SOP 2

**Recommended Standard Operations Procedure**

**for**

# Applying Air Buoyancy Corrections

1.  Introduction

    1.  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.

2.  Prerequisites

    1.  Verify that (true) mass values or corrections are available for
        the standards used in the measurement process. The calibration
        values for the mass standards must be traceable to NIST.

    2.  Verify that the thermometer, barometer, and hygrometer used have
        been calibrated and are in good operating condition as verified
        by periodic tests or cross‑checks with other standards.

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2.  Methodology

    1.  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.

2.  Apparatus/Equipment Required[^1]

    1.  Barometer accurate to ± 66.5 Pa (0.5 mm Hg) to determine air
        pressure.

    2.  Thermometer accurate to ± 0.10 °C to determine air temperature.

    3.  Hygrometer accurate to ± 10 % to determine relative humidity.

3.  Estimating the Magnitude of the Air Buoyancy Correction

    1.  Estimate the magnitude of the air buoyancy correct, *MABC*,
        using the following formula:

$$\text{MABC}\text{  } = \text{  }(\rho_{a}\text{  } - \text{  }\rho_{n})(V_{x}\text{  } - \text{  }V_{s})$$

2.  The equation may also be represented as follows:

$$\text{MABC}\text{  } = \text{  }m_{o}(\rho_{a}\text{  } - \text{  }\rho_{n})\left( \frac{1}{\rho_{x}}\text{  } - \text{  }\frac{1}{\rho_{s}} \right)$$

**Table 1. Variables for *MABC* equation**

  --------------------- -------------------------------------------------
  Variable              Description of Variable

  *ρ~a~*                air density at the time of the measurement in
                        mg/cm^3^

  *ρ~n~*                density of \"normal\" air; i.e., 1.2 mg/cm^3^

  *m~o~*                nominal mass (in grams)

  *V~x~*                volume of the unknown weight, *X* in cm^3^

  *V~s~*                volume of the reference standard, *S* in cm^3^

  *ρ~s~*                density of reference standard, *S* in g/cm^3^

  *ρ~x~*                density of unknown weight, *X* in g/cm^3^
  --------------------- -------------------------------------------------

> The relative magnitude of the correction can be compared to the
> expanded measurement uncertainty to determine the importance of the
> air buoyancy correction and the uncertainty components for a
> particular measurement.

4.  Procedure

    1.  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.

    2.  Determine the air density using the equation given in Section 8
        of the Appendix to this SOP.

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3.  Calculations

    1.  Calculate the mass, *M~x~*, of the unknown weight, *X*, using
        the following equation, where *d* represents the "difference"
        obtained with buoyancy corrections applied to the sensitivity
        weight.

$$M_{x}\text{  } = \text{  }\frac{M_{s}\left( 1\text{  } - \text{  }\frac{\rho_{a}}{\rho_{s}} \right)\text{  } + \text{  }d}{\left( 1\text{  } - \text{  }\frac{\rho_{a}}{\rho_{x}} \right)}$$

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

$$M_{x}\text{  } = \text{  }\frac{M_{s}\left( 1\text{  } - \text{  }\frac{\rho_{a}}{\rho_{s}} \right)\text{  } + \text{  }d\text{  } + \text{  }M_{t_{s}}\left( 1\text{  } - \text{  }\frac{\rho_{a}}{\rho_{t_{s}}} \right)\text{  } - \text{  }M_{t_{x}}\left( 1\text{  } - \text{  }\frac{\rho_{a}}{\rho_{t_{x}}} \right)}{\left( 1\text{  } - \text{  }\frac{\rho_{a}}{\rho_{x}} \right)}$$

## 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

  *M~s~*             \[true\] mass of the reference standard

  $$M_{t_{s}}$$      \[true\] mass of the tare weights carried with S

  $$M_{t_{x}}$$      \[true\] mass of the tare weights carried with X

  *ρ~s~*             density of the reference standard, S

  *ρ~x~*             density of the unknown standard, X

  $$\rho_{t_{s}}$$   density of the tare weights carried with S

  $$\rho_{t_{x}}$$   density of the tare weights carried with X
  ------------------ --------------------------------------------------------

