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SOP 2 

Recommended Standard Operating Procedure

for

Applying Air Buoyancy Corrections

1 Introduction

Purpose

If uncorrected, the effect of air buoyancy on the objects being compared is 

frequently the largest source of error in mass measurement. This SOP provides the 

equations to be used to correct for the buoyant effect of air. The significance of the 

air buoyancy correction depends upon the accuracy required for the measurement, 

the magnitude of the air buoyancy correction relative to other sources of error in 

the overall measurement process, and the precision of the mass measurement. An 

air buoyancy correction should be made in all high accuracy mass determinations. 

The gravimetric volume procedure uses a high accuracy mass determination with 

the corresponding buoyancy corrections. The Appendix to this SOP provides a brief 

theoretical discussion of this subject.

Prerequisites

Verify that (true) mass values or corrections are available for the standards used in 

the measurement process and that they have demonstrated metrological traceability 

to the international system of units (SI), which may be to the SI through a National 

Metrology Institute such as NIST.

Verify that the thermometer, barometer, and hygrometer used have been calibrated, 

and that they have demonstrated metrological traceability to the international 

system of units (SI), which may be to the SI through a National Metrology Institute 

such as NIST, and are in good operating condition as verified by periodic tests or 

cross-checks with other standards.

2 Methodology

Scope, Precision, Accuracy

This procedure is applicable to all weighings using a comparison of mass standards. 

The precision will depend upon the accuracy of the thermometer, barometer, and 

hygrometer used to determine the air density. When the calculations for the air 

density and air buoyancy corrections are made, a sufficient number of decimal 

places must be carried so the error due to the rounding of numbers is negligible 

relative to the error in the measurement process. Typically, carrying six to eight 

decimal places is sufficient, but depends upon the precision of the measurement. 
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Apparatus/Equipment Required1

Calibrated barometer with sufficiently small resolution, stability, and 

uncertainty (e.g., accurate to ± 66.5 Pa (0.5 mmHg)) to determine 

barometric pressure.

Calibrated thermometer with sufficiently small resolution, stability, and 

uncertainty (e.g., accurate to ± 0.10 °C) to determine air temperature.

Calibrated hygrometer with sufficiently small resolution, stability, and 

uncertainty (e.g., accurate to ± 10 percent) to determine relative humidity.

Estimating the Magnitude of the Air Buoyancy Correction

Estimate the magnitude of the air buoyancy correct, MABC, using the 

following formula:

 = ( )( ) MABC ρ ρ a nx s − − V V Eqn. (1)

The equation may also be represented as follows:

1 1 ( ) oa n

x s

MABC m ρ ρ

ρ ρ

  =− −    

Eqn. (2)

Table 1. Variables for MABC equation.

Variable Description of Variable

ρa air density at the time of the measurement in g/cm3

ρn density of "normal" air; i.e., 0.0012 g/cm3

mo nominal mass (in grams)

Vx volume of the unknown weight, X in cm3

Vs volume of the reference standard, S in cm3

ρs density of reference standard, S in g/cm3

;

effective density is used for summations (see 4.6, Eqn. 8)

ρx density of unknown weight, X in g/cm3

;

effective density is used for summations (see 4.6, Eqn. 8)

The relative magnitude of the correction can be compared to the expanded 

measurement uncertainty to determine the importance of the air buoyancy 

correction for a measurement. In some mass calibration procedures, when 

the calculated value is sufficiently small compared to applicable tolerances, 

the value may be treated as an uncorrected systematic error and incorporated 

into the uncertainty calculations. 

 1

The barometer, thermometer, and hygrometer are used to determine the air density at the time of the measurement. 

The air density is used to make an air buoyancy correction. The accuracies specified are recommended for high 

precision calibration. Less accurate equipment can be used with only a small degradation in the overall accuracy of 

the measurement.
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3 Procedure

Record the temperature, pressure, and relative humidity at the start and at the end 

of the measurement process as near the location of the measurement as necessary 

and practical. If these parameters change significantly during the measurement 

process, it may be necessary to wait for more stable operating conditions or to use 

average values to compute the air density. Use of the average environmental values 

may influence the uncertainty of the measurement result and must be evaluated for 

significance. 

