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Chapter: 2 General physics
    Section: 2.1 Measurement of mass, pressure and other mechanical quantities
        SubSection: 2.1.1 Mass, volume and density

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2.1 Measurement of mass, pressure and other mechanical quantities

2.1.1 Mass, volume and density

Correction of weighings for buoyancy

If a substance is weighed in air and found to be balanced by mass standards (‘weights’) of value M, the true mass of the substance is

M + Mda

1

1

d

ds

where d is the density of the substance, ds is the density of the mass standards and da is the density of the air. In practice, the members of an ordinary set of mass standards, including the fractions, are standardized by being assigned values which are equal to the masses of reference standards of density 8000 kg m−3 which they would balance in air. The value 8000 kg m−3 is a conventional value close to the density of the larger weights in a set. The buoyancy of any group of weights from such a set should be calculated by taking this value for ds. The following table has been prepared on this basis, da being taken to be 1.2 kg m−3.

Density of
substance
weighed

Buoyancy correction
(mg per gram
of substance
)

Density of
substance
weighed

Buoyancy correction
(mg per gram
of substance
)

d/kg m−3

ds = 8000 kg m−3
da = 1.2 kg m−3

d/kg m−3

ds = 8000 kg m−3
da = 1.2 kg m−3

500  

+2.250

  6 000

+0.050

1000   

+1.050

  8 000

0     

1500   

+0.650

10 000

−0.030

2000   

+0.450

12 500

−0.054

2500   

+0.330

15 000

−0.070

3000   

+0.250

20 000

−0.090

4000   

+0.150

22 000

−0.096


Density of ambient air (unit = 1 kg m−3) 50% relative humidity, 0.04% CO2 by volume

Air pressure,
P
/kPa
(1 kPa = 10 mb)

Air temperature, t/ºC

6

8

10

12

14

16

18

20

22

24

26

28

30

                           

80

0.997

0.989

0.982

0.974

0.967

0.960

0.953

0.946

0.939

0.931

0.924

0.917

0.910

81

1.009

1.002

0.994

0.987

0.979

0.972

0.965

0.958

0.950

0.943

0.936

0.929

0.922

82

1.022

1.014

1.006

0.999

0.991

0.984

0.977

0.969

0.962

0.955

0.948

0.940

0.933

83

1.034

1.026

1.019

1.011

1.004

0.996

0.989

0.981

0.974

0.967

0.959

0.952

0.945

84

1.046

1.039

1.031

1.023

1.016

1.008

1.001

0.993

0.986

0.978

0.971

0.964

0.956

85

1.059

1.051

1.043

1.036

1.028

1.020

1.013

1.005

0.998

0.990

0.983

0.975

0.968

86

1.071

1.064

1.056

1.048

1.040

1.032

1.025

1.017

1.009

1.002

0.994

0.987

0.979

87

1.084

1.076

1.068

1.060

1.052

1.044

1.037

1.029

1.021

1.014

1.006

0.998

0.991

88

1.096

1.088

1.080

1.072

1.064

1.056

1.049

1.041

1.033

1.025

1.018

1.010

1.002

89

1.109

1.101

1.093

1.084

1.076

1.068

1.061

1.053

1.045

1.037

1.029

1.022

1.014

 

 

 

 

 

 

 

 

 

 

 

 

 

 

90

1.121

1.113

1.105

1.097

1.089

1.081

1.073

1.065

1.057

1.049

1.041

1.033

1.025

91

1.134

1.126

1.117

1.109

1.101

1.093

1.085

1.076

1.068

1.061

1.053

1.045

1.037

92

1.146

1.138

1.129

1.121

1.113

1.105

1.096

1.088

1.080

1.072

1.064

1.056

1.048

93

1.159

1.150

1.142

1.133

1.125

1.117

1.108

1.100

1.092

1.084

1.076

1.068

1.060

94

1.171

1.163

1.154

1.146

1.137

1.129

1.120

1.112

1.104

1.096

1.088

1.079

1.071

 

 

 

 

 

 

 

 

 

 

 

 

 

 

95

1.184

1.175

1.166

1.158

1.149

1.141

1.132

1.124

1.116

1.107

1.099

1.091

1.083

96

1.196

1.188

1.179

1.170

1.161

1.153

1.144

1.136

1.128

1.119

1.111

1.103

1.094

97

1.209

1.200

1.191

1.182

1.174

1.165

1.156

1.143

1.139

1.131

1.122

1.114

1.106

98

1.221

1.212

1.203

1.195

1.186

1.177

1.168

1.160

1.151

1.143

1.134

1.126

1.117

99

1.234

1.225

1.216

1.207

1.198

1.189

1.180

1.172

1.163

1.154

1.146

1.137

1.129

 

