When relatively low orders of accuracy are required, reading mass or weight values directly from the weighing instrument are adequate. Except for the equal arm balance and some torsion balances, most modern weighing instruments have direct readout capability. For most commercial transactions and for simple scientific experiments, this direct readout will provide acceptable results.
In the case of equal arm balances, the balance will have a pointer and a scale. When relatively low accuracy is needed, the pointer and scale are used to indicate when the balance is close to equilibrium.The same is true when using a torsion balance. However, the equal arm balances of smaller (e.g., 30 g)
or larger (e.g., 900 kg) capacity are also used for high-accuracy applications. Only the new generation of electronic balances are equal or better in terms of accuracy and benefit from other features. Weighing is a deceptively simple process. Most people have been making and using weighing measurements
for most of their lives. We have all gone to the market and purchased food that is priced by weight. We have weighed ourselves many times, and most of us have made weight or mass measurements in school. What could be simpler? One places an object on the weighing pan or platform and reads the
result. Unfortunately, the weighing process is very susceptible to error. There are errors caused by imperfections
in the weighing instrument; errors caused by biases in the standards used; errors caused by the
weighing method; errors caused by the operator; and errors caused by environmental factors. In the case
of the equal arm balance, any difference between the lengths of the arms will result in a bias in the
measurement result. Nearly all weighing devices will have some degree of error caused by small amounts
of nonlinearity in the device. All standards have some amount of bias and uncertainty. Mass is the only
base quantity in the International System of Units (SI) defined in relation with a physical artifact. The
international prototype of the kilogram is kept at Sevres in France, under the custody of the International
Bureau of Weights and Measures. All weighing measurements originate from this international standard.
The international prototype of the kilogram is, by international agreement, exact; however, over the last
century, it has changed in value. What one does not know is the exact magnitude or direction of the
change. Finally, environmental factors such as temperature, barometric pressure, and humidity can affect
the weighing process.
There are many weighing techniques used to reduce the errors in the weighing process. The simplest
technique is
substitution weighing
. The substitution technique is used to eliminate some of the errors
introduced by the weighing device. The single-substitution technique is one where a known standard
and an unknown object are both weighed on the same device. The weighing device is only used to
determine the difference between the standard and the unknown. First, the standard is weighed and the
weighing device’s indication is noted. (In the case of an equal arm balance, tare weights are added to the
second pan to bring the balance to equilibrium.) The standard is then removed from the weighing device
and the unknown is placed in the same position. Again, the weighing device’s indication is noted. The
first noted indication is subtracted from the second indication. This gives the difference between the
standard and the unknown. The difference is then added to the known value of the standard to calculate
the value of the unknown object. A variation of this technique is to use a small weight of known value
© 1999 by CRC Press LLC
to offset the weighing device by a small amount. The amount of offset is then divided by the known
value of the small weight to calibrate the readout of the weighing device. The weighing results of this
measurement is calculated as follows:
where
U= value of the unknown
S= known value of the standard
SW = small sensitivity weight used to calibrate the scale divisions
1= first observation (standard)
2= second observation (unknown)
3= third observation (unknown + SW)
These techniques remove most of the errors introduced by the weighing device, and are adequate whenresults to a few tenths of a gram are considered acceptable.If results better than a few tenths of a gram are required, environmental factors begin to causesignificant errors in the weighing process. Differences in density between the standard and the unknown
object and air density combine together to cause significant errors in the weighing process.
It is the buoyant force that generates the confusion in weighing. What is called the “true mass” of an
object is the mass determined in vacuum. The terms “true mass” and “mass in vacuum” are referring to
the same notion of inertial mass or mass in the Newtonian sense. In practical life, the measurements are
performed in the surrounding air environment. Therefore, the objects participating in the measurement
process adhere to the Archimedean principle being lifted with a force equal to the weight of the displaced
volume of air. Applying the buoyancy correction to the measurement requires the introduction of the
term “apparent mass.” The “apparent mass” of an object is defined in terms of “normal temperature”
and “normal air density,” conventionally chosen as 20°C and 1.2 mg cm–3
, respectively. Because of theseconventional values, the “apparent mass” is also called the “conventional mass.” The reference materialis either brass (8.4 g cm–3) or stainless steel (8.0 g cm–3), for which one obtains an “apparent mass versus
brass” and an “apparent mass versus stainless steel,” respectively. The latter is preferred for reporting the“apparent mass” of an object.