Determine the air density using the equation given in Section 8 of the Appendix to 

this SOP. 

4 Calculations

Calculate the mass, Mx, of the unknown weight, X, using the following equation, 

where d represents the “difference” obtained with buoyancy corrections applied to 

the sensitivity weight.

s 1

1

a

s

x

a

x

M d

M

ρ

ρ

ρ

ρ

    − +

  =     −

  Eqn. (3)

If tare weights were carried with X and/or S, use the following equation: 

s 1 1 1

1

s x

s x

a aa

t t

s tt

x

a

x

M dM M

M

ρ ρρ

ρ ρρ

ρ

ρ

    

  − ++ − − −         =     −

 

Eqn. (4)
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Table 2. Variables not previously defined.

Variable Description of Variable

d measured difference between X and the reference standard, S, using one of the 

weighing designs given in other SOPs

Ms [true] mass of the reference standard or summations

s Mt [true] mass of the tare weight(s) carried with S

x Mt [true] mass of the tare weight(s) carried with X

ρs density or effective density of the reference standard(s), S

ρx density or effective density of the unknown standard(s), X

st ρ density or effective density of the tare weight(s) carried with S

x ρt density or effective density of the tare weight(s) carried with X

If reporting the conventional mass2, CMx, compute it using the following. 

0.0012 1

0.0012 1

8.0

x

x

x

M

CM ρ

    −

  =     −   Eqn. (5)

If reporting the apparent mass, AMx, versus brass, which is not common, compute 

it using the following.

 

0.0012 1

0.0012 1

8.3909

x

x

x vs brass

M

AM ρ

    −

  =     −  

Eqn. (6)

The conventional and apparent mass values are related by the following:

 

0.0012 1

8.3909

0.0012 1

8.0

x vs brass

x

AM

CM

    −   =     −  

Eqn. (7)

 2 Conventional Mass: “The conventional value of the result of weighing a body in air is equal to the mass of a 

standard, of conventionally chosen density, at a conventionally chosen temperature, which balances this body at this 

reference temperature in air of conventionally chosen density.” The conventions are: reference density 8.0 g/cm3

; 

reference temperature 20 °C; normal air density 0.0012 g/cm3

. Conventional mass was formerly called “Apparent 

Mass versus 8.0 g/cm3

” in the United States and is not recommended. See OIML D28 (2004).
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Effective density values must be calculated and used for summations of mass 

standards. The subscripts 5, 3, and 2 refer to the individual nominal masses that 

comprise the summation. This approach may also be needed with a 5, 2, 2, 1 

combination. E.g., a summation of 1 kg, might be 500 g, 300 g, and 200 g (or 500 

g, 200 g, 200g, and 100 g). A summation for 100 g would be 50 g, 30 g, and 20 g 

(or the equivalent of a 5221 series). A metric calibration of a 4 oz weight would 

need to add up to 113.389 g (probably 100 g, 10 g, 3 g, 300 mg, 100 mg) and then 

must also address unequal nominal values as shown in the equations in SOP 4, 5, 

and 7 as applicable.:

5 32

5 3 2

532

Effective Density

M MM + 

M M M

ρρρ

+ =         + +       

Eqn. (8)

5 Assignment of Uncertainty 

The uncertainty in determining the air buoyancy correction is usually negligible relative to 

the precision of the measurement process itself. Consequently, the uncertainty for the 

measurement is based upon the uncertainty for the measurement process used and the 

uncertainty associated with the environmental measurement standards. The uncertainty in 

the CIPM3 2007 air density equation is 0.000 026 4 mg/cm3 (or 0.002 2 % of normal air 

density.) See each SOP for uncertainties associated with buoyancy corrections. 