 

 

 

 

 

 

 

 

 

 

 

 

 

100  

1.246

1.237

1.228

1.219

1.210

1.201

1.192

1.184

1.175

1.166

1.157

1.149

1.140

101  

1.259

1.250

1.240

1.231

1.222

1.213

1.204

1.195

1.187

1.178

1.169

1.160

1.152

102  

1.271

1.262

1.253

1.243

1.234

1.225

1.216

1.207

1.198

1.190

1.181

1.172

1.163

103  

1.284

1.274

1.265

1.256

1.246

1.237

1.228

1.219

1.210

1.201

1.192

1.184

1.175

104  

1.296

1.287

1.277

1.268

1.259

1.249

1.240

1.231

1.222

1.213

1.204

1.195

1.186

 

 

 

 

 

 

 

 

 

 

 

 

 

 

105  

1.309

1.299

1.290

1.280

1.271

1.261

1.252

1.243

1.234

1.225

1.216

1.207

1.198

106  

1.321

1.312

1.302

1.292

1.283

1.273

1.264

1.255

1.246

1.236

1.227

1.218

1.209

 

 

 

 

 

 

 

 

 

 

 

 

 

 



Corrections for humidity (unit = 1 kg m-3)

Relative
humidity,
R
%

Air temperature, t/°C

10

20

30

 

 

 

 

20

+0.002

+0.003

+0.006

25

+0.001

+0.003

+0.005

30

+0.001

+0.002

+0.004

35

+0.001

+0.002

+0.003

40

+0.001

+0.001

+0.002

 

 

 

 

45

0     

+0.001

+0.001

50

0     

0     

0     

55

0     

−0.001

−0.001

60

−0.001

−0.001

−0.002

65

−0.001

−0.002

−0.003

 

 

 

 

70

−0.001

−0.002

−0.004

75

−0.001

−0.003

−0.005

80

−0.002

−0.003

−0.006

 

 

 

 


The figures in the main table relate to air of 50% relative humidity and containing 0.04% carbon dioxide by volume. For other states of humidity a correction should be applied from the subsidiary table. The tables are based on the following expression recommended by an international working group formed by the International Bureau of Weights and Measures (BIPM) (Giacomo, 1982) and later revised to incorporate a re-working of some of the data, and the adoption of the International Temperature Scale of 1990 (Davis, 1992). The differences in numerical results due to this revision are small.
The density of moist air, ρ, is:

   

 ρ =

pMa

1 − xv

1 − 

Mv

 

ZRT

Ma



Where   p is the air pressure in pascals

           Ma is the molar mass of dry air in kg mol−1

             Z is the compressibility factor of air

             R is the Molar Gas Constant in J mol−1 K−1

             T is the (thermodynamic) temperature in kelvin

           Xv is the mole fraction of water vapour

          Mv is the molar mass of water vapour in kg mol−1


 R and Mv are constants, as is Ma for air of a defined composition. Values for these constants are:


      R = 8.314510 J mol−1 K−1

     Mv = 18.015 × 10−3 kg mol−1

     Ma = 28.963 5 × 10−3 kg mol−1

Thus the expression reduces to

 ρ = 

3.483 49 × 10−3 p

 [1 − 0.3780 xv]

ZT


where x v and Z are given by the following expressions:

xv = hf(p,t)

Psv

 × 10−2

p

   Where h is the relative humidity (%)

      f(p,t) is the enhancement factor at ambient temperature and pressure

      f = 1.000 62 + 3.14 × 10−8p + 5.6 × 10−7 t2

     where p is the ambient pressure in pascals

     and    t is the ambient temperature in degrees Celsius

              psv is the saturation vapour pressure at ambient temperature, in pascals

 psv = e(AT2 + BT + C + D/T)

  where the values of the constant terms are:

    A = 1.237 8847 × 10−5 K−2

    B = −1.912 1316 × 10−2 K−1

    C = 33.937 110 47

    D = −6.343 1645 × 103 K


and T is the thermodynamic temperature in Kelvin

Z = 1 −  

p

[a0 + a1t + a2t2 + (b0 + b1t)xv + (c0 + c1t) xv2

] + 

p2

 [d + exv2

]

