A twenty-first century molar mass for dry air.
Researchers, practitioners, and educators in the fields of agricultural and food science engineering, air conditioning, atmospheric physics, drying and dehumidification, gas turbines, compressors and expanders, meteorology, psychrometrics, and standards make numerous psychrometric (moist air) calculations that are based in part on the molar mass of dry air. Dry air is a mixture of nitrogen, oxygen, argon, [CO.sub.2], and eight or more minor constituents called trace gases. The molar mass of dry air is calculated as the sum of the products of the mole ratio of each gas times its molar mass.
In the last half of the twentieth century, the following changes took place that resulted in an increase in the molar mass of dry air:
* The scientific community changed from the Oxygen-16 to the Carbon-12 reference for the molar mass of elements and compounds in 1960.
* The molar masses of the basic chemical elements were updated by the International Union of Pure and Applied Chemistry (IUPAC) (Wieser 2005).
* [CO.sub.2] in the atmosphere has increased from 314 [[mu]mol.[mol.sup.-1]]] (~1955) to 379 [[mu]mol.[mol.sup.-1]] (Keeling and Whorf 2005a, 2005b). The 65 [[mu]mol.[mol.sup.-1]] increase in [CO.sub.2] in this time span is accompanied by a decrease in [O.sub.2] because combustion and respiration processes combine a carbon atom with [O.sub.2] from the atmosphere to produce [CO.sub.2] (Park et al. 2004).
* The stated argon mole fraction in air has changed from 9340 [[mu]mol.[mol.sup.-1]] at the start of the twentieth century to 9170 [[mu]mol.[mol.sup.-1]] at mid-century to 9332 [[mu]mol.[mol.sup.-1]] (Park et al. 2004). NASA apparently did not accept the mid-century value, as the U.S. Standard Atmosphere 1976 document (NOAA/NASA 1976) used the 9340 [[mu]mol.[mol.sup.-1]] value.
* With more accurate calculations and data, the scientific community has revised their best estimate of the universal gas constant from 8.31441 [J.[mol.sup.-1].[K.sup.-1]] to 8.314510 and finally to [8.314472 (1545.349 ft.lb.[mol.sup.-1].[degrees][R.sup.-1])], which is the value recommended by the Committee on Data for Science and Technology (of the International Council for Science headquartered at Paris, France) (Mohr and Taylor 1999) and is now officially listed by IUPAC and the National Institute of Standards and Technology (NIST).
A sampling of dry-air molar mass values used by researchers over the last 30 years is shown in Figure 1. Different values result from different assumed or measured compositions for atmospheric air. Some models of Earth's atmosphere include [CO.sub.2] and others do not. [CO.sub.2] is the fourth most abundant atmospheric gas and is currently increasing at an annual rate of approximately [1.9 [mu]mol.[mol.sup.-1]/year] (Keeling and Whorf 2005a, 2005b). The current rate of increase in [CO.sub.2] results in an increase in the molar mass of dry air of 0.0001 [kg.[kmol.sup.-1]] (lb.[lbmol.sup.-1]) every four to five years. (Note: Throughout the rest of this paper, the units associated with molar mass values have been omitted.)
[FIGURE 1 OMITTED]
The NIST Boulder (Lemmon et al. 2000; Lemmon and Jacobsen 2004) and the VDI-4670 (VDI 2003) models of dry air use similar but not identical atmospheric air models made up of the three gases: nitrogen, oxygen, and argon. Neither contain [CO.sub.2]. Most of the remaining models include 314 [[mu]mol.[mol.sup.-1]] of [CO.sub.2], which is representative of the middle of the twentieth century.
REASONS FOR UPDATING THE MOLAR MASS OF DRY AIR
The effect of using an [M.sub.da] value of 28.966 [kg.[kmol.sup.-1]] (lb.[lbmol.sup.-1]) for psychrometric calculations will be small, e.g., changes in property and process calculations will be less than 0.5%. Nevertheless, a consistent rational value will be beneficial.
Students in meteorology, thermodynamics, fluid flow, metrology, and other fields should appreciate a consistent rational value based on the latest composition of the atmosphere.
Researchers and others who develop mathematical models of the atmosphere or psychrometric processes can update the majority of their code except for certain functions or routines that use equations curve fit to measured property data. If functions or routines that use these curve-fit equations include either (1) the gas constant for dry air, (2) the molar mass of dry air, or (3) a ratio based on [M.sub.da], then the function or routine should not be changed because such a change will result in a change in the value of the underlying measured property data, which obviously creates errors in subsequent calculations. Researchers and programmers in the period through the year 2058 will appreciate having a standard value for the molar mass of dry air.