Table 3. Tolerances for measurements related to air density estimation.

Uncertainty of air density values in % of air density

Variable ± 0.1 % of air 

density

± 1.0 % of air density Recommended 

(Section 2.2)

Air pressure (Pa) ± 101 ± 1010 ± 66.5

Air pressure (mmHg) ± 0.76 ± 7.6 ± 0.5

Air temperature ( °C) ± 0.29 ± 2.9 ± 0.1

Relative Humidity (%) ± 11.3 ............ ± 10

 3 CIPM References: A Picard, R.S. Davis, M. Gläser, K. Fujii, Metrologia 45:149-155 (2008).
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Appendix A

Based on “The Basic Theory of Air Buoyancy Corrections”

by Richard S. Davis4

1 Introduction 

In performing measurements of mass, the balance or scale used acts as a force transducer. 

The force produced by an object to be weighed in air has two components: one proportional 

to the mass of the object, and the other proportional to its volume. The latter component, 

or buoyant force, may under some circumstances be large enough to require correction. 

The following shows under what circumstances buoyancy corrections are required as well 

as how they are made. 

2 Scope

The method for applying buoyancy corrections presented below applies to mass 

measurements made in air. The density of air is computed from auxiliary measurements of 

temperature, pressure and relative humidity after which the buoyancy corrections are 

calculated directly from the Principle of Archimedes. The following weighing situations 

are considered. 

Two-Pan Balance

Single-Pan Balance

With Built-In Weights

With Electronic Control 

3 Summary of Method

In general, buoyancy corrections are applied to mass measurements by calculating the 

difference in volume between the unknown weight and the standard, multiplying this 

volume difference by the density of air at the balance or scale, and adding the product to 

the mass of the standard. The density of air is computed from an equation of state using 

measured values for the temperature, pressure and relative humidity of the air.

4 Significance and Use 

Buoyancy corrections generally must be applied when determining the mass of an 

unknown object to high accuracy. The corrections may become important even at modest 

accuracies if the unknown object whose mass is to be determined has a density that differs 

widely from that of the standards (weighing of water, for example). Many mass standards 

are calibrated in terms of a so-called "apparent mass" [conventional mass] scale (See 

 4

Richard S. Davis, formerly of the National Institute of Standards and Technology, Mass Group, International 

Bureau of Weights and Measures (BIPM), retired.
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Chapter 7.3, Handbook 1451). Use of this scale does not indiscriminately eliminate the 

need for buoyancy corrections as is sometimes assumed. 

5 Terminology 

Weighing by Substitution 

Substitution weighing is the procedure by which one determines the assembly of 

standard weights that will produce nearly the same reading on a one-pan balance as 

does the unknown object. The balance thus serves as a comparator. A two-pan 

balance may be used in this mode if one of the pans contains a counterpoise and the 

standards and unknown are substituted on the second pan. (See SOP No. 3.)

Mass

The term "mass" is always used in the strict Newtonian sense as a property intrinsic 

to matter. Mass is the proportionality constant between a force on a material object 

and its resulting acceleration. This property is sometimes referred to as "true mass", 

"vacuum mass", or "mass in vacuum" to distinguish it from conventional [apparent] 

mass.

Conventional [Apparent] Mass

The mass of material of a specified density that would exactly balance the unknown 

object if the weighing were carried out at a temperature of 20 °C in air of density 

0.0012 g/cm3. The mass, MN, of an object, N, is related to its apparent mass MN,A

by the equation:

,

0.0012 1

0.0012 1

N

N

N A

B

M

M ρ

ρ

    −

  =     −

 

Eqn. (9)

Table 4. Variables for conventional (apparent) mass equation.