2.  If reporting the conventional mass[^2], *CM~x~*, compute it using
    the following.

$\text{CM}_{x}\text{  } = \text{  }\frac{M_{x}\left( 1\text{  } - \text{  }\frac{0\text{.}\text{0012}}{\rho_{x}} \right)}{\left( 1\text{  } - \text{  }\frac{0\text{.}\text{0012}}{8\text{.}0} \right)}$

3.  If reporting the apparent mass, *AM~x~*, versus brass, compute it
    using the following.

$\text{AM}_{x\text{vs}\text{brass}}\text{  } = \text{  }\frac{M_{x}\left( 1\text{  } - \text{  }\frac{0\text{.}\text{0012}}{\rho_{x}} \right)}{\left( 1\text{  } - \text{  }\frac{0\text{.}\text{0012}}{8\text{.}\text{3909}} \right)}$

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

$$\text{CM}_{x}\text{  } = \text{  }\frac{\text{AM}_{x\text{vs}\text{brass}}\left( 1\text{  } - \text{  }\frac{0\text{.}\text{0012}}{8\text{.}\text{3909}} \right)}{\left( 1\text{  } - \text{  }\frac{0\text{.}\text{0012}}{8\text{.}0} \right)}$$

3.  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. The uncertainty
> in the air density equation as given in numerous periodicals is 0.0012
> mg/cm^3^ (or about 0.1 % of normal air density.)

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

  ------------------- --------------- ------------------ -----------------
                      Uncertainty of                     
                      air density                        
                      values in % of                     
                      air density                        

  Variable            ± 0.1 % of air  ± 1.0 % of air     Recommended
                      density         density            (Section 2.2)

  Air pressure (Pa)   ± 101           ± 1010             ± 66.5

  Air pressure (mm    ± 0.76          ± 7.6              ± 0.5
  Hg)                                                    

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

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

**Appendix A**

**Based on "The Basic Theory of Air Buoyancy Corrections"**

**by Richard S. Davis**[^3]

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.

2.1. Two‑Pan Balance

2.2. Single‑Pan Balance

1.  With Built‑In Weights

2.  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 Chapter 7.3, Handbook 145). Use of this scale does not
indiscriminately eliminate the need for buoyancy corrections as is
sometimes assumed.

5.  Terminology

    1.  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.)

2.  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.

3.  Conventional \[Apparent\] Mass[^4]

> 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/cm^3^. The mass, *M~N~*, of an
> object, *N*, is related to its apparent mass *M~N,A~* by the equation:

$$M_{N,A}\text{  } = \text{  }\frac{M_{N}\left( 1\text{  } - \text{  }\frac{0\text{.}\text{0012}}{\rho_{N}} \right)}{\left( 1\text{  } - \text{  }\frac{0\text{.}\text{0012}}{\rho_{B}} \right)}$$

**Table A-1. Variables for conventional (apparent) mass equation**

  ---------------------- ------------------------------------------------
  Variable               Description

  *ρ~N~*                 density of the object *N* at 20 °C in g/cm^3^

  *ρ~B~*                 density of the conventional (apparent) mass
                         scale at 20 °C in g/cm^3^
  ---------------------- ------------------------------------------------

> There are at present two apparent mass scales in wide use. The older
> is based on\
> *ρ~B~* = 8.4000 g/cm^3^ at 0 °C with a coefficient of volumetric
> expansion of 0.000054/ °C and the more recent (Conventional Mass)
> specifies *ρ~B~* = 8.0000 g/cm^3^ at 20 °C. The quantity *M~N,A~* is a
> function of the particular conventional or apparent mass scale, which
> has been used in its calculation. OIML IR 33 only recognizes
> Conventional Mass.

