WP 1
Project Info 4
Gas mixture composition: Conversions
The composition of a gas mixture is determined as
qualitative composition (determined by defined
analytes in the gas mixture and the
complementary gas that is matrix gas) or
quantitative composition (composition of the matrix
gas and all analytes are quantitatively known). The
matrix gas can be for example air if a gas mixture
in air is studied. Information about the gas
composition is needed when synthetic gas
mixtures are produced or unknown gas mixtures
are analysed.
of moles of the gas mixture is the sum of moles of
analytes in the gas mixture.
The gas mixture composition is expressed as
mole, mass, or volume fractions as well as mole,
mass, or volume concentrations. Conversions of
gas mixtures are based on ideal gas law at the
simplest case. Dependence among the ideal gas
volume, temperature and pressure has to be
considered in accurate calculations. Therefore the
amount of analytes in the gas mixture depends on
state of the gas mixture and analytes.
where
1 Fractions
Element contents are often presented as mole,
mass, or volume fractions in gas production.
1.1 Mole fraction xi
The mole fraction of an analyte i in gas mixture is
solved by dividing the number of moles of the
analyte i by the total number of moles of the gas
mixture.
𝑥𝑖 =
where
xi
ni
ns
𝑛𝑖
𝑘 𝑛𝑘
=
𝑛𝑖
𝑛𝑆
,
(1)
mole fraction of the analyte i in the
gas mixture [-],
number of moles of the analyte i
[mol],
total number of moles of the gas
mixture [mol].
Mole fractions are independent of pressure or
temperature of the gas mixture. The total number
1.2 Mass fraction wi
If masses of analytes are m1, m2, …, mN, the mass
fraction of the analyte in the gas mixture is
calculated by the equation (2).
𝑤𝑖 =
wi
mi
mS
𝑚𝑖
𝑚𝑖
=
,
𝑚𝑆
𝑘 𝑚𝑘
(2)
mass fraction of the analyte i in the
gas mixture [-],
mass of the analyte i [kg],
total mass of the gas mixture [kg].
1.3 Volume fraction ϕi
The volume fraction of the analyte i in the gas
mixture is calculated by dividing the volume of the
analyte by the sum of component volumes in the
gas mixture.
𝜙𝑖 =
𝑉𝑖
,
𝑘 𝑉𝑘
3
The volume fraction of the gas mixture depends on
temperature and pressure, therefore pressure and
temperature have to be defined.
2 Concentration
Results of an analysis are usually presented as
mole, mass, or volume concentrations. The
concentration of the gas depends on temperature
and pressure, therefore they have to be defined.
2.1 Mole concentration ci
The mole concentration is calculated by the
equation (4).
𝑐𝑖 =
𝑛𝑖
,
𝑉𝑆
(4)
where
ci
VS
mole concentration of the analyte i
in the gas mixture [mol / m3],
total volume (sample volume) of the
gas mixture in the specific pressure
and temperature [m3].
2.2 Mass concentration βi
The mass concentration of the analyte i in the gas
mixture is calculated by the equation (5).
𝛽𝑖 =
where
βi
𝑚𝑖
,
𝑉𝑆
(5)
𝑁
𝑚𝑆 =
𝑚𝑘 .
𝑘=1
3.3 Volume of gas mixture Vs
Although mass or moles of the gas mixture are sum
of masses or moles of analytes in the gas mixture,
is the total volume of the gas mixture not additive
and it is not precisely sum of volumes of analytes. If
in the gas mixture there are analytes 1, 2, …, N
and their volumes are V1, V2, …, VN, can the total
volume of the gas mixture be calculated by the
equation (9).
mass concentration of the analyte i
[kg/m3].
𝑁
𝑉𝑆 = 𝑓𝑆
2.3 Volume concentration si
where
si
Vi
𝑉𝑖
,
𝑉𝑆
𝑉𝑘 ,
(9)
𝑘=1
The volume concentration of the analyte i is
calculated by the equation (6), if the volume of the
analyte in the gas mixture is Vi.