Variable Description

ρN density of the object N at 20 °C in g/cm3

ρB

density of the conventional (apparent) mass scale at 20 °C 

in g/cm3

In the past, there were two apparent mass scales in wide use. The older is based on 

ρB = 8.4000 g/cm3 at 0 °C with a coefficient of volumetric expansion of 

0.000054 /°C and the more recent (Conventional Mass) specifies ρB = 8.0000 g/cm3

at 20 °C. The quantity MN,A is a function of the particular conventional or apparent 

mass scale, which has been used in its calculation. OIML D28 only recognizes 

Conventional Mass.
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Sensitivity

The response of a balance under load to an additional small weight:

1

sensitivity 

a

sw

sw a sw sw

M

M V

R R

ρ

ρ ρ

    − −   = = ∆ ∆ Eqn. (10)

Table 5. Variables for sensitivity equation.

Variable Description

sensitivity balance sensitivity (mass per division)

Msw mass of the small, additional weight

ρa density of the air

ρsw density of the small, additional weight

Vsw volume of the small, additional weight

∆R change in balance reading due to the addition of the small weight, balance 

deflection

6 Apparatus

In order to ascertain the density of air at the balance, the following measuring instruments 

are necessary: thermometer, barometer, and hygrometer. Ideally, these instruments should 

be placed in or next to the balance case (as near the measurement location as is practical). 

It may only be practical for the thermometer or temperature sensor to actually be placed 

inside the balance chamber. 

7 Procedure

Weigh the unknown object as directed by the balance manufacturer or in accordance with 

accepted procedure. Record the temperature, pressure and relative humidity of the air in 

the balance at the time of weighing (generally immediately before and immediately after 

weighings). Do not correct the barometric pressure to sea level. 

8 Calculations 

Air density, Option A (Option B is preferred)

The density of air, in g/cm3, can be approximated for lesser accuracy from the 

following formula:5

[ ( ) ]

( )

3 x 10

273.15 

0.464 60 0.003 796 0 −













+

− = tP U es ρa

Eqn. (11)

 5

Jones, F.E., "The Air Density Equation and the Transfer of the Mass Unit," Journal of Research, National Institute 

of Standards and Technology, Vol. 83, 1978, p. 419. 
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Table 6. Variables for air density equation.

Variable Description

ρa density of air, g/cm3

P barometric pressure, mmHg

U % relative humidity, entered as a whole number

t temperature, °C

es 1.314 6 x 109 x e[-5 315.56/(t + 273.15)]

Note: es can be written as follows in a spreadsheet and in some calculators:

1.3146E9*@EXP(-5315.56/(t+273.15))

Small errors (of order 0.01 %) in this equation occur for locations well above sea 

level or under conditions in which the concentration of carbon dioxide differs 

greatly from the global average. See the references for a more general formulation 

of the equation. 

Air density, Option B – Preferred

The density of air should be calculated with the following formulae.

6

1 1 a v

a v

a

pM M

x

ZRT M ρ

    = −−        

Eqn. (12)

 = /100 ( ) sv

v

p xh f

p

Eqn. (13)

( ) ( ) ( )

2

2 2 2

0 1 2 01 01 2 1 vv v

p p Z a a t a t b b t x c c t x d ex

T T

=− + + + + + + + +     Eqn. (14)

 6 CIPM References: Giacomo, P. Metrologia 18: 33-40 (1982), Davis, R.S., Metrologia 29: 67-70 (1992), A Picard, 

R.S. Davis, M. Gläser, K. Fujii, Metrologia 45:149-155 (2008).
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Table 7. Variables for CIPM air density equation.