4.  Sensitivity

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

$$\text{sensitivity}\text{  } = \text{  }\frac{M_{\text{sw}}\text{  } - \text{  }\rho_{a}V_{\text{sw}}}{\Delta R}\text{  } = \text{  }\frac{M_{\text{sw}}\left( 1\text{  } - \text{  }\frac{\rho_{a}}{\rho_{\text{sw}}} \right)}{\Delta R}$$

## Table A-2. Variables for sensitivity equation

  --------------- -------------------------------------------------------
  Variable        Description

  *sensitivity*   balance sensitivity (mass per division)

  *M~sw~*         mass of the small, additional weight

  *ρ~a~*          density of the air

  *ρ~sw~*         density of the small, additional weight

  *V~sw~*         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. A calculator or computer
will be extremely useful for this procedure.

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. Do
not correct the barometric pressure to sea level.

8.  Calculation

> 8.1 Air density, Option A (Option B is preferred)
>
> The density of air, in g/cm^3^, can be *approximated for lesser
> accuracy* from the following formula:[^5]

$\rho_{a}\text{  } = \text{  }\left\{ \frac{0\text{.}\text{464}\text{60}\left\lbrack P\text{  } - \text{  }\left( 0\text{.}\text{003}\text{796}0\text{  }U\text{  }e_{s} \right) \right\rbrack}{\left( \text{273}\text{.}\text{15}\text{  } + \text{  }t \right)} \right\}\text{ x }\text{10}^{- 3}$
(1)

**Table A-3. Variables for air density equation**

  ------------- ---------------------------------------------------------
  Variable      Description

  *ρ~a~*        density of air, g/cm^3^

  *P*           barometric pressure, mm Hg

  *U*           \% relative humidity, entered as a whole number

  *t*           temperature, °C

  *e~s~*        1.314 6 x 10^9^ x e^\[-5\ 315.56/(t\ +\ 273.15)\]^
  ------------- ---------------------------------------------------------

> Note: e~s~ 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.

2.  Air density, Option B -- Preferred

> $\begin{array}{r}
> \rho\text{  } = \text{  }\frac{\text{pM}_{a}}{\text{ZRT}}\text{   }\left( 1\text{  } - \text{  }0\text{.}\text{3780}x_{v} \right) \\
>  \\
> x_{v}\text{  } = \text{  }\left( \frac{h}{\text{100}} \right)\text{  }f\text{  }\frac{\rho_{\text{sv}}}{p} \\
>  \\
> Z\text{  } = \text{  }1\text{  } - \text{  }\frac{p}{T}\left\lbrack a_{0}\text{  } + \text{  }a_{1}t\text{  } + \text{  }a_{2}t^{2}\text{  } + \text{  }\left( b_{0}\text{  } + \text{  }b_{1}t \right)x_{v}\text{  } + \text{  }\left( c_{0}\text{  } + \text{  }c_{1}t \right)x_{v^{2}} \right\rbrack\text{  } + \text{  }\frac{p^{2}}{T^{2}}\text{  }\left( d\text{  } + \text{  }\text{ex}_{v^{2}} \right)
> \end{array}$The density of air should be calculated with the following
> formula.[^6]

