has a large influence on the
calibration result. Thus high accuracy of temperature measurement is also important [2].
Volume measurements using gravimetric method require weighing the contents of a liquid of
known temperature and density. The liquid mass measurements need to be converted into
volume from the knowledge or determination of densities of the liquid. The conversion from
mass to volume must take into account the density of the liquid, air buoyancy as well as the
evaporation during the weighing cycle [4]. Volume at the measured temperature can be
calculated from
𝑉𝑖 = (𝑀𝑖 + 𝑒)𝑍
Where
𝑀𝑖
𝑒
𝑍
(1)
is the mass as read on the balance or differences in balance
indication
average evaporation loss during the weighing cycle
is a conversion factor incorporating density of water, air
density at test temperature and barometric pressure
Evaporation that occurs during the measurements depends on temperature, humidity and cycle
time. It has noticeable effect on small volume measurements less than 50 µL [5].
Volume measurements require that reference temperature (generally 20 oC) be reported as
volume depends on the temperature. Reference temperature is normally the temperature at which
volume measure is intended to contain or deliver its nominal volume. Thus volume at measured
temperature would generally require to be converted to reference temperature such as at 20 oC,
therefore equation 1 would become
𝑉20 = (𝑀𝑖 + 𝑒)𝑍[1 − 𝛾(𝑡 − 20)]
(2)
2.1 Estimation of the Z factor
The Z factor is not just equal to the density of liquid adjusted to the local temperature and
pressure. It also takes into account the air density and density of weights used to calibrate the
balance.
Z factor is calculated as
Z = [(𝜌
1
𝐿 −𝜌𝐴
𝜌
)(1 − 𝜌𝐴)]
𝐵
(3)
Where
𝜌𝐴 - density of air
𝜌𝐿 - density of the liquid
𝜌𝐵 - density of the weights used for balance calibration
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The air density (𝜌𝐴 ) to be considered is (in principle) the density of the air inside the volumetric
instrument and displaced when the instrument is filled with liquid. It is generally assumed that
the ambient air density does not change significantly between and during weighing. This ensures
that buoyancy effect exerted on the instrument is constant. If the ambient air changes, the true
mass of the instrument must be determined for each weighing [3]. The environmental or ambient
conditions influence the air density determination.
Density of the liquid can be determined or obtained from formulae given in literature. Batista and
Paton [1] detailed differences in volume as a result of using different density formulae which are
common in practice.
Density of the weights/mass pieces is presented in the calibration certificate of the weights or
alternatively according to OIML R111-1, it can be deduced from the weight class used for
balance calibration [7].
3. Typical Examples of Z factor Calculation
Values of the conversion factor Z (µL/mg) as a function of temperature and pressure as given in
literature is mainly for distilled water. When other liquids other than distilled water are used or
measured, correction factors (Z factors) for the specific liquid would have to be determined. The
conversion factors presented in tables 1 to 4 below present typical examples of corrections to be
applied for different liquids at specific temperatures and atmospheric pressure.
Table 1: Values for Z factor as a function of temperature and pressure for distilled water
Distilled water 𝜌𝑤 (999.10 to 997.04)kg/m3
Temperature ( C)
Atmospheric Pressure (hPa)
800
864
1012
15
1.0017
1.0018
1.0020
20
1.0026
1.0027
1.0029
25
1.0038
1.0039
1.0040
o
1067
1.0020
1.0029
1.0041
Table 2: Values for Z factor as a function of temperature and pressure for sea water
Sea water 𝜌𝑤 (1026.021 to 1023.3873)kg/m3
Temperature (oC)
Atmospheric Pressure (hPa)
800
864
1012
15
0.9754
0.9755
0.9757
20
0.9766
0.9766
0.9768
25
0.9779
0.9780
0.9781
𝜌𝑤 sourced from [6]
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1067
0.9757
0.9768
0.9782
Table 3: Values for Z factor as a function of temperature and pressure for Toluene
o
Temperature ( C)
15
20
25
Toluene 𝜌𝑤 (869 to 862.3)kg/m3
Atmospheric Pressure (hPa)
800
864
1012
1.1545
1.1546
1.1549
1.1547
1.1547
1.1550
1.1608
1.1609
1.1611
1067
1.1549
1.1550
1.1612
Table 4: Values for Z factor as a function of temperature and pressure for Benzene
o
Temperature ( C)
15
20
25
Benzene 𝜌𝑤 (884 to 873.8)kg/m3
Atmospheric Pressure (hPa)
800
864
1012
1.1323
1.1324
1.1326
1.1387
1.1388
1.1390
1.1455
1.1456
1.1458
1067
1.1327
1.1391
1.1459
Effect of Z factor on the volume measurement indicated as the difference from nominal; where
positive indicate an increase in volume while negative sign indicate a decrease in volume.
Table 5 below provides various volume measures at a temperature of 20oC and atmospheric
pressure of 864 hPa corrected with respective Z factors.
Table 5: Effect of Z factors on volume
Measurement
(𝑀𝑖 + 𝑒)
0.01
0.05
0.1
0.25
0.5
50
100
250
500
4.
Z (Distilled
H20)
V (ml)
0.000027
0.000135
0.00027
0.000675
0.00135
0.135
0.27
0.675
1.350
Z (Sea H20)
Z (Toluene)
Z (Benzene)
V (ml)
-0.000234
-0.00117
-0.00234
-0.00585
-0.0117
-1.17
-2.34
-5.85
-11.7
V (ml)
0.001547
0.007735
0.01547
0.038675
0.07735
7.735
15.470
38.675
77.350
V (ml)
0.001388
0.00694
0.01388
0.0347
0.0694
6.940
13.88
34.7
69.4
Conclusions
It could be seen in table 5 above that incorrect use of correction factor can have a significant
influence in volume determination. Table 5 also showed that the effect increases with an increase
in nominal volume. It is therefore critical for one to ensure that the volume correction factors
applied are correct for the intended liquid.
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5.
References:
1. Batista E and Paton R (2007). The selection of water property formulae for volume and
flow calibration. BIPM and IOP Publishing Ltd, Metrologia 44 (2007) 453 – 463.
2. B F van der Merwe, Calibration of volume standards, November 2008, NML06-0122
3. EURAMET cg-19. Calibration guide: Guidelines on the determination of uncertainty in
gravimetric volume calibration, version 2.1 (03/2012), ISBN 978-3-942992-24-4
4. Gilson guide to pipetting, Second Edition.
5. ISO 8655-6:2002(E), International Standard, Piston-operated volumetric apparatus, Part
6: Gravimetric methods for the determination of measurement error
6. ITTC- Recommended Procedures: Fresh water and Seawater properties.(2011) Revision
02
7. OIML R111-1. Edition 2004 (E). International Recommendation. Weights of class E1, E2,
F1, F2, M1, M1-2, M2-3, and M3. Part 1: Metrological and technical requirements
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