Researchers who update real moist air psychrometric models for air-conditioning, drying, combustion turbine, and meteorology processes will see a small benefit in the comparison and validation of new models when both models use the same value for the molar mass of dry air.
Logic is perhaps the best reason for using a standard value for the twenty-first century molar mass of dry air. The composition of dry air is accurately known, as are values for the molar mass of its constituents.
CALCULATING THE MOLAR MASS OF DRY AIR
The molar mass of dry air is calculated as the sum of the products of the mole ratio of each gas times its molar mass. It is a simple matter to access the IUPAC or NIST Web sites and obtain the molar mass of each constituent of atmospheric air. It is more difficult to find an up-to-date table listing the abundance of each constituent gas in the atmosphere.
Hundreds of textbooks and Web pages provide tables listing the composition of dry air. Unfortunately, few of these provide a source reference or the year for which the abundance values were determined. The reference year is important for [CO.sub.2] and [O.sub.2] because [CO.sub.2] is increasing at the expense of [O.sub.2]. It is also important because of the apparent erroneous value of argon reported in the mid-twentieth century.
Fortunately, a straightforward and accurate means is available for determining the composition of the atmosphere if one has a value for the past, present, or projected abundance of [CO.sub.2]. The underlying reasoning for this statement is covered in the following paragraphs, after which the abundance of [CO.sub.2] will be addressed.
Instead of tediously listing 14 or more components of atmospheric dry air, the molar mass of dry air can be evaluated as the sum of three sets of components with a total of [1.00000 mol.[mol.sup.-1]]:
Two reacting gases, [O.sub.2] and [CO.sub.2] [[psi].sub.O2] + [[psi].sub.CO2] = 0.20982 mol.[mol.sup.-1] (1)
Two inert gases, [N.sub.2] and Ar [[psi].sub.N2] + [[psi].sub.Ar] = 0.79015 mol.[mol.sup.-1] (2)
Trace gases [[psi].sub.Trace] = 0.00003 mol.[mol.sup.-1] (3)
Equations 1 and 2 are presented in Park et al. (2004). They differ slightly from earlier Comite International des Poids et Mesures (International Committee for Weights and Measures) 81/91 equations (Giacomo 1981; Davis 1992), which used (a) [[psi].sub.Ar] = 0.00914, (b) [SIGMA] ([[psi].sub.N2] + [[psi].sub.Ar]) = 0.79018, and (c) [summation] ([[psi].sub.O2] + [[psi].sub.CO2]) = 0.20979.
The resulting 2008 composition of dry air and the calculation of [M.sub.da] are shown in Table 1, which includes the molar mass of each gas, its abundance, and its contribution to [M.sub.da]. The 2008 composition uses the mean 2004 National Oceanic and Atmospheric Administration Mauna Loa [CO.sub.2] concentration (Keeling and Whorf 2005a, 2005b), increased by four years times 1.9 [[mu]mol.[mol.sup.-1]], the Park et al. (2004) value of 9332 [[mu]mol.[mol.sup.-1]] for Ar, and the resulting values of [N.sub.2] and [O.sub.2] from Equations 1 and 2. The individual trace gas abundances are from Table 1 of Park et al. (2004). The results based on IUPAC molar masses (Wieser 2005) are displayed in Table 1. The results are displayed with more significant digits than is customary so that readers may check their own tables or calculations.
Table 1. The 2008 Composition of Dry Air and Calculation of [M.sub.da] Constituent Molar Mole Fraction Contribution Mass M [[psi].sub.i] (M.[psi].sub.i]) kg/kmol [[psi].sub.i]. kg/kmol (lb/lbmol) [10.sup.6] (lb/lbmol) Two Inert Gases and Two Reacting Gases [N.sub.2] 28.0134 780818 21.873367 (1) [O.sub.2] 31.9988 209435 6.701669 (2) Ar (1) 39.9480 9332 0.372795 [CO.sub.2] 44.0100 385 0.016944 (2) Subtotals 999970 28.964774 Eight Trace Gases Ne 20.1797 18.2 0.000367 He 4.002602 5.2 0.000021 [CH.sub.4] 16.04246 1.5 0.000024 Kr 83.798 1.1 0.000092 [H.sub.2] 2.01588 0.5 0.000001 [N.sub.2]O 44.0128 0.3 0.000013 CO 28.0101 0.2 0.000006 Xe 131.293 0.1 0.000013 Subtotals 27.1 0.000537 Trace to 30 19.8254 2.9 0.000057 [mu]mol [mol.sup.-1] Grand Totals 1000000 28.965369 Notes: (1.) [[psi].sub.N2] + [[psi].sub.Ar] = 790150 [[mu]mol.[mol.sup.-1]]; "Two Inert Gases." (2.) [[psi].sub.O2] + [[psi].sub.CO2] = 209820 [[mu]mol.[mol.sup.-1]]; "Two Reacting Gases." (3.) Mean molar mass of the eight trace gases is 19.8254. (4.) Adding 2.9 [[mu]mol.[mol.sup.-1]] of trace gases brings the total of all gases to unity. (5.) 30 [[mu]mol.[mol.sup.-1]] trace gases times 19.8254 = 0.0005948.