#### Table A-4. Variables for CIPM air density equation

+---------------+------------------------------------------------------+
| Variable      | Description                                          |
+---------------+------------------------------------------------------+
| *M~a~*        | molar mass of the air within laboratory 28.963 5 x   |
|               | 10^-3^ kg/mol                                        |
+---------------+------------------------------------------------------+
| *p*           | ambient barometric pressure in Pascal                |
+---------------+------------------------------------------------------+
| *T*           | ambient temperature in Kelvin                        |
+---------------+------------------------------------------------------+
| *R*           | universal gas constant: 8.314 510 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               |
+---------------+------------------------------------------------------+
| *p~sv~*       | 1 Pascal x exp (AT^2^ + 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 10^3^ K                               |
+---------------+------------------------------------------------------+
| *a~0~*        | 1.581 23 x 10^-6^ K Pa^-1^                           |
+---------------+------------------------------------------------------+
| *a~1~*        | -2.933 1 x 10^-8^ Pa^-1^                             |
+---------------+------------------------------------------------------+
| *a~2~*        | 1.104 3 x 10^-10^ K^-1^ Pa^-1^                       |
+---------------+------------------------------------------------------+
| *b~0~*        | 5.707 x 10^-6^ K Pa^-1^                              |
+---------------+------------------------------------------------------+
| *b~1~*        | -2.051 x 10^-8^ Pa^-1^                               |
+---------------+------------------------------------------------------+
| *c~0~*        | 1.989 8 x 10^-4^ K Pa^-1^                            |
+---------------+------------------------------------------------------+
| *c~1~*        | -2.376 x 10^-6^ Pa^-1^                               |
+---------------+------------------------------------------------------+
| ### *d*       | 1.83 x 10^-11^ K^2^ Pa^-2^                           |
+---------------+------------------------------------------------------+
| *e*           | -0.765 x 10^-8^ K^2^ Pa^-2^                          |
+---------------+------------------------------------------------------+

> Calculate the density of air at the balance during the weighing. Then
> determine the mass of the unknown, *M~x~*, as follows:

3.  If a two‑pan balance is used:

$$\begin{array}{r}
M_{x}\text{  } = \text{  }M_{s}\text{  } + \text{  }\rho_{a}\left( V_{x}\text{  } - \text{  }V_{s} \right)\text{  } + \text{  sensitivity }{\Delta M}_{\text{opt}}\text{                       }(3a) \\
 \\
\text{or} \\
 \\
M_{x}\text{  } = \text{   }\frac{M_{s}\left( 1\text{  } - \text{  }\frac{\rho_{a}}{\rho_{s}} \right)\text{  } + \text{  sensitivity }{\Delta M}_{\text{opt}}}{\left( 1\text{  } - \text{  }\frac{\rho_{a}}{\rho_{x}} \right)}\text{                                }(\text{3b}) \\
 \\
\text{or} \\
 \\
M_{x}\text{  } = \text{  }\frac{M_{s}\text{  } - \text{  }\rho_{a}V_{s}\text{  } + \text{  }\text{sensitivity}\text{  }{\Delta M}_{\text{opt}}}{\left( 1\text{  } - \text{  }\frac{\rho_{a}}{\rho_{x}} \right)}\text{                                    }(\text{3c})
\end{array}$$

##### Table A-5. Variables not previously defined

  ------------------ ----------------------------------------------------
  Variable           Description

  *M~x~*             mass of the unknown object

  *M~s~*             mass of the standard weights

  *ρ~s~*             density of the standard weights,
                     $\frac{M_{S}}{V_{S}}$

  *∆M~opt~*          off-balance indication read on the optical scale

  *V~x~*             volume of the unknown object

  *V~s~*             volume of the standard weights

  *ρ~x~*             density of the unknown object, $\frac{M_{X}}{V_{X}}$
  ------------------ ----------------------------------------------------

> 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 *M~x~* incurred by ignoring the buoyancy correction is
> *ρ~a~ (V~x~ ‑ V~s~)*. To estimate quickly whether such an error is of
> consequence in a particular measurement, (assume *ρ~a~ = 1.2 x
> 10^‑3^g/cm^3^*).
>
> If the mass and volumes of the standards have been adjusted to a
> conventional mass scale, then

$M_{x}\text{  } \approx \text{  }\frac{\text{CM}_{s}\left( 1\text{  } - \text{  }\frac{\rho_{a}}{\rho_{B}} \right)\text{  } + \text{  }\text{sensitivity}{\Delta M}_{\text{opt}}}{\left( 1\text{  } - \text{  }\frac{\rho_{a}}{\rho_{x}} \right)}$
(4)

**Table A-6. Variables not previously defined**

  ------------------------ ----------------------------------------------
  Variable                 Description