Calibration reports from the National Institute of Standards and Technology will report mass in threeways: True Mass, Apparent Mass versus Brass, and Apparent Mass versus Stainless Steel. Conventional
mass is defined as the mass of an object with a density of 8.0 g cm–3, at 20°C, in air with a density of1.2 mg cm–3. However, most scientific weighings are of materials with densities that are different from8.0 g cm–3. This results in significant measurement errors.As an example, use the case of a chemist weighing 1 liter of water. The chemist will first weigh a mass
standard, a 1 kg weight made of stainless steel; then the chemist will weigh the water. The 1 kg mass
standard made of 8.0 g cm–3stainless steel will have a volume of 125 cm3. The same mass of water willhave a volume approximately equal to 1000 cm3(Volume = Mass/Density). The mass standard will displace 125 cm3
of air, which will exert a buoyant force of 150 mg (125 cm3´1.2 mg cm –3). However,the water will displace 1000 cm3air, which will exert a buoyant force of 1200 mg (1000 cm3 1.2 mg cm –3 ).
Thus, the chemist has introduced a significant error into the measurement by not taking the differing
densities and air buoyancy into consideration.
Using 1.2 mg cm
–3
for the density of air is adequate for measurements made close to sea level; it must
be noted that air density decreases with altitude. For example, the air density in Denver, CO, is approximately
0.98 mg cm
–3
. Therefore, to make accurate mass measurements, one must measure the air density
at the time of the measurement if environmental errors in the measurement are to be reduced.
Air density can be calculated to an acceptable value using the following equations:
(20.5)
U = S + (O -O )( ) (O -O ) 2 1 3 2 SW
rA s @ 0.0034848 (t + 273.15)(P -0.0037960 ´U ´ e )
© 1999 by CRC Press LLC
where
r
A
= air density in mg cm
–3
t
= temperature in
°
C
P
= barometric pressure in pascals
U
= relative humidity in percent
e
s
= saturation vapor pressure
(20.6)
where
e
@
2.7182818
t
= temperature in
°
C
To apply an air buoyancy correction to the single substitution technique, use the following formulae:
(20.7)
where
M
u
= mass of the unknown (in a vacuum)
M
s
= mass of the standard (in a vacuum)
M
sw
= mass of the sensitivity weight
r
A
= air density
r
s
= density of the standard
r
u
= density of the unknown
r
sw
= density of the sensitivity weight
O
1
= first observation (standard)
O
2
= second observation (unknown)
O
3
= third observation (unknown + SW)
(20.8)
where CM = conventional mass
M
u
= mass of the unknown in a vacuum
r
u
= density of the unknown
When very precise measurements are needed, the double-substitution technique coupled with an air
buoyancy correction will provide acceptable results for nearly all scientific applications. The doublesubstitution
technique is similar to the single-substitution technique using the sensitivity weight. In the
double-substitution technique, the sensitivity weight is weighed with both the mass standard and the
unknown. The main advantage of this technique over single substitution is that any drift in the weighing
device is accounted for in the technique. Because of the precision of this weighing technique, it is only
appropriate to use it on precision balances or mass comparators. As in the case of single substitution,
one places the standard on the balance pan and takes a reading. The standard is then removed and the
unknown object is placed on the balance pan and a second reading is taken. The third step is to add the
small sensitivity weight to the pan with the unknown object and take a third reading. Then remove the
unknown object and return the standard to the pan with the sensitivity weight and take a fourth reading.
The mass is calculated using the following formulae:
(20.9)
e e
t
s @ ( ´ ) ´ (- ( + )) 1 7526 1011 5315 56 273 15
.
. .