𝜎𝑖 =
8
The mixing factor (fs) of the gas mixture can be
defined in the specific temperature and pressure by
dividing the total volume of the gas mixture by the
sum of volumes of each analytes in the gas mixture
before mixing.
(6)
volume concentration of the analyte
i [m3/m3],
volume of the analyte i in the gas
mixture in the specific pressure and
temperature [m3].
The volume concentration, like the volume fraction,
(weakly) depends on temperature and pressure,
therefore the state of the gas mixture and its
components has to be considered in accurate
conversions.
The mixing factor in most of gas mixtures can be
ignored in normal air pressure and normal room
temperature (fS ≈ 1).
3.4 Molar mass of gas mixture Ms
The number of moles of a compound can be
calculated by equation (10).
𝑛=
where
𝑚
,
𝑀
M
3 Gas mixture
If in the sample S of the gas mixture there is N
number of gas analytes, the number of the mole of
the gas mixture is the total number of moles of
analytes.
𝑁
𝑁
𝑀𝑆 =
MS
(7)
𝑘=1
3.2 Mass of the gas mixture ms
The mass of the gas mixture is the sum of masses
of each analyte in the gas mixture.
𝑥𝑖 𝑀𝑖 ,
(11)
𝑖=1
where
𝑛𝑘 .
molar mass of a pure gas or a gas
mixture [mol/kg].
The molar mass of a pure compound is calculated
by its elemental composition. The molar mass of
the gas mixture is calculated by the equation (11).
3.1 Number of mole of gas mixture ns
𝑛𝑆 =
(10)
Mi
molar mass of the gas mixture
(apparent molar mass) [kg/mol],
molar mass of the analyte i in the
gas mixture [kg/mol].
If the composition of the gas mixture is expressed
as mass fractions, molar mass of the gas mixture is
calculated by the equation (12).
1
=
𝑀𝑆
𝑁 𝑤𝑖
𝑖=1 𝑀
𝑖
,
Respectively:
1
𝑀𝑆 = 𝑓
𝑆
where
𝑁
𝑖=1 𝜙𝑖 𝑀𝑖
,
(13)
mixing factor (≈ 1),
volume fraction of the analyte i in
the gas mixture.
fS
fi
3.5 Molar volume Vm
The molar volume of a pure gas or a gas mixture in
the specific reference state is calculated by the
equation (14).
𝑉𝑚 =
where
V
Z
(12)
𝑛
,
𝑉
The compression factor describes the difference
between a real gas or a gas mixture and an ideal
gas. The factor can be defined as ration of volumes
of the real gas and the ideal gas at the same
conditions.
𝑍=
8.3144621
R  Tref
Vm 

Z  pref
J
mol  K
 ( 273.15 0)  K
1.0 1.0 atm
L
 22.414
mol
3.6 Gas density r
𝑍𝑆 = 𝑓𝑆
where
V
volume of gas or gas mixture in the
specific temperature and pressure
[m3].
3.7 General equation of state
The equation of state for real gas is expressed by
equation (16).
𝑛=
where
p
T
R
𝑝𝑉
,
𝑍𝑅𝑇
(16)
pressure of the gas or the gas
mixture [Pa],
absolute temperature of the gas or
the gas mixture [K],
molar gas constant (=8,3144621
J/(mol K) ),
𝑥𝑖 𝑍𝑖 ,
(18)
𝑖=1
where
Zi
compression factor of the analyte i
in the gas mixture.
In most cases the compression factor of the gas
mixture is considered as one and compression
factor of the analyte i is approximated by using its
virial coefficients.
𝑍𝑖 = 𝑍𝑖 𝑝, 𝑇 ≈ 1 + 𝐵𝑖′ 𝑇 𝑝,
where
(15)
(17)
𝑁
The density of gas or gas mixture is calculated by
the equation (15).