Note that the 999970 [mu]mol.[mol.sup.-1] total of the first four gases plus the 27.1 [mu]mol.[mol.sup.-1] of trace gases does not equal unity. In order to reach unity, 2.9 [mu]mol.[mol.sup.-1] of trace gases have been added to the mean molar mass of the eight trace gases.
The total contribution of the trace gases (including the 2.9 [[mu]mol.[mol.sup.-1] adjustment) to the molar mass of dry air should be viewed with proper perspective. The total contribution of the trace gases to [M.sub.da] is 0.0006. Readers may not agree with adding the 2.9 [[[mu]mol.[mol.sup.-1]] to the trace gases and may also suggest changes to the abundance of one or more of the trace gases. Some may want to include other trace gases, such as chlorofluorocarbons and hydrochlorofluorocarbons; however, the abundances of these are significantly less than the abundance of xenon and will not impact [M.sub.da]. The perspective is that it is doubtful that any changes to the abundance of the trace gases will change their total contribution to [M.sub.da] from the [0.0006 kg.[kmol.sup.-1]] (lb.[lbmol.sup.-1]) value.
Equation 1 is solved for [[psi].sub.O2] using the locally measured abundance of [CO.sub.2] or the NOAA Mauna Loa observatory mean measured concentration of [CO.sub.2] (Figure 2). The 2008 [CO.sub.2] value is 385 [[[mu]mol.[mol.sup.-1] ]([[psi].sub.CO2] = 0.000385). The concentration of [CO.sub.2] has been measured for more than 50 years at Mauna Loa and agrees within 2 [[mu]mol.[mol.sup.-1] with nine other stations in the Oak Ridge National Laboratory Carbon Dioxide Information Analysis Center Scripps Institution of Oceanography (La Jolla, California) sampling network (Keeling and Whorf 2005a, 2005b). Equation 1 assumes that increases in [CO.sub.2] result from combustion and respiration processes, and the increase is perfectly correlated by a decrease in [O.sub.2] (Park et al. 2004).
[FIGURE 2 OMITTED]
[CO.sub.2] increases in the colder months in the northern hemisphere due to an increase in combustion processes for heating combined with reduced photosynthesis from plants; [CO.sub.2] decreases in the warmer months when these effects are reversed. The southern hemisphere has similar cycles, but they are offset by six months. This paper uses the computed annual mean abundance of [CO.sub.2]. Some research may require the actual abundance of [CO.sub.2] at the site.
Equation 2 is solved for [[psi].sub.N2] using 0.009332 [kmol.[kmol.sup.-1]] for [[psi].sub.Ar] determined by Park et al. (2004) at the Korea Research Institute of Standards and Science (KRISS), Daejeon, Korea, from samples at Anmyeon Island, Korea, and at Niwot Ridge, Colorado (collected by NOAA, Boulder, Colorado). Subsequently, Sutour et al. (2006) at Laboratoire National de Metrologie et d'Essais (LNE), Paris, France, determined a mole fraction in atmospheric air of 0.009323 for argon.
Equation 3 is based on the eight trace gases listed in Table 1 in Park et al. (2004), which agrees closely with Harrison (1965) and which was used by Hyland and Wexler (1983) and Hyland et al. (1983). The sum of the [[psi].sub.Trace] gases from this table is 27.1 [[mu]mol.[mol.sup.-1]]. The total contribution to the molar mass of dry air is 0.000537, which, divided by the 27.1 [[mu]mol.[mol.sup.-1]], gives a mean value for the molar mass of the trace gases of 19.8254. The third equation has been rounded up to 30 [[mu]mol.[mol.sup.-1]] (0.00003 mol.[mol.sup.-1]) so that the sum of the three sets of components is unity.
