SOP No. 3

Recommended Standard Operations Procedure

for

Weighing by Double Substitution

Using an Equal-Arm Balance

I. Introduc tion

I. i . Purpose

The double substitution procedure is one in which a standard and an

unknown weight are intercompared twice to determine the average

difference between the two weighings. Any effects of inequality of

arms and linear drift are eliminated by the weighing sequence used.

Accordingly, the procedure is especially useful for high accuracy

calibrations.

I. 2. Prerequisites

I. 2. I. Mass standards must be available

certificates traceable to NBS.

with calibration

I. 2.2. The balance used must be in good operating condition as

verified by a valid control chart or preliminary experiments

to ascertain its performance quality.

1.2. 3. The operator must be experienced in precision weighing

techniques.

2. Methodology

2. I. Scope, Precision, Accuracy

This method is applicable to all weighings utilizing an equal arm

balance. Because considerable effort is involved, it is most useful

for calibrations of the highest accuracy. The precision will depend

upon the sensitivity of the balance and the care exercised in making

the required weighings. The accuracy will depend on the accuracy of

calibration of the standard weights and the precision of the

intercomparison.

2 . 2 . Sumary

The standard is balanced with respect to a counterweight. The

standard is then replaced by the test weight which is weighed with

respect to the same counterweight. A small weight (called a

sensitivity weight) is added to the test weight and these are

weighed. The standard and the same sensitivity weight are then

weighed. The latter two weighings provide both second weighings of

SOP 3 - I
the standard and test weights as well as a determination of the

sensitivity of the balance under the load conditions. All weighings

are made at regularly spaced time intervals to average out any

effects due to linear instrument drift.

2.3. Apparatus/Equipment

2.3. I. Precision equal-arm balance with sufficient capacity and

sensitivity for the calibrations planned. The index scale of

the balance is conveniently numbered from 0 to 20 with 10 as

the center division, although other numbering systems such as

o to 200 are possible. A system in which the center division

is 0 is not recommended since the negative readings that

result can cause observational and/or computational problems.

The graduations are so numbered that the addition of a small

weight to the left arm will increase the scale reading.

2.3.2. Standard weights with valid calibrations, traceable to NBS.

The sensitivity weight may be a calibrated secondary standard. The use of secondary standards as sensitivity weights

reduces wear on the primary mass standards.

2.3.3. Counterweights, (uncalibrated) , of approximately the same

mass as the standard weights. Lead shot in a suitable

container is useful for this purpose.

2.3.4. Small calibration weights (usually decimal fractions) to be

used as tare weights.

2.3.5. Equipment capable of loading and unloading weights on the

balance without daage to either (especially important in the

case of large weights).

2.3.6. Stop watch or other timing device to observe time of each

trial measurement.

2.4. Symbols

The following symbols are used in this procedure:

S - standard weight

X - weight calibrated

T counterweight

t small calibrated weight. A subscript s or x is used to

indicate the larger weight with which it is associated.

sw - small calibrated weight used to evaluate the sensitivity of the

balance. M - the mass of a specific weight. Subscripts s, x, t, sw, are

used to identify the weight

AM - the apparent mass of a specific weight. Subscripts s, x, t,

sw. are used to identify the weight.

SOP 3 - 2
2.5. Procedure

2.5. I. Preliminary Procedure

Conduct preliminary measurements (without recording data) to

determine the values for T, ts ' tx, and sw for use in

measurement procedures, 2.5.2, or 2.5.3. This will serve to

warm up the balance and facilitate the actual measurements in

which the trials should be observed at regu¡ar time intervals. Estimate the time required to complete a trial (see

2.5.2) and maintain essentially equal intervals between

successive trials (see 2.5.2.5). In a series of calibrations, this preliminary procedure is only required when new

values need to be determined for T, ts' tx, and sw.

2.5.2. Measurement Procedure, Option A

Measurement No. Left Pan (arm) Right Pan (arm) Observation

i S + ts T °l

2 X + tx T °2

3 X+tx+ sw T °3 4 S + ts + sw T °4

The turning points, 0l' 02, 03 ' and 04 are observed as

described in GMP No. I. All observations should be recorded

on suitable data sheets, such as those in the Appendix.

2.5.2.1. Observation 1. Place the standard weight(s), S,

equivalent to the nominal weight of the test

weight, on the left pan (arm) of the balance and

add a slight excess of counter weight (T) to the

right pan (arm).. Add tare weight, ts' to the left

pan (arm) to obtain an approximate balance. Record

the sum of the turning points and the time.

2.5.2.2. Observation 2. Remove weight(s) S and replace with

test weight, X. Adjust tare weight, tx, to obtain

an approximate balance condition within one

division of the sum. of turning points obtained for

0l (for a 0 to 20 division graduated scale). (If S

and X are approximately equal., tx may equal ts.)