𝑚
𝜌= ,
𝑉
𝑉𝑟𝑒𝑎𝑙
𝑛𝑍𝑅𝑇 𝑝
=
.
𝑉𝑖𝑑𝑒𝑎𝑙
𝑛𝑅𝑇 𝑝
The compression factor of the ideal gas is defined
as one (Z=1). The compression factor of most pure
gases differ only slightly from one in normal room
temperature and air pressure, so in most cases Z ≈
1. The compression factor of the gas or the gas
mixture is calculated by the equation (18) by using
compression factors of analytes in the gas mixture
and composition of the gas mixture.
volume of gas or gas mixture in
the reference state (pref, Tref) [m3].
For example the molar volume of an ideal mixture
composed by ideal gases (1 atm, 0 °C) is
calculated by the equation of state as follows:
compression factor of the gas or the
gas mixture.
(19)
𝑍𝑖 𝑝, 𝑇 compression factor of the analyte i
in temperature T and pressure p.
𝐵𝑖′ 𝑇 is obtained from the series development of
compression factor as for pressure. In the appendix
C of the Standard EN-ISO 14912:2006 is
presented virial coefficients for most common
gases.
3.8 Dependency of variables M, Z, ρ
Dependency between variables M, Z, ρ is
expressed in the equation (20) which is obtained by
combining equations (10), (15) and (16).
𝜌∙𝑍=
𝑀∙𝑝
,
𝑅∙𝑇
(20)
From the equation (20) is noticed that maximum
two variables can be independent of other
variables. Conversion factors are presented by
known variables M and Z in Standard EN ISO
14912:2006. Some other variables are used in
some other cases. For example calculating of a
flue gas composition is based of conversion factors
that are calculated with variables M and ρ.
𝜎𝑖 𝑝𝑟𝑒𝑓 𝑇𝑟𝑒𝑓
𝑍𝑖 𝑝𝑟𝑒𝑓 , 𝑇𝑟𝑒𝑓
𝑍𝑆 𝑝, 𝑇
=
∙
𝑍𝑆 (𝑝𝑟𝑒𝑓 , 𝑇𝑟𝑒𝑓 )
𝑍𝑖 𝑝, 𝑇
4 State conversions
𝜙𝑖 𝑝𝑟𝑒𝑓 𝑇𝑟𝑒𝑓 =
𝑍𝑆 𝑝, 𝑇
State conversions are used to convert in the
specific state defined variables which are
presented in chapters 1 and 2 to similar variables
in another state.
𝑍𝑆 𝑝𝑟𝑒𝑓 , 𝑇𝑟𝑒𝑓
where
4.1 Conversion to a reference state
The mole fraction and the mass fraction are only
variables that do not depend on the temperature or
the pressure of the gas. Conversion of other
variables from the initial state (p, T) to the
reference state (pref, Tref) is based on the
conversion of the volume of the gas mixture to
reference conditions. Conversion factors used in
state conversions are derived from the general
equation of state (16).
𝑝𝑉
𝑍𝑅𝑇
=
𝑉𝑟𝑒𝑓 =
where
𝑝𝑟𝑒𝑓 𝑉𝑟𝑒𝑓
𝑍𝑟𝑒𝑓 𝑅 𝑇𝑟𝑒𝑓
𝑝
∙
𝑝𝑟𝑒𝑓
𝑝
𝑝𝑟𝑒𝑓
𝑇𝑟𝑒𝑓
𝑇
𝑍(𝑝𝑟𝑒𝑓 ,𝑇𝑟𝑒𝑓 )
⇒
𝑇𝑟𝑒𝑓
𝑍(𝑝𝑟𝑒𝑓 , 𝑇𝑟𝑒𝑓 )
⋅
∙ 𝑉,
𝑇
𝑍(𝑝, 𝑇)
(21)
correction factor for pressure,
correction
factor
for
temperature,
correction
𝑍(𝑝,𝑇)
factor
for
compression factor.
