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Unless otherwise stated this page contains Version 1.0 content (Read more about versions) 2.1 Measurement of mass, pressure and other mechanical quantities2.1.1 Mass, volume and densityCorrection 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% CO_{2} 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
reworking 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.
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^{−8}p + 5.6 × 10^{−7} t^{2} 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 × 10^{3} 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: a_{0} = 1.581 23 × 10^{−6} K Pa^{−1} a_{1} = −2.933 1 × 10^{−8} Pa^{−1} a_{2} = 1.104 3 × 10^{−10} K^{−1} Pa^{−1} b_{0} = 5.707 × 10^{−6} K Pa^{−1} b_{1} = −2.051 × 10^{−8} Pa^{−1} c_{0} = 1.989 8 × 10^{−4} K Pa^{−1} c_{1} = −2.376 × 10^{−6} Pa^{−1} d = 1.83 × 10^{−11} K^{2}Pa^{−2} e = −0.765 × 10^{−8} K^{2} Pa^{−2} Correction for variation in CO_{2} content For a carbon dioxide (CO_{2}) content of other
than 0.04% by volume, the
correct value of air density is obtained by multiplying the value obtained by
assuming a CO_{2} content of 0.04% (400 ppm) by the correction
factor {1 + [0.4147(CO_{2} − 400)] ×
10^{−6}} where CO_{2} is the CO_{2} content of the air in
ppm. Density of dry CO_{2}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 CO_{2} 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 cm^{3} 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:
where ρ_{w} is the density of the liquid at t °C, in grams per cm^{3} ρ_{a} is the density of the air, in grams per cm^{3} ρ_{b} is the reference density of the weights used to calibrate the balance, in grams per cm^{3} (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.
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 cm^{3} 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 cm^{3} 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. Temperature correction. To obtain from the reading R (density or relative density) of a sodaglass 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. 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) D.R.Armitage 
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