  *CM~s~*                  conventional mass of the standard

  The symbol ≈ signifies   
  an approximation         
  ------------------------ ----------------------------------------------

4.  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 scale[^8]. Determine which apparent mass
    scale has been used and calculate the mass of the unknown from the
    equation

$M_{x}\text{  } \approx \text{  }\frac{M_{D}\left( 1\text{  } - \text{  }\frac{\rho_{a}}{\rho_{B}} \right)\text{  } + \text{  }\text{sensitivity}M_{\text{opt}}}{\left( 1\text{  } - \text{  }\frac{\rho_{a}}{\rho_{x}} \right)}$
(5)

## Table A-7. Variables not previously defined

  ------------------------ ----------------------------------------------
  Variable                 Description

  *M~D~*                   mass indicated by dial or digital readings

  *M~opt~*                 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:

$$M_{x}\text{  } = \text{  }M_{s}\text{  } + \text{  }\rho_{a}\left( V_{x}\text{  } - \text{  }V_{s} \right)\text{  } + \text{  }\text{sensitivity}\Delta M'_{\text{opt}}$$

**Table A-8. 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.

8.5 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.

###### The mass of an unknown object weighed on the balance is then 

> $M_{x}\text{  } = \text{  }\frac{M_{R}\left( 1\text{  } - \text{  }\frac{\rho_{a}}{\rho_{c}} \right)}{\left( 1\text{  } - \text{  }\frac{\rho_{a}}{\rho_{x}} \right)}$
> (6)

## Table A-9. Variables not previously defined

  ---------------- ------------------------------------------------------
  Variable         Description

  *M~R~*           readout displayed on the balance

  *ρ~a~*           density of air at the time of balance calibration

  *ρ~c~*           density of the standard used to calibrate the balance
                   (or *ρ~B~* 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.

6.  Top‑loading balances may be considered a form of single‑pan balance
    and the appropriate procedure for buoyancy correction followed.

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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:

$$\text{δρ}_{a}\left( V_{x}\text{  } - \text{  }V_{s} \right)$$

## Table A-10. Variables for above equation

+--------------+-------------------------------------------------------+
| Variable     | Description                                           |
+--------------+-------------------------------------------------------+
| *δρ~a~*      | random error of evaluation of *ρ~a~*                  |
+--------------+-------------------------------------------------------+
| *V~s~*       | volume of standards, if weighing by substitution      |
|              |                                                       |
|              | *V~s~ = M~D~ / D~B~*, if using the built-in weights   |
|              | on a single pan balance.                              |
+--------------+-------------------------------------------------------+
| *V~x~*       | volume of object weighed                              |
+--------------+-------------------------------------------------------+

> The quantity, *δρ~a~* will have contributions from the measurements of
> temperature, pressure and relative humidity which are required for the
> calculation of *ρ~a~*. Equation (1) may be used to estimate the
> effects of imprecision in measurements of *P*, *t*, and *U*. It is
> unrealistic to expect *δρ~a~ /ρ~a~* ever to be less than 0.05 % even
> using the best techniques available.

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 *ρ~a~(V~x~ ‑ V~s~)*. While this quantity
> may be significant to a measurement, day-to-day variation in *ρ~a~*
> (usually no more than 3 %) may not be noticeable and hence need not be
> measured. For the most accurate work, not only must *ρ~a~* be
> accurately determined, but the volumes of the unknown and standard may
> have to be measured to better than 0.05 % ‑ the minimum systematic
> uncertainty attainable in the calculation of *ρ~a~*.
>
> If the standards have been calibrated in terms of conventional mass,
> complete neglect of buoyancy corrections will produce an error in the
> measured result *M~x~* of order:
>
> $\text{CM}_{s}\left\lbrack \left( 1\text{.}2x\text{10}^{- 3}\text{  } - \text{  }\rho_{a} \right)\left( \frac{1}{\rho_{s}}\text{  } - \text{  }\frac{1}{\rho_{B}} \right)\text{  } + \text{  }\left( \frac{\rho_{B}\text{  } - \text{  }\rho_{x}}{\rho_{B}} \right)\left( \frac{\rho_{a}}{\rho_{x}} \right) \right\rbrack$
> (7)

This error is often unacceptably large.