Conversion factors are used to convert variables
from the initial state to the reference state.
𝑥𝑖 (𝑝𝑟𝑒𝑓 𝑇𝑟𝑒𝑓 ) = 𝑥𝑖 (𝑝, 𝑇),
(22)
𝑤𝑖 𝑝𝑟𝑒𝑓 𝑇𝑟𝑒𝑓 = 𝑤𝑖 𝑝, 𝑇 ,
(23)
𝑐𝑖 (𝑝𝑟𝑒𝑓 𝑇𝑟𝑒𝑓 )
𝑝𝑟𝑒𝑓
𝑇
𝑍𝑆 𝑝, 𝑇
=
𝑐 (𝑝, 𝑇), (24)
𝑝
𝑇𝑟𝑒𝑓 𝑍𝑆 (𝑝𝑟𝑒𝑓 , 𝑇𝑟𝑒𝑓 ) 𝑖
𝛽𝑖 𝑝𝑟𝑒𝑓 𝑇𝑟𝑒𝑓
𝑝𝑟𝑒𝑓
𝑇
=
𝑝
𝑇𝑟𝑒𝑓
𝑓𝑆 𝑝𝑟𝑒𝑓 , 𝑇𝑟𝑒𝑓
𝑓𝑆 𝑝, 𝑇
𝑍𝑖 𝑝𝑟𝑒𝑓 , 𝑇𝑟𝑒𝑓
𝑍𝑖 𝑝, 𝑇
𝜎𝑖 𝑝, 𝑇 ,
(26)
𝜙𝑖 (𝑝, 𝑇),
(27)
𝑐𝑖 (𝑝, 𝑇) mole concentration of the analyte i
in the initial state (p,T),
𝑐𝑖 (𝑝𝑟𝑒𝑓 𝑇𝑟𝑒𝑓 ) mole concentration of the
analyte i in the reference state
(pref, Tref),
𝛽𝑖 (𝑝, 𝑇) mass concentration of the analyte
i in the initial state (p,T),
𝜎𝑖 (𝑝, 𝑇) volume concentration of the
analyte i in the initial state (p,T),
𝜎𝑖 𝑝𝑟𝑒𝑓 𝑇𝑟𝑒𝑓 volume concentration of the
analyte i in the reference state
(pref, Tref),
𝜙𝑖 (𝑝, 𝑇) volume fraction of the analyte i in
the initial state (p,T),
𝜙𝑖 (𝑝𝑟𝑒𝑓 𝑇𝑟𝑒𝑓 ) volume fraction of the analyte
i in the refenrence state (pref, Tref),
𝑍𝑖 𝑝, 𝑇 compression factor of the analyte i
in the initial state (p,T),
𝑍𝑖 (𝑝𝑟𝑒𝑓 , 𝑇𝑟𝑒𝑓 ) compression factor of the
analyte i in the reference state
(pref, Tref),
𝑍𝑆 𝑝, 𝑇 compression factor of the gas
mixture S in the initial state (p,T),
𝑍𝑆 (𝑝𝑟𝑒𝑓 , 𝑇𝑟𝑒𝑓 ) compression factor of the
gas mixture S in the reference
state (pref, Tref),
𝑓𝑆 𝑝, 𝑇 mixing factor of the gas mixture S
in the initial state (p,T),
𝑓𝑆 𝑝𝑟𝑒𝑓 , 𝑇𝑟𝑒𝑓 mixing factor of the gas
mixture S in the reference state
(pref, Tref).
Approximations can be used in conversion
equations (24) - (27) instead of accurate mixing
and compression factors.
4.1.1 Ideal mixture of ideal gases
𝑍𝑆 𝑝, 𝑇
𝑍𝑆 𝑝𝑟𝑒𝑓 , 𝑇𝑟𝑒𝑓
𝛽𝑖 𝑝, 𝑇 , (25)
Compression and mixing factors of ideal gases in
ideal mixture can be assumed as one. Mole
fractions,
volume
fractions
and
volume
concentrations of analytes in ideal mixture of ideal
gases are equal (𝑥𝑖 = 𝜙𝑖 = 𝜎𝑖 ).