Variable Description

Ma molar mass of the air within laboratory 28.965 46 x 10-3 kg/mol

Mv 18.01528(17) x10-3 kg/mol

p ambient barometric pressure in Pascal

T ambient temperature in Kelvin

R universal gas constant: 8.314 472(15) J mol-1 K-1

h relative humidity in %

f 1.000 62 + (3.14 x 10-8) p + (5.6 x 10-7) t

2

t ambient temperature in degrees Celsius

psv 1 Pascal x exp (AT2 + BT + C + D/T)

A 1.237 884 7 x 10-5 K-2

B -1.912 131 6 x 10-2 K-1

C 33.937 110 47

D -6.343 164 5 x 103 K

a0 1.581 23 x 10-6 K Pa-1

a1 -2.933 1 x 10-8 Pa-1

a2 1.104 3 x 10-10 K-1 Pa-1

b0 5.707 x 10-6 K Pa-1

b1 -2.051 x 10-8 Pa-1

c0 1.989 8 x 10-4 K Pa-1

c1 -2.376 x 10-6 Pa-1

d 1.83 x 10-11 K2 Pa-2

e -0.765 x 10-8 K2 Pa-2

Calculate the average density of air at the balance during the weighing. Then 

determine the mass of the unknown, Mx, as follows: 

If a two-pan balance is used, use one of the following equations:

 ( ) sensitivity M M VV x s a x s = + −+ ∆ ρ Mopt Eqn. (15)

1 sensitivity 

1

a

s opt

s

x

a

x

M M

M

ρ

ρ

ρ

ρ

    −+ ∆

  =     −

 

Eqn. (16)

 

1

s a s opt

x

a

x

M V sensitivity M M ρ

ρ

ρ

−+ ∆ =     −

 

Eqn. (17)
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Table 8. Variables not previously defined.

Variable Description

Mx mass of the unknown object

Ms mass of the standard weights

ρs density of the standard weights, 

S

S

V

M

∆Mopt off-balance indication read on the optical scale

Vx volume of the unknown object

Vs volume of the standard weights

ρx density of the unknown object, 

X

X

V

M

Volumes and densities are, in general, a function of temperature. The thermal 

coefficients of volumetric expansion of the unknown object and the standard may 

be significant in very accurate work.7 The coefficient of volumetric expansion is 

usually estimated as three times the linear coefficient of expansion of the weight 

material. 

The error in Mx incurred by ignoring the buoyancy correction is ρa (Vx - Vs). To 

estimate quickly whether such an error is of consequence in a particular 

measurement, (assume ρa = 1.2 x 10-3 g/cm3

). 

If the mass and volumes of the standards have been adjusted to a conventional mass 

scale, then:

1 sensitivity 

 

 1

a

s opt

B

x

a

x

CM M

M

ρ

ρ

ρ

ρ

    −+ ∆

  ≈     −

 

Eqn. (18)

Table A-6. Variables not previously defined.

Variable Description

CMs conventional mass of the standard

The symbol ≈ signifies an approximation

 7

In general, Vt = V20 [1 + ß(t - 20 °C)] where t is the temperature of the weight, V20 is the volume at 20 °C, and ß is 

the cubical coefficient of expansion. 
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If a single-pan balance with built-in weights is used, it is probable that the built-in 

weights have been adjusted on an apparent mass or conventional mass scale8. 

Determine which apparent mass scale has been used and calculate the mass of the 

unknown from the equation

1 sensitivity 

 

1 

a

D opt

B

x

a

x

M M

M

ρ

ρ

ρ

ρ

    − +

  ≈     −

 

Eqn. (19)

Table 9. Variables not previously defined.

Variable Description

MD mass indicated by dial or digital readings

Mopt mass indicated on the optical scale when present

If the balance has been used only as a comparator, that is, to compare the mass of 

the unknown object with that of some external standard, then: 

 = ( ) sensitivity ' M M VV x s a x s + −+ ∆ ρ M opt Eqn. (20)

Table 10. Variable not previously defined.

Variable Description

∆M’opt

difference in optical scale reading between observations of the standard 

and the unknown

For some balances, operation requires that the user restore the balance to null by 

means of a manually controlled dial. The portion of the mass reading controlled by 

this dial should be treated, for purposes of buoyancy corrections, as an optical scale.