M M A O O M O O u s s SW A SW A u = - ( )+ - ( ) - ( ) - ( ( ) æ
è
öø
1 1 (1- ) 2 1 3 2 r r r r r r
CM u u = M (1-0.0012 r ) 0.99985
M
M O O O O M O O
u
S A S SW A SW
A u
=
( ( - )+ ( - + - ) ( ( - ) ( - )
( - )
1 2 1
1
2 1 3 4 3 2 r r r r
r r
© 1999 by CRC Press LLC
where
M
u
= mass of the unknown (in a vacuum)
M
s
= mass of the standard (in a vacuum)
M
sw
= mass of the sensitivity weight
r
A
= air density
r
s
= density of the standard
r
u
= density of the unknown
r
sw
= density of the sensitivity weight
O
1
= first observation (standard)
O
2
= second observation (unknown)
O
3
= third observation (unknown + sensitivity weight)
O
4
= fourth observation (standard + sensitivity weight)
(20.10)
where CM = conventional mass
M
u
= mass of the unknown in a vacuum
r
u
= density of the unknown
To achieve the highest levels of accuracy, advanced weighing designs have been developed. These
advanced weighing designs incorporate redundant weighing, drift compensation, statistical checks, and
multiple standards. The simplest of these designs is the three-in-one design. It uses two standards to
calibrate one unknown weight. In its simplest form, one would perform three double substitutions. The
first compares the first standard and the unknown weight; the second double substitution compares the
first standard against the second standard, which is called the check standard; and the third and final
comparison compares the second (or check standard) against the unknown weight. These comparisons
would then result in the following:
O
1
= reading with standard on the balance
O
2
= reading with unknown on the balance
O
3
= reading with unknown and sensitivity weight on the balance
O
4
= reading with standard and sensitivity weight on the balance
O
5
= reading with standard on the balance
O
6 = reading with check standard on the balance
O7 = reading with check standard and sensitivity weight on the balance
O8 = reading with standard and sensitivity weight on the balance
O9 = reading with check standard on the balance
O10 = reading with unknown on the balance
O11 = reading with unknown and sensitivity weight on the balance
O12 = reading with check standard and sensitivity weight on the balance
The measured differences are calculated using the following formulae:
(20.11)
(20.12)
(20.13)
CM= Mu (1-0.0012 ru ) 0.99985
a = [(O -O +O -O ) ]´ [M ( - ) O -O ] 1 2 4 3 3 2 2 1 SW A SW r r
b = [(O -O +O -O ) ]´ [M ( - ) O -O ] 5 6 8 7 7 6 2 1 SW A SW r r
c = [(O -O +O -O ) ]´ [M ( - ) O -O ] 9 10 12 11 11 10 2 1 SW A SW r r
© 1999 by CRC Press LLC
where a = difference between standard and unknown
b = difference between standard and check standard
c = difference between check standard and unknown
Msw = mass of sensitivity weight
rA = air density calculated using Equations 20.5 and 20.6
rsw = density of sensitivity weight
The least-squares measured difference is computed for the unknown from:
(20.14)
Using the least-squares measured difference, the mass of the unknown is computed as:
(20.15)
where U = mass of unknown
S = mass of the standard
du = least-squares measured difference of the unknown
rA = air density calculated using Equations 20.5 and 20.6
rS = density of the standard
rU = density of the unknown
The conventional mass of the unknown is now calculated as:
(20.16)
where CU = conventional mass
U = mass of unknown
rU = density of unknown
The least-squares measured difference is now computed for the check standard as:
(20.17)
Using the least-squares measured difference, the mass of the check standard is computed from:
(20.18)
where CS = mass of check standard
s = mass of the standard
dCS = least-squares measured difference of the check standard
rA = air density calculated using Equations 20.5 and 20.6
rS = density of the standard
rCS = density of unknown
The mass of the check standard must lie within the control limits for the check standard. If it is out of
the control limits, the measurement must be repeated.
The short-term standard deviation of the process is now computed:
(20.19)
du = (-2a -b -c) 3
U = (S(1-r r )+ d ) (1-r r ) A S u A U
CU U =U(1-0.0012 r ) 0.99985
d a b c CS = (- -2 - ) 3
CS CS A S CS A = (S(1-r r )+ d ) (1-r r )
Short-termstandard deviation = 0.577(a -b + c)