Tables or lists of atmospheric abundances frequently do not sum to unity. The data source for this paper leaves 2.9 [[mu]mol.[mol.sup.-1] unaccounted. This small deviation may be due to rounding, other very minor trace elements, a combination of the previous two causes, or some other cause. In order to bring the total abundance to unity, the 2.9 [[mu]mol.[mol.sup.-1] must be multiplied by a representative molar mass. This paper uses the weighted molar mass of the eight trace components, and this results in an [M.sub.da] increase of 0.000057 kg.[kmol.sup.-1] (lb.[lbmol.sup.-1]). Another alternative is to use the weighted molar mass of the three major components (28.9647831 kg.[kmol.sup.-1] [lb.[lbmol.sup.-1]]), which results in an [M.sub.da] increase of 0.000084 kg.[kmol.sup.-1] (lb.[lbmol.sup.-1]), which adds an additional 0.000027 kg.[kmol.sup.-1] (lb.[lbmol.sup.-1]). This last increase is equivalent to the increase of an additional 2.25 [[mu]mol.[mol.sup.-1]] of [CO.sub.2], which is slightly greater than the current 1.9 [[mu]mol.[mol.sup.-1]] annual increase of [CO.sub.2] in the atmosphere. This change does not alter the recommended value of [M.sub.da] for the first half of the twenty-first century.
As shown in Table 2, this same information can be shown in compacted form without loss of accuracy by lumping the trace gases into a single line. This demonstrates the tiny effect that trace gases have on [M.sub.da]. Focus can then be concentrated on the two inert gases and the two reacting gases.
Table 2. The 2008 Composition of Dry Air and Calculation of [M.sub.da] Constituent Molar Mole Fraction Contribution Mass M [[psi].sub.i] (M .[[psi].sub.i]) kg/kmol [[psi].sub.i] kg/kmol (lb/lbmol) . (lb/lbmol) [10.sup.6] [N.sub.2] 28.0134 780818 21.873367 (1) [O.sub.2] 31.9988 209435 6.701669 (2) Ar (1) 39.9480 9332 0.372795 [CO.sub.2] 44.0100 385 0.016944 (2) 8 trace 19.8254 30 0.000595 gases Grand 1000000 28.965369 Totals Notes: (1.) [[psi].sub.N2] + [[psi].sub.Ar] = 790150 [[mu]mol.[mol.sup.-1]; "Two Inert Gases." (2.) [[psi].sub.O2] + [[psi].sub.CO2] = 209820 [[mu]mol.[mol.sup.-1]; "Two Reacting Gases."
A SIMPLE (BUT NOT SIMPLISTIC) EQUATION
It is highly probable that the concentrations of Ar and the eight minor trace gases will remain as shown in the tables. With the concentration of Ar fixed, this also sets the concentration of [N.sub.2]. Based on these assumptions, Equation 4 gives exactly the same results as the table or spreadsheet calculations above:
[M.sub.da] = 28.960745 + [[CO.sub.2],.sub.[[mu]mol.[mol.sup.-1] *] 12.0107 / 1000000 (4)
Equation 4 or table calculations with [CO.sub.2] set to zero result in a molar mass of dry air of 28.960745 kg.[kmol.sup.-1] (lb.[lbmol.sup.-1]). The second term of the equation converts [CO.sub.2], [[mu]mol.[mol.sup.-1] to mole fraction [CO.sub.2], which is multiplied by 12.0107 (the IUPAC value for carbon). Some may challenge the use of 12.0107 rather than 44.0095; however, the equation is correct as written. The explanation is based on the fact that in combustion or respiration, carbon combines with [O.sub.2] to produce [CO.sub.2]. Every added molecule or mole fraction of [CO.sub.2] causes an offsetting reduction in the mole fraction of [O.sub.2]. The total amount of [O.sub.2], whether in the form of [O.sub.2] or the [O.sub.2] in [CO.sub.2], is already accounted for in the table or Equation 4, and, as a consequence, it is only necessary to account for the change in the abundance of carbon.
RECOMMENDED TWENTY-FIRST CENTURY VALUE FOR THE MOLAR MASS OF DRY AIR
An [M.sub.da] value of 28.966 kg.[kmol.sup.-1] (lb.[lbmol.sup.-1]) is suggested for the fields of meteorology, drying and dehumidification, air conditioning, gas turbines, compressors and expanders, and agricultural and food science engineering. When the computed value of [M.sub.da] is rounded to three decimal places, the value 28.966 kg.[kmol.sup.-1] (lb.[lbmol.sup.-1]) is correct through 2058, assuming that the future increase of [CO.sub.2] is 1.9 [[mu]mol.[mol.sup.-1] per year. The exact value of 28.9660 kg.[kmol.sup.-1] (lb.[lbmol.sup.-1]) is projected to occur in 2036. If the local abundance of [CO.sub.2] is significantly different than the current mean Mauna Loa value, then researchers modeling some processes may wish to test the effect of the difference in [CO.sub.2].