If a single-pan balance with full-range electronic control is used, the following 

should be noted. As part of its calibration, the electronic gain has been adjusted by 

means of a weight of known mass. For example, if the range of electronic control 

is 100 g, the electronics have been adjusted so that a 100 g standard weight produces 

an indication of precisely 100 g. This procedure effectively builds an apparent mass 

calibration into the balance. The reference density of the apparent mass scale is the 

density of the standard mass used for the calibration and the reference air density 

is the air density at the time of calibration. 

 8

Schoonover, R. M. and Jones, F. E., "Air Buoyancy in High-Accuracy Weighing on Analytical Balances," Anal. 

Chem., 53, 1981, p. 900. 
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The mass of an unknown object weighed on the balance is then 

1

 

1

a

R

c

x

a

x

M

M

ρ

ρ

ρ

ρ

    −

  =     −

 

Eqn. (21)

Table 11. Variables not previously defined.

Variable Description

MR readout displayed on the balance

ρa density of air at the time of balance calibration

ρc

density of the standard used to calibrate the balance (or ρB if the apparent 

[conventional] mass of the standard was used instead of the true mass

If the balance includes both an electronic control system and built-in 

weights, the buoyancy considerations for the built-in weights are as 

described in section 8.2 and the considerations for the electronically 

determined mass are those given directly above. 

Top-loading balances may be considered a form of single-pan balance and the 

appropriate procedure for buoyancy correction followed. 

9 Precision

The contribution of the random error of the evaluation of air density to the precision of 

mass measurement may be estimated as follows: 

For mechanical balances, or electronic balances used in weighing by substitution, the 

contribution is:

( ) δρax s V V− Eqn. (22)

Table 12. Variables for above equation.

Variable Description

δρa random error of evaluation of ρa

Vs

volume of standards, if weighing by substitution

Vs = MD / DB, if using the built-in weights on a single pan balance.

Vx volume of object weighed

The quantity, δρa will have contributions from the measurements of temperature, pressure 

and relative humidity which are required for the calculation of ρa. Eqn. (11) may be used 

to estimate the effects of imprecision in measurements of P, t, and U. It is unrealistic to 

expect δρa /ρa ever to be less than 0.05 % even using the best techniques available. 
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10 Accuracy 

Inattention to problems of buoyancy most often results in systematic errors. For a 

substitution weighing, for example, the buoyancy correction is of the order of ρa(Vx - Vs). 

While this quantity may be significant to a measurement, day-to-day variation in ρa

(usually no more than 3 %) may not be noticeable and hence need not be measured. For 

the most accurate work, not only must ρa be accurately determined, but the volumes of the 

unknown and standard may have to be measured to better than 0.05 % which is the 

minimum systematic uncertainty attainable in the calculation of ρa.

If the standards have been calibrated in terms of conventional mass, complete neglect of 

buoyancy corrections will produce an error in the measured result Mx of order:

( ) 



 



































 − + 













 − − − CM x 

x

a

B

B x

s B

s a ρ

ρ

ρ

ρ ρ

ρ ρ

ρ

1 1 1.2 10 3 Eqn. (23)

This error is often unacceptably large. 

Use of Eqn. (18), on the other hand, introduces only an error of approximately 

( ) 













 − − − CM x 

s B

s a ρ ρ

ρ

1 1 1.2 10 3 Eqn. (24)

It is a requirement for manufacture that the actual density of standard weights be near 

enough to the assumed density of the apparent mass scale to which they are adjusted that 

the magnitude of (8) will always be small under normal conditions in laboratories near sea 

level. 

The fact that there are two apparent mass scales widely used - one based on density 

8.0 g/cm3 and an older one based on 8.4 g/cm3 - means that some caution is required on the 

part of the user. Conventional mass is generally preferred and reported for all calibrations 

where mass standards will be used to calibrate weighing instruments. For the most accurate 

work, the apparent mass scale should be abandoned in favor of substitution weighing with 

standards of known mass, density, and volume. 

The user must decide the accuracy required of the particular mass measurement and choose 

a buoyancy correction technique commensurate with that accuracy.