Record the sum of the turning points and the time.

2.5.2.3. Observation 3. Add a small sensitivity weight, sw,

to the weights of observation 2. The value of sw

should be such as to change the turning points by

about 4 divisions on a 0 to 20 scale (40 divisions

on a 0 to 200 scale). Record the sum of the

turning points and the time.

SOP 3 - 3

~
2.5.2.4. Observation 4. Remove weight X and tx and replace

with Sand ts. Weight sw remains on the balance

pan. Record the sum of the turning points and the

time.

2.5.2.5. Calculate the time intervals between the successive

trials. These should not differ from one another

by more than ~ 20% . If this difference is

exceeded, reject the data and take a new series of

measurements that will so agree.

2.5.3. Measurement Procedure, Option B

Measurement No. LeftPan (arm) Right Pan (arm) Observation

i X + tx T

°l 2 S + ts T 02 3 S + ts + sw T 03 4 X+tx+ sw T 04

Measurements for Option B are made as described in Option A

except that X, S, tx' and ts are interchanged appropriately.

3. Calculations

3.1. Calculate the sums of the turning points, 01, 02, 03 and 04 (Ref. GMP

No.1) .

3.2. Calculate the correction, Cx' required for the test weight, as

follows, according to the optional sequence used. In each case, Cs

. is the apparent mass correction required for the standard weight

used, including that for the tare weights as appropriate.

3.2.1. No air buoyancy correction. Calculate 

the apparent mass correction, Cx, for the test weight as follows, according to

the optional sequence used. In each case, the appropriate

apparent mass corrections for the standard weight(s), Cs' the

tare weights AMts and Motx, and the sensitivity weights,

AKsw' are included. The symbols Ns and Nx refer to the

nominal values of S and X, respectively.

3.2.1.1. Optional Sequence A

(°i-°i+03-04) Cx - Cs + AMt - AMt + 2 s x

(AMsw)

+ N s - Nx (03-02)

3.2.1.2. Optional Sequence B

(Oi -02 + 04-03)

Cx - Cs + AMts - AMtx +

(AMsw)

2

+ Ns - Nx

(03-02)

SOP 3-4
3.2.2. Air Buoyancy Correction

3.2.2.1. Calculate the air density, PA' as described in

section 8 of the Appendix to SOP No. 2 or obtain

from Table 9.9.

3.2.2.2. Calculate the mass Mx of the test weight X and its

mass correction Cx using the mass of the standard

weight(s), the tare weights and the sensitivity

weights according to the optional sequence used.

3.2.2.2.1. Optional Sequence A

Msw (1 _ PA )

Ms(l

PA ) ( PA )

Mtx ( 1 - ~)+

(02 -01+03 -04) Psw

-- +Mt 1--- .

Ps s Pts Ptx 2 03 -02

Mx -

(1 - ::)

3.2.2.2.2. Optional Sequence B

MS(l

Mx -

_ PA) + Mt (1 _ Ps s --Mtl--+ · PA ) ( PA) (°1-°2+°4-°3) Msw (1 - ::J

Pts x Pex 2 03-02

(1 - ::)

3.2.2.3. Calculate the mass correction Cx' as follows :

Cx-Mx-Nx

where Nx is the nominal value for X.

3.2.2.4. Calculate the appar~nt mass of X (AM), versus the

desired reference density of 8.0 g/cm3 or brass.

It is recommended that the apparent mass versus 8.0

g/cm3 be reported unless otherwise requested. The

density, Px' must be in g/cm3.

3.2.2.4.1. Apparent mass versus 8.0 g/cm3

(1 _ . 0~:2 )

AM vs 8.0 - Mx 0.999850

SOP 3 - 5
3.2.2.4.2 . Apparent mass versus brass

1 - .0012

Px

AM vs brass - Mx .999857

4. Assignent of Uncertainty

The limits of uncertainty, U, include estimtes of the uncertainty of the

mass standards used, Us' plus th~ uncertainty of measurement, Um, at the

99.73% level of confidence. The latter is estimated by

ts

where s is the standard deviation of measurement and t is obtained from

Table 9.3.

Then U - ~ (Us + ts)

4. I. Precision of Measurement Known from Control Chart Performance. (See

SOP No.9.)

The value for s is obtained from the control chart limits and current

knowledge that the measurements are in a state of statistical

contro1. This will need to be ascertained by measurement of at least

one check standard while the above measurements are in progress.

Use the value of t (corresponding to a probability level of 99.73%)

from Table 9.3 appropriate for the number of degrees of freedom, v,

on which the control limits of the control chart are based.