Designers of air-conditioning and dehumidification systems for high-occupancy interior spaces who desire extreme accuracy should evaluate the effect of increased levels of [CO.sub.2], e.g., a 771 [[mu]mol.[mol.sup.-1]] [CO.sub.2] abundance results in an [M.sub.da] value of exactly 28.97 kg.[kmol.sup.-1] (lb.[lbmol.sup.-1]) and an [[M.sub.wv] / [M.sub.da]] value of 0.6218594.
[M.sub.da] IN METEOROLOGY AND PSYCHROMETRICS
The value for the molar mass of dry air occurs in equations for the calculations of density, specific volume, enthalpy, and entropy, and the conversion between properties expressed in molar units, mass-based units, and [mass.sub.da]-based units. [M.sub.da] appears in many equations in combination with the molar mass of [H.sub.2]O. The scientific community has standardized on the Vienna Standard Mean Ocean Water (of the International Atomic Energy Agency) isotopic composition of water, which results in a molar mass of 18.015268. The ratio [M.sub.wv] / [M.sub.da] is found in the calculation of the humidity ratio using the equation W = ([M.sub.wv] / [M.sub.da]) . [[P.sub.wv] / ([P.sub.bar] - [P.sub.wv])]. That ratio (18.015268 / 28.966) equals 0.6219453, and it occurs so often that some give it the symbol [epsilon], the lowercase Greek symbol for epsilon. Readers may also recognize the reciprocal of [epsilon] that equals 1.60786, the quantity (1 - [epsilon]) that equals 0.37805, and the quantity (1 / [epsilon] - 1) that equals 0.60786. This latter value rounded to 0.61 appears in many equations used by meteorologists and scientists in the field of atmospheric physics.
A twenty-first century value for the molar mass of dry air of 28.966 kg.[kmol.sup.-1] (lb.[lbmol.sup.-1]) is recommended for most psychrometric calculations. This value is supported by Park et al. (2004) and Sutour et al. (2006) measurements and accurate determinations of the abundance of argon in the troposphere and by the new equations for the inert gases ([N.sub.2] and Ar) and the reacting gases ([O.sub.2] and [CO.sub.2]) (Park et al. 2004).
Trace gases should not be ignored; however, lumping all trace gases into a single line entry focuses attention on the four most abundant atmospheric constituents. The 0.0006 [kg.[kmol.sup.-1]] (lb.[lbmol.sup.-1]) contribution of the eight or more trace gases to [M.sub.da] is unlikely to change even with new determinations of the abundance of one or more of the trace gases.
SYMBOLS AND ABBREVIATIONS
Ar = argon
[CH.sub.4] = methane
CO = carbon monoxide
[CO.sub.2] = carbon dioxide
[epsilon] = epsilon
[H.sub.2] = hydrogen
He = helium
Kr = krypton
[M.sub.da] = molar mass of dry air, kg [kmol.sup.-1] (lb [lbmol.sup.-1])
[M.sub.H2O] or [M.sub.wv] = molar mass of [H.sub.2]O, kg [kmol.sup.-1] (lb [lbmol.sup.-1])
[N.sub.2] = nitrogen
[N.sub.2]O = nitrous oxide
Ne = neon
[O.sub.2] = oxygen
[P.sub.bar] = barometric pressure
[P.sub.wv] = partial pressure of water vapor
W = humidity ratio, [kg.sub.wv].[kg.sub.da.sup.-1] ([lb.sub.wv].[lb.sub.da.sup.-1])
Xe = xenon
[psi] = mole fraction
Note: [P.sub.bar] and [P.sub.wv] units must be identical
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Received February 14, 2008; accepted June 9, 2008
Donald P. Gatley, PE
Student Member ASHRAE
Hans-Joachim Kretzschmar, Dr.-Ing.
Donald P. Gatley is a retired consulting engineer, Atlanta, GA. Sebastian Herrmann is a mechanical engineering graduate from Zittau/Goerlitz University of Applied Sciences and is currently a PhD candidate and research assistant at the University of Rostock, Germany. Hans-Joachim Kretzschmar is a professor of technical thermodynamics at Zittau/Goerlitz University of Applied Sciences, Zittau, Germany.
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|Author:||Gatley, Donald P.; Herrmann, Sebastian; Kretzschmar, Hans-Joachim|
|Publication:||HVAC & R Research|
|Date:||Sep 1, 2008|
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