The same considerations, which apply to the accuracy of buoyancy corrections in weighing 

by substitution, are easily extended to the other types of weighing indicated above. 

There are many factors, which affect the accuracy of a mass measurement. The above has 

dealt only with those arising from problems of buoyancy.
SOP 2 – 2019 Page 15 of 19

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Appendix B

Examples

Example 1: 

The weight set of Table B-1 is used with an equal-arm balance to find the mass of a piece of 

single-crystal silicon. The following weights were used to balance the silicon: 10 g, 3 g. The 

balance pointer showed the silicon side to be light by 3.5 divisions. The 10 mg weight of Table 1 

was used to find the sensitivity of the balance. When the weight was added, the pointer moved 

10.3 divisions. At the time of the weighing, the following measurements were taken: 

P = 748.1 mmHg

t = 22.3 °C

U = 37 % relative humidity

What is the mass of the silicon? 

Answer:

Using the equation for es found in Table 6, or Table B-2, calculate

( )

5 315.56 

9 22.3 273.15 es 1.314 6 10 x xe 20.194 mmHg

  −   +   = =

Then, using Eqn. (11), calculate the air density, (ρa)

[ ( )( )( ) ]

( )

3 10

273.15 22.3

0.464 60 748.1 0.003 796 0 37 20.2 −













+

− ρa =

ρa = 1.171 94 x 10-3 g/cm3

The density of silicon at 20 °C is 2.329 1 g/cm3 and its coefficient of linear expansion is 

0.000 002 6 /°C. 

Making use of Eqns. (10) and (17) and Table B-1:

10.000 130 3.000 046 13.000 176 g Ms = +=

( ) 3 Vs = 1.267 44+0.380 23 1.647 67 cm =

Calculate the balance sensitivity in g/division: 

Sensitivity= ( ) ( ) 3 3 3 0.010 002 77 g 0.003 70 cm 1.171 94 10 g/cm g/division 10.3 divisions

x − −
SOP 2 – 2019 Page 16 of 19

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( ) 3 0.009 998 4 g 0.000 970 72 g/division = 0.970 724 10 g/division 10.3 divisions

x − = =

( )( ) ( )( ) 3 3 3 3

3 3

3

13.000 176 g 1.647 67 cm 1.171 94 10 g/cm 0.970 724 10 g/division 3.5 divisions 1.171 94 10 g/cm 1 

2.329 1 g/cm

x

x x

M

x

− −

−

 − +−        =     −

 

Mx = 13.001 389 g

Note that for this example the thermal expansion is insignificant. 

Example 2: 

Let us again consider the weighing performed in Example 1. This time, all we know about our 

weight set is that it has been adjusted to the 8.4 apparent mass scale at the best available 

commercial tolerance.

Using Eqn. (4),

3 3 3 3

3 3

3 3

3

1.171 94 10 g/cm 1.171 94 10 g/cm 3.5 divisions 13.00 g 1 0.010 g 1 8.390 9 g/cm 8.390 9 g/cm 10.3 divisions

1.171 94 10 g/cm 1 

2.329 1 g/cm

x

x x

M

x

− −

−

         −   −     + −             =     −

 

  

 

 

 

Mx = 13.001 329 g

For routine weighing, it is sometimes satisfactory to assume that the temperature is 20 °C and the 

density of air is 1.2 x 10-3 g/cm3. Had this been done, the computed value for the silicon would be

3 3 3 3

3 3

3 3

3

1.2 10 g/cm 1.2 10 g/cm 3.5 divisions 13.00 g 1 0.010 g 1 8.390 9 g/cm 8.390 9 g/cm 10.3 divisions

1.2 10 g/cm 1 

2.329 1 g/cm

x

x x

M

x

− −

−

          −       − +−                =         −

   

Mx = 13.001 442 g

which is within 55 μg of the answer found in Example 1. 
SOP 2 – 2019 Page 17 of 19

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Example 3: 

Another piece of silicon is measured on a single-pan microbalance. The balance weights were 

adjusted by the manufacturer to the conventional mass scale. The sensitivity of the balance has 

been determined to be exactly 1.000. This particular laboratory is well above sea level. At the time 

of the weighing, the following measurements were recorded: 

P = 612.3 mmHg

t = 23.4 °C 

U = 23 % relative humidity

The balance reading was 15.00 g on the built-in weights and 0.000 358 g on the optical screen. 

