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"K" Value and Best Way to Calculate Gas Concentration

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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. 푷 ∙ 푲 푪 = (푬풒풖풂풕풊풐풏 ퟏ) 푭 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. ′ (푽풎′) ∙ (푷 ) ∙ (푻풂풎풃) 푲 = ′ (푬풒풖풂풕풊풐풏 ퟐ) 푴 ∙ (푻 ) ∙ (푷풂풎풃) 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. −9 −9 10 푔 퐿푒푚푖푡푡푒푑 푝푒푟푚 푔푎푠 10 푔 퐿푒푚푖푡푡푒푑 푝푒푟푚 푔푎푠 ( 푚푖푛 ) ∙ ( 푔 ) ( 푚푖푛 ) ∙ ( 푔 ) 푪 = −3 = −3 10 퐿푑푖푙푢푡푒푑 푔푎푠 10 퐿푑푖푙푢푡푒푑 푔푎푠 푚푖푛 푚푖푛 −9 (10 ) ∙ (퐿푒푚푖푡푡푒푑 푝푒푟푚 푔푎푠) = −3 (10 )퐿푑푖푙푢푡푒푑 푔푎푠 (퐿푒푚푖푡푡푒푑 푝푒푟푚 푔푎푠) = 6 = 푃퐴푅푇푆 푃퐸푅 푀퐼퐿퐿퐼푂푁 퐵푌 푉푂퐿푈푀퐸 (10 )퐿푑푖푙푢푡푒푑 푔푎푠 Basis of Equation 2 Equation 2 can be re-written as shown below. ퟏ 푲 = ( ) ∙ 푲∗ (푬풒풖풂풕풊풐풏 ퟐ) 푴 Where ( ) ( ′) ( ) ∗ 푽풎′ ∙ 푷 ∙ 푻풂풎풃 푲 = ′ (푬풒풖풂풕풊풐풏 ퟐ풃) (푻 ) ∙ (푷풂풎풃) 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. Page 2 of 6 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. 푃푉 = 푛푅푇 (푬풒풖풂풕풊풐풏 ퟑ) where P = pressure in atm V = volume in L n = mol 퐿 ∙푎푡푚 R = gas constant 0.08206 퐾푒푙푣푖푛 ∙푚표푙 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 퐿 ∙푎푡푚 퐿 ∙푎푡푚 R = gas constant 0.08206 R = gas constant 0.08206 퐾푒푙푣푖푛 ∙푚표푙 퐾푒푙푣푖푛 ∙푚표푙 Page 3 of 6 Equation 3 transforms into Equation 4 as given below. 푃′푉′ 푃푎푚푏푉푎푚푏 = (푬풒풖풂풕풊풐풏 ퟒ) 푛′푇′ 푛푎푚푏푇푎푚푏 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*: 푉푚 푃′ 푃푎푚푏푉푎푚푏 = 푇′ 푛푎푚푏푇푎푚푏 푉푚 푃′푇푎푚푏 푉푎푚푏 ∗ = = 퐾 푇′푃푎푚푏 푛푎푚푏 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. ′ ′ 푽 푹푻 푽 = = (푬풒풖풂풕풊풐풏 ퟑ풃) 풎 풏 푷′ Page 4 of 6 ( ) ( ′) ( ) ∗ 푽풎′ ∙ 푷 ∙ 푻풂풎풃 푲 = ′ (푬풒풖풂풕풊풐풏 ퟐ풃) (푻 ) ∙ (푷풂풎풃) Substituting the above expression for Vm’ above into Equation 2b yields the equation below. 푹푻′ ′ ( ) ∙ (푷 ) ∙ (푻풂풎풃) ∗ 푷′ 푲 = ′ (푬풒풖풂풕풊풐풏 ퟐ풃) (푻 ) ∙ (푷풂풎풃) ( ) ( ) ∗ 푹 ∙ 푻풂풎풃 푲 = (푬풒풖풂풕풊풐풏 ퟐ풃) (푷풂풎풃) This shows that K* is indeed molar volume at ambient conditions. ( ) ( ) ∗ 푹 ∙ 푻풂풎풃 푽풂풎풃 푲 = = (푬풒풖풂풕풊풐풏 ퟐ풃) (푷풂풎풃) 풏풂풎풃 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. ퟏ 푹 ∙ 푻풂풎풃 푲 = ( ) ∙ (푬풒풖풂풕풊풐풏 ퟐ − 풔) 푴 푷풂풎풃 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. 푷 ∙ 푹 ∙ 푻풂풎풃 푪 = (푬풒풖풂풕풊풐풏 ퟏ − 풔) 푭 ∙ 푴 ∙ 푷풂풎풃 Page 5 of 6 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 Page 6 of 6 .
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