Mole and mass concentrations are converted to the
reference state by using following conversion
factors.
𝑐𝑖 (𝑝𝑟𝑒𝑓 𝑇𝑟𝑒𝑓 ) =
𝛽𝑖 (𝑝𝑟𝑒𝑓 𝑇𝑟𝑒𝑓 ) =
𝑝𝑟𝑒𝑓
𝑝
𝑝𝑟𝑒𝑓
𝑝
𝑇
𝑐 (𝑝, 𝑇), (28)
𝑇𝑟𝑒𝑓 𝑖
𝑇
𝛽 (𝑝, 𝑇), (29)
𝑇𝑟𝑒𝑓 𝑖
Approximation of ideal gases in ideal mixture can
be applied to all kinds of gas mixtures. A relative
error of the approximation is usually under 1 % and
for permanent gases (N2, O2, CO2,…)
it is
negligible.
4.1.2 Ideal mixture of real gases
Table 1. Conversion between gas composition variables.
Desired
value
𝑥𝑖
𝑥𝑖
𝜙𝑖
𝑤𝑖
𝑐𝑖
𝜎𝑖
The mixing factor of real gases in an ideal mixture
is one therefore the volume fraction and the volume
concentration of analyte are equal. An
approximation of real gases in an deal mixture is
suitable for all kinds of real gas mixtures. In the
most cases a relative error of the approximation is
below 0,3 %.
5 Conversions between mix variables
Initial
value
𝛽𝑖
1
𝑓𝑆 𝑍𝑖
Initial
value
Initial
value
Initial
value
Initial
value
Initial
value
𝜙𝑖
𝑤𝑖
𝑐𝑖
𝜎𝑖
𝛽𝑖
𝑍𝑆
𝑀𝑆
𝑍𝑆
𝑍𝑆
𝑍𝑆
𝑓𝑆 𝑍𝑖
𝑀𝑖
𝛼
𝑍𝑖
𝛼𝑀𝑖
𝑓𝑆 𝑀𝑆 𝑍𝑖
𝑓𝑆 𝑍𝑖
𝑍𝑆
1
𝑀𝑖
𝑍𝑆 𝑀𝑖
𝑀𝑆
𝑓𝑆 𝑀𝑆 𝑍𝑖
1
𝛼
𝛼
𝛼𝑀𝑆
𝑍𝑆 𝑀𝑖
𝛼
𝑓𝑆 𝑍𝑖
𝛼𝑀𝑆
𝑍𝑆 𝑀𝑖
𝑍𝑆 𝑀𝑖
𝑍𝑆
𝛼𝑀𝑆
𝑀𝑆 𝑍𝑖
𝛼𝑀𝑆
𝛼
1
𝑍𝑖
𝑀𝑖
𝑍𝑆
𝑓𝑆 𝑍𝑖
𝑍𝑆 𝑀𝑖
1
𝑍𝑖
1
𝑀𝑆 𝑍𝑖
𝑍𝑖
𝑍𝑆
𝑓𝑆
𝑍𝑆 𝑀𝑖
𝛼
𝛼𝑀𝑖
𝛼𝑀𝑖
𝛼𝑀𝑆
𝑍𝑆
𝑓𝑆 𝑍𝑖
𝑍𝑆
𝑥𝑖 mole fraction of the analyte i
𝜙𝑖 volume fraction of the analyte i
𝑤𝑖 mass fraction of the analyte i
𝑐𝑖 mole concentration of the
analyte i
𝜎𝑖 volume concentration of the
analyte i
𝛽𝑖 mass concentration of the
analyte i
𝑀𝑖 molar mass of the analyte i
𝑓𝑆
𝑀𝑖
1
𝛼𝑀𝑖
𝑍𝑖
𝑍𝑖
𝛼𝑀𝑖
1
𝑍𝑖 compression factor of
analyte i
𝑀𝑆 molar mass of the
mixture S
𝑍𝑆 compression factor of
gas mixture S
𝑓𝑆 mixing factor of the
mixture S
𝑝
𝛼=
𝑅𝑇
the
gas
the
gas
Variables in equations (1) – (6) can be converted to
another variable only in the same state
NB: All variables are in the same state (pressure and
(temperature and pressure).
temperature).