####### Use of equation (4), on the other hand, introduces only an error of approximately 

$\text{CM}_{s}\left( 1\text{.}2x\text{10}^{- 3}\text{  } - \text{  }\rho_{a} \right)\left( \frac{1}{\rho_{s}}\text{  } - \text{  }\frac{1}{\rho_{B}} \right)$
(8)

> 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/cm^3^ and an older one based on 8.4 g/cm^3^ -
> 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]{.underline} accurate work, the apparent
> mass scale should be abandoned in favor of substitution weighing with
> standards of known mass 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.

**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 mm Hg*

*t = 22.3 °C*

*U = 37 % relative humidity*

What is the mass of the silicon?

**Answer:**

From Eq. 1, or Table B-2, calculate

$$e_{s}\text{  } = \text{  }1\text{.}\text{314}6x\text{10}^{9}x\text{  }e^{\left( \frac{- 5\text{315}\text{.}\text{56}}{\left( \text{22}\text{.}3\text{  } + \text{  }\text{273}\text{.}\text{15} \right)} \right)}\text{  } = \text{  }\text{20}\text{.}\text{194}\text{ mm Hg}$$

$$\rho_{a}\text{  } = \text{  }\left\{ \frac{0\text{.}\text{464}\text{60}\left\lbrack \text{748}\text{.}1\text{  } - \text{  }\left( 0\text{.}\text{003}\text{796}0 \right)\left( \text{37} \right)\left( \text{20}\text{.}2 \right) \right\rbrack}{\left( \text{273}\text{.}\text{15}\text{  } + \text{  }\text{22}\text{.}3 \right)} \right\}\text{10}^{- 3}$$

*ρ~a~ =* 1.171 94 x 10^-3^ g/cm^3^

The density of silicon at 20 °C is 2.329 1 g/cm^3^ and its coefficient
of linear expansion is 0.000 002 6 /°C.

Make use of Eq. 3c and Table B-1.

$$M_{s}\text{  } = \text{  }\text{10}\text{.}\text{000}\text{126}4\text{  } + \text{  }3\text{.}\text{000}\text{045}9\text{  } = \text{  }\text{13}\text{.}\text{000}\text{172}\text{ g}$$

$V_{s}\text{  } = \text{  }\left( 1\text{.}\text{267}\text{  } + \text{  }0\text{.}\text{380} \right)\text{  } = \text{  }1\text{.}\text{647}\text{ cm}^{3}$

Calculate the sensitivity:

$$\text{sensitivity  } = \text{  }\frac{0\text{.}\text{010}\text{003}g\text{  } - \text{  }\left( 0\text{.}\text{003}\text{70} \right)\left( 1\text{.}\text{172}x\text{10}^{- 3} \right)}{\text{10}\text{.}3}$$

$$\text{sensitivity  } = \text{  }\frac{\left( 0\text{.}\text{009}\text{999} \right)}{\text{10}\text{.}3}\text{  } = \text{  }0\text{.}\text{000}\text{971}\text{ g/division  } = \text{  }0\text{.}\text{971}\text{ mg/division}$$

$$M_{x}\text{  } = \text{  }\frac{\left\lbrack \text{13}\text{.}\text{000}\text{172}\text{  } - \text{  }\left( 1\text{.}\text{647} \right)\left( 1\text{.}\text{172}x\text{10}^{- 3} \right) \right\rbrack\text{  } + \text{  }\left\lbrack \left( 0\text{.}\text{971}x\text{10}^{- 3} \right)\left( - \text{  }3\text{.}5 \right) \right\rbrack}{\left( 1\text{  } - \text{  }\frac{1\text{.}\text{172}x\text{10}^{- 3}}{2\text{.}\text{329}1} \right)}$$

*M~x~ =* 13.001 385 g

Note that the thermal expansion is insignificant in this example.