T

T2


   where p is the ambient pressure in pascals

     T is the temperature in Kelvin

     t is the ambient temperature in degrees Celsius

     xv is the mole fraction of water vapour


and the values of the constant terms are:

   a0 = 1.581 23 × 10−6 K Pa−1

   a1 = −2.933 1 × 10−8 Pa−1

   a2 = 1.104 3 × 10−10 K−1 Pa−1

   b0 = 5.707 × 10−6 K Pa−1

   b1 = −2.051 × 10−8 Pa−1

   c0 = 1.989 8 × 10−4 K Pa−1

   c1 = −2.376 × 10−6 Pa−1

   d = 1.83 × 10−11 K2Pa−2

   e = −0.765 × 10−8 K2 Pa−2


Correction for variation in CO2 content

For a carbon dioxide (CO2) content of other than 0.04% by volume, the correct value of air density is obtained by multiplying the value obtained by assuming a CO2 content of 0.04% (400 ppm) by the correction factor

{1 + [0.4147(CO2 − 400)] × 10−6}

where CO2 is the CO2 content of the air in ppm.

Density of dry CO2-free air

The density in kg m−3 of dry air free from carbon dioxide may be derived from the BIPM expression referred to above, taking the mole fractions of CO2 and water vapour as zero. The expression then becomes

 ρ =

3.482 91 × 10−3 p

ZT

where

 Z =1 −  

p

[a0 + a1t + a2t2] + 

 p2 

T 

 T2 


Reduction of gaseous volumes to s.t.p.

The volume at s.t.p. (0°C and 1 atm) = vp/[101.325(1 + γt)], where v and t°C are the observed volume and temperature, p is the observed pressure in kPa (1 kPa = 10 mb) and γ is the coefficient of cubical expansion at constant pressure of the gas concerned. For ‘permanent’ gases γ = 0.003 67 ± 0.000 01 approximately. For coefficients of other gases, see section 3.5.

If it is desired to find the volume of dry gas at s.t.p., v and p being measured when the gas contains water vapour whose pressure is e kPa, p in the above expression must be replaced by (pe).

Volumetric calibration of vessels with water or mercury

The volume content (Vt) of a vessel in cm3 at the temperature of determination (t°C, liquid and air at the same temperature) and at a pressure of 1 atm (101.325 kPa) is given by

   Vt = (ILIE) f

where    IL is the balance reading of the vessel with liquid, in grams

      IE is the balance reading of the empty vessel, in grams

and    f is a factor including allowances for the density of the liquid and the appropriate buoyancy corrections, thus:

f  =
1 1 − ρa
ρwρa ρb

where ρw is the density of the liquid at t °C, in grams per cm3

   ρa is the density of the air, in grams per cm3

   ρb is the reference density of the weights used to calibrate the balance, in grams per cm3 (usually 8.0 g cm−3)


The following table gives values of f for water and mercury for a range of temperatures. Density tables are given in section 2.2.1 for water and mercury, and earlier in this section for air. The f values tabulated assume air of 50% relative humidity and unsaturated water; neither of these factors affects the tabulated values significantly.

Temperature of liquid, t/ºC.

10

11

12

13

14

15

  f

Water     .     .     .

1.001 39

1.001 49

1.001 59

1.001 71

1.001 84

1.001 98

Mercury .     .     . 

  0.073 685

  0.073 698

  0.073 712

  0.073 725

  0.073 738

  0.073 752

Temperature of liquid, t/ºC        .     .     .

16

17

18

19

20

  f   

Water       .     .     .     .     .     . 

1.002 13

1.002 30

1.002 47

1.002 66

1.002 86

Mercury    .     .     .     .     .     .

   0.073 765

   0.073 779

   0.073 792

   0.073 805

   0.073 819

Temperature of liquid, t/ºC        .     .     . 

21

22

23

24

25

  f  

Water       .     .     .     .     .     .

1.003 07

1.003 30

1.003 53

1.003 77

1.004 03

Mercury    .     .     .     .     .     . 

   0.073 832

   0.073 846

   0.073 859

   0.073 872

   0.073 886


   An additional buoyancy correction is required when weighings with water are made at air pressures other than 101.325 kPa (Kell, Whalley, 1975). The value of f (for water) must be increased ( + ), or decreased (−), by the following amounts (unit 0.000 01):

Pressure kPa (1 kPa = 10 mb)

97

98

99

100

101

102

103

104

105

Correction to f(water)   .       .