5.1 Conversions between gas composition
variables
In the table 1 is presented conversion factors
calculated by the molar mass M and the
compression factor Z. The initial value of the
variable is multiplied by the conversion factor to get
the desired value of different variable. For example
the mole concentration ci of the analyte i in the gas
mixture can be converted to the mass fraction wi as
follows.
𝑤𝑖 =
𝑍𝑆 𝑀𝑖
𝑅𝑇𝑍𝑆 𝑀𝑖
𝑐𝑖 =
𝑐𝑖 ,
𝛼𝑀𝑆
𝑝𝑀𝑆
5.1.1 Ideal mixture of ideal gases
(28)
Table 2. Conversion factors of ideal mixture of ideal gases.
Desired Initial
Initial
Initial
Initial
Initial
Initial
value
value
value
value
value
value
value
𝑥𝑖
𝜙𝑖
𝑤𝑖
𝑐𝑖
𝜎𝑖
𝛽𝑖
𝑥𝑖
1
1
𝑀𝑆
𝑀𝑖
1
𝛼
1
1
𝛼𝑀𝑖
𝜙𝑖
1
1
𝑀𝑆
𝑀𝑖
1
𝛼
1
1
𝛼𝑀𝑆
𝑤𝑖
𝑀𝑖
𝑀𝑆
𝑀𝑖
𝑀𝑆
1
𝑀𝑖
𝛼𝑀𝑆
𝑀𝑖
𝑀𝑆
1
𝛼𝑀𝑆
𝑐𝑖
𝛼
𝛼
𝛼𝑀𝑆
𝑀𝑖
1
𝛼
1
𝑀𝑖
𝜎𝑖
1
1
𝑀𝑆
𝑀𝑖
1
𝛼
1
1
𝛼𝑀𝑖
Compression and mixing factors of ideal gases in
𝛼𝑀𝑖
𝛼𝑀𝑖
𝛼𝑀𝑆
𝑀𝑖
𝛼𝑀𝑖
𝛽𝑖
ideal mixture can be assumed as one. Mole
fractions,
volume
fractions
and
volume NB: All variables are in the same state (pressure and
concentrations of analytes in an ideal mixture of temperature).
ideal gases are equal (𝑥𝑖 = 𝜙𝑖 = 𝜎𝑖 ).
1
5.1.2 Discharge gas mixtures
Discharge gas analytes (trace gas mixtures) in
discharge gas mixtures occur typically in small
contents, in the grade ppm or ppb. Discharge or
trace gas mixtures contain complementary gas (1)
and discharge or trace gas analytes (2, 3, …, N).
The complementary gas can be a pure gas (e.g.
N2) or a known gas mixture (e.g. air). The
compression factor, the density and the molar
mass of the gas mixture can be assumed same as
for the complementary gas (ZS = Z1, ρS = ρ1, MS =
M1) because the total amount of trace gas analytes
is very small. Furthermore the mixing factor can be
assumed as one.
The relative uncertainty of the conversion of
discharge gas mixtures, with complementary gas
content over 90 %, is typically 0.5 %.
References
SFS-EN
ISO
14912:2006
Gas
analysis.
Conversion of gas mixture composition data.
ISO 14912:2003(E), Gas analysis. Conversion of
gas mixture composition data
Author:
March 2013
Kari Pieniniemi
kari.pieniniemi@centria.fi