**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 Eq. (4),

$$M_{x}\text{  } = \text{  }\left\{ \frac{\left\lbrack \text{13}\text{.}\text{00}\left( 1\text{  } - \text{  }\frac{1\text{.}\text{172}x\text{10}^{- 3}}{8\text{.}\text{390}9} \right) \right\rbrack\text{  } + \text{  }\left\lbrack 0\text{.}\text{010}\left( 1\text{  } - \text{  }\frac{1\text{.}\text{172}x\text{10}^{- 3}}{8\text{.}\text{390}9} \right) \right\rbrack\left( \frac{- 3\text{.}5}{\text{10}\text{.}3} \right)}{\left( 1\text{  } - \text{  }\frac{1\text{.}\text{172}x\text{10}^{- 3}}{2\text{.}\text{329}1} \right)} \right\}$$

*M~x~* = 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/cm^3^. Had
this been done, the computed value for the silicon would be

$$M_{x}\text{  } = \text{  }\left\{ \frac{\left\lbrack \text{13}\text{.}\text{00}\left( 1\text{  } - \text{  }\frac{1\text{.}2x\text{10}^{- 3}}{8\text{.}\text{390}9} \right) \right\rbrack\text{  } + \text{  }\left\lbrack 0\text{.}\text{010}\left( 1\text{  } - \text{  }\frac{1\text{.}2x\text{10}^{- 3}}{8\text{.}\text{390}9} \right) \right\rbrack\left( \frac{- 3\text{.}5}{\text{10}\text{.}3} \right)}{\left( 1\text{  } - \text{  }\frac{1\text{.}2x\text{10}^{- 3}}{2\text{.}\text{329}1} \right)} \right\}$$

*M~x~ =* 13.001 442 g

which is within 100 µg of the answer found in Example 1.

**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 mm Hg

*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 *e~s~* and *ρ~a~*:

*e~s~ =* 21.59 mm Hg

*ρ~a~ =* 0.956 x 10^‑3^ g/cm^3^

Then, use Eq. 5:

$$M_{x}\text{  } = \text{  }\frac{\left( \text{15}\text{.}\text{00}\text{  } + \text{  }0\text{.}\text{000}\text{358} \right)\left( 1\text{  } - \text{  }\frac{0\text{.}\text{956}x\text{10}^{- 3}}{8\text{.}0} \right)}{\left( 1\text{  } - \text{  }\frac{0\text{.}\text{956}x\text{10}^{- 3}}{2\text{.}\text{329}1} \right)}$$

*M~x~* = 15.004 724 g

**Example 4:**

The built‑in weights in Example 3 are actually stainless steel of
density 7.78 g/cm^3^ at 20 °C. What is the approximate error caused by
using the apparent mass scale?

**Answer:**

Using (7), the error is approximately

$\text{15}g\left( 0\text{.}2x\text{10}^{- 3}\frac{g}{\text{cm}^{3}} \right)\left( \frac{1}{7\text{.}\text{78}\frac{g}{\text{cm}^{3}}}\text{  } - \text{  }\frac{1}{8\text{.}0\frac{g}{\text{cm}^{3}}} \right)\text{  } = \text{  }\text{11}\mu g$

This discrepancy, though larger than the precision of the best
analytical balances, is actually well within the tolerance of Class 1
weights.