−4

−3

−2

−1

0

+1

+2

+3

+4

The additional buoyancy correction for mercury is insignificant. The above tables give the volume content of the vessel at the temperature of determination (t). At any other temperature (T) the volume VT is given by

VT = Vt(l + γ(Tt))

where γ is the coefficient of cubical expansion of the material of the vessel.

   Example. Let the apparent mass of water contained in a vessel at 10 °C, when weighed in air at 10 °C and pressure 102 kPa, be 100.00 g.

Then the volume of vessel at 10 °C is

   100.00(1.001 39 + 0.000 01) = 100.14 cm3

The same vessel, if made of glass (γ = 0.000 027 K−1, assumed), would contain at 20 °C

   100.14 (1 + 0.000 027 (20 − 10)) = 100.17 cm3

Comprehensive correction tables are given in BS 1797: 1987. Further information is given in BS 6696: 1986.

Measurement of density

The density of a solid or liquid specimen is normally found by determining in air the apparent mass of a particular volume of the specimen and the apparent mass of an equal volume of water. The latter is obtained in the case of a solid specimen by weighing it when immersed in water and thus measuring its apparent decrease in mass due to the water displaced, and in the case of a liquid by using the same container (bottle or pyknometer) for the water as for the liquid. The ratio, r, of the two apparent masses thus determined is approximately equal to the density of the specimen in g cm−3 (1 g cm−3 = 1 000 kg m−3); the true density is equal to r(dwda) + da, where dw is the density of the water at its observed temperature t°C (see section 2.2.1) and da is the density of the air (see Density of Ambient Air); the result will be in the same units as those used for dw and da. The correction to be applied to r at 20 °C and 1 atm is of the order of 1 part in 350 of the density. For temperatures ranging from 12 °C to 25 °C and pressures from 97 kPa (970 mb) to 104 kPa, the correction ranges over about 1 part in 400 of the density.

r is usually termed the relative density in air with reference to water, and is denoted by d t/t' in air, where t and t' are the temperatures of the material and water respectively (usually t = t'). The true relative density, viz. d t/t' in vacuo, is equal to r(1 − da) + da when da is expressed in g cm−3.

Hydrometers

   • Density hydrometers usually indicate density, kg m−3 (or g cm−3) at 20°C or 15°C when readings are taken at 20 °C or 15 °C respectively. In hot countries 27 °C may be used.
   • Relative density (or specific gravity) hydrometers usually indicate density at 60 °F relative to water at 60 °F (d 60/60 °F) when readings are taken at 60 °F.
   • Twaddle hydrometers indicate (d 60/60 °F – 1) 200 when readings are taken at 60 °F.
   • Baumé scales are related to relative density by various arbitrary formulae.

Temperature correction. To obtain from the reading R (density or relative density) of a soda-glass hydrometer, standard at t °C, the density or relative density d θ °C/t °C, of the liquid at the temperature θ °C of observation, subtract R × 0.000 025 (θ − t).

Surface tension. The surface tensions of aqueous solutions are often reduced considerably (20 mN m−1 or more) by contaminating films. Change of reading due to −1 mN m−1 change in surface tension = +4000/nld kg m−3, where n = density (kg m−3) or relative density × 1000 (i.e. ‘degrees’), l = length of scale (mm) corresponding to 10 kg m−3 or 10 ‘degrees’ relative density, and d = stem diameter (mm).

Further information on density hydrometers is given in BS 718:1991 and on relative density hydrometers in ISO 650.

References

BS 718:1991, A specification for density hydrometers, British Standards Institute.
BS 1797:1987, A schedule for tables for use in the calibration of volumetric glassware, British Standards Institute.
BS 6696:1986, A method for use and testing of volumetric glassware, British Standards Institute.
ISO 650, Relative density 60/60 degrees F hydrometers for general purposes, International Standards Organisation.
R. S. Davis (1992) Equation for the determination of the density of moist air (1981/91), Metrologia, 29, 67–70.
P. Giacomo (1982) Equation for the determination of the density of moist air (1981), Metrologia, 18, 33–40.
G. S. Kell and E. Whalley (1975) Reanalysis of the density of liquid water in the range 0–150 °C and 0–1 kbar, Journal of Chemical Physics 62(9), 3496–3503.

Further information is given in the following volumes in the series ‘NPL Notes on Applied Science’, regrettably now out of print:

No. 6   Volumetric glassware (1957)
No. 7   Balances, weights and precise laboratory weighing (1962)
No. 25 Hydrometers and hydrometry (1961)

D.R.Armitage

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