What is the mass of the silicon? 

Answer: 

First, calculate es and ρa: 

es = 21.627 mmHg

ρa = 0.956 32 x 10-3 g/cm3

Then, use Eqn. (3):

( )

3 3

3

3 3

3

0.956 32 10 g/cm 15.00g 0.000 358 g 1 8.0 g/cm

0.956 32 10 g/cm 1

2.329 1 g/cm

x

x

M

x

−

−

 

+ −     =     −

 

Mx = 15.004 726 g

Example 4: 

The built-in weights in Example 3 are actually stainless steel of density 7.78 g/cm3 at 20 °C. What 

is the approximate error caused by using the apparent mass scale? 

Answer: 

Using Eqn. (2), the error is approximately:

( ) 33 33

3 3

1 1 15 g 0.956 32 10 g/cm 1.2 10 g/cm 12.92 μg 7.78 g/cm 8.0 g/cm

x x − −   −   − =−

 

This discrepancy, though larger than the precision of the best analytical balances, is actually well 

within the tolerance of ASTM Class 1 weights. 
SOP 2 – 2019 Page 18 of 19

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Table B-1. Example of calibration certificate data.

Mass

(g)

Uncertainty

(g)

Vol at 20 °C

(cm3

)

Cubical 

Coefficient of 

Expansion (/°C)

100.000 941 0.000 025 12.674 39 0.000 045

50.000 463 0.000 016 6.337 19 0.000 045

30.000 293 0.000 014 3.802 32 0.000 045 

20.000 158 0.000 011 2.534 87 0.000 045

10.000 130 0.000 013 1.267 44 0.000 045

5.000 042 0.000 0069 0.633 72 0.000 045

3.000 046 0.000 0046 0.380 23 0.000 045

2.000 006 3 0.000 003 3 0.253 49 0.000 045

1.000 014 4 0.000 003 0 0.126 74 0.000 045

0.499 953 8 0.000 001 6 0.030 12 0.000 020

0.299 961 4 0.000 001 2 0.018 07 0.000 020

0.199 949 84 0.000 000 87 0.012 05 0.000 020

0.099 963 78 0.000 000 91 0.006 02 0.000 020

0.049 986 59 0.000 000 72 0.003 01 0.000 020

0.029 991 00 0.000 000 77 0.001 81 0.000 020

0.020 005 70 0.000 000 66 0.007 41 0.000 069

0.010 002 77 0.000 000 86 0.003 70 0.000 069

0.004 997 06 0.000 000 70 0.001 85 0.000 069

0.003 002 99 0.000 000 76 0.001 11 0.000 069

0.002 001 97 0.000 000 66 0.000 74 0.000 069

0.001 000 83 0.000 000 86 0.000 37 0.000 069
SOP 2 – 2019 Page 19 of 19

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Table B-2. es approximation in terms of temperature.

Temperature (°C) es (mmHg)

18 15.48

18.5 15.97

19 16.48

19.5 17

20 17.54

20.5 18.09

21 18.65

21.5 19.23

22 19.83

22.5 20.44

23 21.07

23.5 21.72

24 22.38

24.5 23.07

25 23.77

25.5 24.49

26 25.23

26.5 25.99

27 26.77

27.5 27.57

28 28.39

28.5 29.23

29 30.09

29.5 30.98

1 Handbook 145 is out of print. The archival version of Section 7 is available as a scanned PDF file on the NIST 

website.

 
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