Rigorous discussion of "K" value and best way to calculate gas concentration
The general formula for concentration, C of a gas that is emitted from a permeation tube in
unit of ppmv (parts per million by volume) is given in Equation 1 below.
𝑪=
𝑷∙𝑲
𝑭
(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏)
where
P = emission rate (or permeation rate) in nanograms/minute or ng/min at a reference condition
(either STP or NTP). This value is calculated using information provided in a certificate by the
manufacturer (Kin-Tek or Vici). See separate (future) document for explanation of calculations
for P. An explanation of the reference points (STP and NTP) is also given below in page 5.
F = Total Flow rate of the calibration mixture (perm tube gas + dilution gas) in mL/min or 10-3
L/min or cc/min (note that cc = mL). This flow rate is measured in ambient conditions. In
practice, the flow rate from the (or out of the) mixing tee is adjusted to this F.
K = constant which converts the permeation rate at STP (Standard Temperature and Pressure
273.15 Kelvin and 1.00 atm for Kin-Tek perm tubes ) or NTP (Normal Temperature and Pressure
which is 298.15 Kelvin and 1.00 atm for Vici perm tubes) INTO ambient conditions rate. This
conversion is necessary to be consistent since the denominator, F is rate in ambient conditions.
K converts the mass (nanogram or 10-9 g) dimension of P into volume (nanoliters or 10-9 L). K
can be calculated using Equation 2 below. See Page 6 below for an explanation of the difference
between Kin-Tek’s and Vici’s K.
𝑲=
(𝑽𝒎 ′) ∙ (𝑷′ ) ∙ (𝑻𝒂𝒎𝒃 )
𝑴 ∙ (𝑻′ ) ∙ (𝑷𝒂𝒎𝒃 )
(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐)
where
Vm’ = molar volume at STP or NTP in L/mol
P’ = pressure at STP or NTP in atm
T’ = temperature at STP or NTP in Kelvin
Tamb = ambient temperature in Kelvin as measured in the stream of diluted gas
Page 1 of 6
Pamb = ambient (atmospheric) pressure in atm
M = molar mass of gas in perm tube in g/mol
Note that the unit of K is Lemitted perm gas/gram hence K is a multiplier that converts mass into
volume at ambient conditions.
The basis of Equation 2 is explained in pages 3 - 4 below.
Dimensional analysis of Equation 1 can be used to show how concentration is calculated as
given below.
𝐿𝑒𝑚𝑖𝑡𝑡𝑒𝑑 𝑝𝑒𝑟𝑚 𝑔𝑎𝑠
𝐿𝑒𝑚𝑖𝑡𝑡𝑒𝑑 𝑝𝑒𝑟𝑚 𝑔𝑎𝑠
10−9 𝑔
10−9 𝑔
( 𝑚𝑖𝑛 ) ∙ (
)
(
)
𝑔
𝑚𝑖𝑛 ) ∙ (
𝑔
𝑪=
=
10−3 𝐿𝑑𝑖𝑙𝑢𝑡𝑒𝑑 𝑔𝑎𝑠
10−3 𝐿𝑑𝑖𝑙𝑢𝑡𝑒𝑑 𝑔𝑎𝑠
𝑚𝑖𝑛
𝑚𝑖𝑛
(10−9 ) ∙ (𝐿𝑒𝑚𝑖𝑡𝑡𝑒𝑑 𝑝𝑒𝑟𝑚 𝑔𝑎𝑠 )
=
(10−3 )𝐿𝑑𝑖𝑙𝑢𝑡𝑒𝑑 𝑔𝑎𝑠
=
(𝐿𝑒𝑚𝑖𝑡𝑡𝑒𝑑 𝑝𝑒𝑟𝑚 𝑔𝑎𝑠 )
= 𝑃𝐴𝑅𝑇𝑆 𝑃𝐸𝑅 𝑀𝐼𝐿𝐿𝐼𝑂𝑁 𝐵𝑌 𝑉𝑂𝐿𝑈𝑀𝐸
(106 )𝐿𝑑𝑖𝑙𝑢𝑡𝑒𝑑 𝑔𝑎𝑠
Basis of Equation 2
Equation 2 can be re-written as shown below.
𝟏
𝑲 = ( ) ∙ 𝑲∗
𝑴
(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐)
Where
𝑲∗ =
(𝑽𝒎 ′) ∙ (𝑷′ ) ∙ (𝑻𝒂𝒎𝒃 )
(𝑻′ ) ∙ (𝑷𝒂𝒎𝒃 )
(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐𝒃)
The term (1/M, molar mass) is used to convert the mass of gas dimension of P into mol of gas. A
condition is then used as reference point. The reference point is either STP (Kin-Tek) or NTP
(Vici). Then, the mol of gas is said to be mol of gas at STP or NTP.
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When mol of gas at STP or NTP is multiplied by K* – this converts the mol of gas at STP or NTP
into volume at ambient conditions to be consistent with the denominator term, F which is
measured at ambient conditions.
K* can be derived from the ideal gas law as given in Equation 3.