## Table B-1. Example of calibration report data

+-----------------+-----------------+-----------------+-----------------+
| Mass            | Uncertainty     | Vol at 20 °C    | Coefficient of  |
|                 |                 |                 | Expansion       |
| \(g\)           | \(g\)           | (cm^3^)         |                 |
+-----------------+-----------------+-----------------+-----------------+
| 100.00094070    | 0.00002535      | 12.67439        | 0.000045        |
+-----------------+-----------------+-----------------+-----------------+
| 50.00046277     | 0.00001550      | 6.33719         | 0.000045        |
+-----------------+-----------------+-----------------+-----------------+
| 30.00029259     | 0.00001361      | 3.80232         | 0.000045        |
+-----------------+-----------------+-----------------+-----------------+
| 20.00015779     | 0.00001113      | 2.53487         | 0.000045        |
+-----------------+-----------------+-----------------+-----------------+
| 10.00012644     | 0.00001330      | 1.26744         | 0.000045        |
+-----------------+-----------------+-----------------+-----------------+
| 5.00004198      | 0.00000688      | 0.63372         | 0.000045        |
+-----------------+-----------------+-----------------+-----------------+
| 3.00004588      | 0.00000459      | 0.38023         | 0.000045        |
+-----------------+-----------------+-----------------+-----------------+
| 2.00000627      | 0.00000333      | 0.25349         | 0.000045        |
+-----------------+-----------------+-----------------+-----------------+
| 1.00001438      | 0.00000300      | 0.12674         | 0.000045        |
+-----------------+-----------------+-----------------+-----------------+
| 0.49995376      | 0.00000160      | 0.03012         | 0.000020        |
+-----------------+-----------------+-----------------+-----------------+
| 0.29996145      | 0.00000115      | 0.01807         | 0.000020        |
+-----------------+-----------------+-----------------+-----------------+
| 0.19994984      | 0.00000087      | 0.01205         | 0.000020        |
+-----------------+-----------------+-----------------+-----------------+
| 0.09996378      | 0.00000091      | 0.00602         | 0.000020        |
+-----------------+-----------------+-----------------+-----------------+
| 0.04998659      | 0.00000072      | 0.00301         | 0.000020        |
+-----------------+-----------------+-----------------+-----------------+
| 0.02999100      | 0.00000077      | 0.00181         | 0.000020        |
+-----------------+-----------------+-----------------+-----------------+
| 0.02000570      | 0.00000066      | 0.00741         | 0.000069        |
+-----------------+-----------------+-----------------+-----------------+
| 0.01000277      | 0.00000086      | 0.00370         | 0.000069        |
+-----------------+-----------------+-----------------+-----------------+
| 0.00499706      | 0.00000070      | 0.00185         | 0.000069        |
+-----------------+-----------------+-----------------+-----------------+
| 0.00300299      | 0.00000076      | 0.00111         | 0.000069        |
+-----------------+-----------------+-----------------+-----------------+
| 0.00200197      | 0.00000066      | 0.00074         | 0.000069        |
+-----------------+-----------------+-----------------+-----------------+
| 0.00100083      | 0.00000086      | 0.00037         | 0.000069        |
+-----------------+-----------------+-----------------+-----------------+

**Table B-2. *e~s~*** approximation in terms of temperature

  ----------------------------------- -----------------------------------
  Temperature (°C)                    *e~s~* (mm Hg)

  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]: 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.

[^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/cm^3^; reference temperature 20 °C; *normal*
    air density 0.0012 g/cm3. Conventional mass was formerly called
    "Apparent Mass versus 8.0 g/cm^3^" in the United States. *See OIML
    IR 33 (1973, 1979), under revision.*

[^3]: Richard S. Davis, formerly of the National Institute of Standards
    and Technology, Mass Group.

[^4]: Pontius, P. E., Mass and Mass Values, NBS Monograph 122, 1974, pp
    12-23, 26-33 described the concept of apparent mass. The term
    conventional mass is described by OIML R 33, the Conventional Value
    of Mass in Air.

[^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.

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

[^7]: In general, V(t) = V~20~ \[1 + ß(t ‑ 20 °C)\] where t is the
    temperature of the weight, V~20~ is the volume at 20 °C, and ß is
    the coefficient of volumetric expansion.

[^8]: Schoonover, R. M. and Jones, F. E., \"Air Buoyancy in
    High‑Accuracy Weighing on Analytical Balances,\" Anal. Chem.,
    [53]{.underline}, 1981, p. 900.