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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

where d is the density of the substance, d_s is the density of the mass standards and d_a 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 d_s. The following table has been prepared on this basis, d_a being taken to be 1.2 kg m⁻³.

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

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:

Where   p is the air pressure in pascals

           M_a 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

           X_v is the mole fraction of water vapour

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

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

      R = 8.314510 J mol⁻¹ K⁻¹

     M_v = 18.015 × 10⁻³ kg mol⁻¹

     M_a = 28.963 5 × 10⁻³ kg mol⁻¹

Thus the expression reduces to

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

   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

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

 p_sv = 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

   where p is the ambient pressure in pascals

     T is the temperature in Kelvin

     t is the ambient temperature in degrees Celsius

     x_v 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

where

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 (p – e).

Volumetric calibration of vessels with water or mercury

The volume content (V_t) 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

   V_t = (I_L− I_E) f

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

      I_E 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
		ρw − ρa				ρb

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

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

   ρ_b 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):

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 V_T is given by

V_T = V_t(l + γ(T − t))

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(d_w − d_a) + d_a, where d_w is the density of the water at its observed temperature t°C (see section 2.2.1) and d_a is the density of the air (see Density of Ambient Air); the result will be in the same units as those used for d_w and d_a. 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 − d_a) + d_a when d_a 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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