𝑃𝑉 = 𝑛𝑅𝑇
(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟑)
where
P = pressure in atm
V = volume in L
n = mol
𝐿 ∙𝑎𝑡𝑚
R = gas constant 0.08206 𝐾𝑒𝑙𝑣𝑖𝑛 ∙𝑚𝑜𝑙
T = temperature in Kelvin
For two states or two conditions:
Condition 1 = STP or NTP
Condition 2 = ambient
P = P’
P = Pamb
T = T’
T = Pamb
n = n’
n = namb
V = V’
V = Vamb
𝐿 ∙𝑎𝑡𝑚
R = gas constant 0.08206 𝐾𝑒𝑙𝑣𝑖𝑛 ∙𝑚𝑜𝑙
𝐿 ∙𝑎𝑡𝑚
R = gas constant 0.08206 𝐾𝑒𝑙𝑣𝑖𝑛 ∙𝑚𝑜𝑙
Page 3 of 6
Equation 3 transforms into Equation 4 as given below.
𝑃′𝑉′
𝑃𝑎𝑚𝑏 𝑉𝑎𝑚𝑏
=
𝑛′𝑇′
𝑛𝑎𝑚𝑏 𝑇𝑎𝑚𝑏
(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟒)
Note that (V’/n’) = Vm’ (or molar volume at STP or NTP) so that Equation 4 can be re-written as
given below to derive the expression for K*:
𝑉𝑚 𝑃′
𝑃𝑎𝑚𝑏 𝑉𝑎𝑚𝑏
=
𝑇′
𝑛𝑎𝑚𝑏 𝑇𝑎𝑚𝑏
𝑉𝑚 𝑃′𝑇𝑎𝑚𝑏
𝑉𝑎𝑚𝑏
=
= 𝐾∗
𝑇′𝑃𝑎𝑚𝑏
𝑛𝑎𝑚𝑏
K* is in reality the molar volume at ambient conditions. Multiplying mol of gas by K* converts
this mol of gas into volume of gas at ambient conditions.
So that, K is actually a converter of mass of gas dimension of P into mol of gas (by virtue of the
(1/M) term) AND THEN finally to volume of gas at ambient conditions (by virtue of K*).
Reference Points: STP or NTP
Why does Kin-Tek uses STP and Vici uses NTP? We do not know. What really matters are the
ambient conditions. Note that Vm’ (molar volume at STP or NTP) can be expressed in terms of
ideal gas law (Equation 3b). Combining and simplifying Equations 3b and 2b reveals that the
Vm’, T’ and P’ “simplifies” into the gas constant, R as shown below.
𝑽′𝒎 =
𝑽
𝒏
=
𝑹𝑻′
𝑷′
(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟑𝒃)
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𝑲∗ =
(𝑽𝒎 ′) ∙ (𝑷′ ) ∙ (𝑻𝒂𝒎𝒃 )
(𝑻′ ) ∙ (𝑷𝒂𝒎𝒃 )
(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐𝒃)
Substituting the above expression for Vm’ above into Equation 2b yields the equation below.
(
𝑲∗ =
𝑹𝑻′
) ∙ (𝑷′ ) ∙ (𝑻𝒂𝒎𝒃 )
𝑷′
(𝑻′ ) ∙ (𝑷𝒂𝒎𝒃 )
𝑲∗ =
(𝑹) ∙ (𝑻𝒂𝒎𝒃 )
(𝑷𝒂𝒎𝒃 )
(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐𝒃)
(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐𝒃)
This shows that K* is indeed molar volume at ambient conditions.
𝑲∗ =
(𝑹) ∙ (𝑻𝒂𝒎𝒃 )
𝑽𝒂𝒎𝒃
=
(𝑷𝒂𝒎𝒃 )
𝒏𝒂𝒎𝒃
(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐𝒃)
Substituting (RTamb / Pamb) into Equation 2 yields a simplified expression for K (Equation 2-s (“s”
stands for simplified)). Note that Equation 2-s shows that K can be expressed using terms for
ambient conditions ONLY.
𝟏
𝑹 ∙ 𝑻𝒂𝒎𝒃
𝑲= ( ) ∙
𝑴
𝑷𝒂𝒎𝒃
(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐 − 𝒔)
Equation 2-s shows that Kin-Tek and Vici uses the same method to calculate K and C. Finally,
Equations 1 and 2-s can be combined to yield Equation 1-s. This Equation 1-s can be used to
directly calculate C regardless of the manufacturer of the permeation tube. Note that this
equation requires ambient (atmospheric) pressure and ambient temperature. Ambient
temperature is the measured temperature of the stream of diluted gas.
𝑪=
𝑷 ∙ 𝑹 ∙ 𝑻𝒂𝒎𝒃
𝑭 ∙ 𝑴 ∙ 𝑷𝒂𝒎𝒃
(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏 − 𝒔)
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The Difference between Kin-Tek’s and Vici’s “K”
Kin-Tek uses the symbol “K” in their technical note in a very similar way as Equation 1. Vici uses
a different form of K in their technical note. In a way, Vici splits this K into two parts. The first
part is using the molar volume at NTP and the MW (i.e. 24.46/MW). The second part is for the
conversion of the total flow from ambient conditions to NTP conditions.
So how does the K as explained above compare to that of Vici’s K? The recommended K above
is used to convert mass of gas to volume of gas at ambient conditions. Vici’s “K” converts the
mass of gas into volume of gas at NTP so that requires the ambient total flow (in the
denominator) to be converted to NTP conditions as well. Using either way will yield the same
concentration value. For the purpose of consistency, it is recommended that Equation 1-s be
used for all concentration calculations when calibrating using a Kin-Tek or a Vici permeation
tube.
© 2016 Interscan Corporation
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