4.2. Precision Estimated from Series of Measurements

Measure a stable test object at least 7 times, no two measurements of

which may be made on a single day. Calculate the mean and the

standard deviation in the conventional manner. The latter is the

value of s that is used in Section 4.l. In this case select the

value for t from Table 9.3 based on t~e number of degrees of freedom

involved in computing s.

Note: Repetitive measurements made on the same day estimate only the

short-term standard deviation.

5. Report

5.1. Report results as described in SOP No.1, Preparation of

CalibrationlTest Reports.

SOP 3 - 6
Appendix

Double Substitution Data Sheet

for Equal-Arm Balance (Option A)

Test No.: Sheet No.: Date:

Item Identification: Balance:

Standard Identification: Observer:

Temperature: Pressure: Re1. Hum.:

,

Cs (Cs)- ~ AMsw(Msw)-

Ps- Px- Psw -

AMt (Mt )

s s

Pt -

s

Time

~ AMt (Mt ) -

x x

~

Pt -

x

Balance standard deviation -

Measurement Weights Turning Points

No.

Left Arm Low High Sum

1 S + ts 01 -

2 X + tx 02 -

3 X+tx+sw 03 -

4 S + ts + sw 04 -

Time

SOP 3-7
Appendix

Double Substitution Data Sheet.

for Equal-Arm Balance (Option B)

Test No. : Sheet No. : Date:

Item Identification: Balance:

Standard Identification: Observer:

Temperature: Pressure: Re1. Hum.:

, Cs(Cs)- ~ AKsw(Msw)-

Ps- Px- Psw -

AMt (Mt ) -

s s

Pt -

s

Time

~ AKt (Mt ) -

x x ~

Pt -

x

Balance standard deviation _

Measurement Weights Turning Points

No.

Left Arm Low High Sum

1 X + tx °l -

2 S + ts 02 -

3 S + ts + sw 03 -

4 X+tx+sw 04 -

Time

SOP 3 - 8
Appendix

Double Substitution Data Sheet

for Equal-Arm Balance (Option A)

Test No.: 137 Sheet No.: 1 Date: 8/27 /86

Item Identification: 1000 lb No. 623 Balance: Russell

Standard Identification: #11 and #22 Observer: HO

Temperature: 23.5°C Pressure: 746.1 mm HI! ReI. Hum. : 45%

,

~'(Cs). 0.0047 1b :t

3

ps. 8.0 g/cm px.

.0030 lb

3

7.0 g/cm

~w(Msw).

psw.

o . 01 lb

3

7.8 i/cm

~ (Mt ). 3.625:t .0000751b ~ (Mt ). 3.642lb :t negligible s s 3 x3x

Time 10: 45 a Balance standard deviation . O. 0018 lb

Pt. 8. 0 ~/cm Pt. 7.8 g/cm s x

Measurement Weights Turning Points

No.

Left Arm Low High Sum

IS.3 i S + ts -5 15.2 0l - 20.4

5.2 15.2

"'

2 X + tx 5.9 l4.1 02 - 20.0

5.9 14.1

17 .6

3 X+tx+sw -6 17 .5 03 - 23.65

6.2 17.4

6.3 16.3

4 S + ts + sw 6.4 16.2 04 - 22.6

6.4

Time 11:05 a

SOP 3 - 9
Pa - 1.163 mg/cm3 - 0.001163 g/cm3

MS(l

Mx .

PA) ( PA ) ( PA ) (0i-01+03-04) -- +Mt 1-- -Mt 1-- + Ps S Pts x Ptx 2

PA

MSW(l - ~)

Psw

.

03 -02

1

Mx -rOOO.0047 1 - 8.0 + 3.625 1- r (.001163) (

Px

.001163) _ 3.642 (1 _ .0011~~) 8.0 7.8

+

20.0 - 20.4 + 23.65 - 22.6

2

( . 001163 )j

.01 1-

7.8

23.65 20.0 (1

_ .001163)

7.0

.

Mx -

999.8593243 + 3.6244730 - 3.6414570 + .0008903

.99983386

Mx - 1000.009372 lb

Uncertainty - ts + Us Degrees of freedom in s - 20

Uncertainty . 3.422 (.0018) + (.0030 + .000075)

Uncertainty - 0.0092346 lb

Mx . 1000.0094 :t 0.0092 lb

ex . Mx - Nx - 0.0094 :t 0.0092 lb

(1 _ .;:12)

AM vs 8.0 - Mx 0.999850

(1 _ .0012) 7.0

AM vs 8.0 - 1000.0094 .999850

AM vs. 8.0 - 999. 9880 :t .0092 lb

Cx . AM vs '8.0 - Nx

Cx . 999.9880 - 1000 - -.0120 :t .0092 lb

SOP 3-10