All equations#

Calculation of amount fractions for linear detection systems#

(1)#\[\chi _{\mathrm{sample}}=\frac{A_{\mathrm{sample}}-A_{\mathrm{blank}}}{V_{\mathrm{sample}}}*f_{\mathrm{calib}}\]

\(\chi _{\mathrm{sample}}\): Amount fraction of a substance in a sample
\(A_{\mathrm{sample}}\): Peak area of sample measurement
\(A_{\mathrm{blank}}\): Possible blank area value determined in zero gas measurements
\(V_{\mathrm{sample}}\): Volume of measured samples
\(f_{\mathrm{calib}}\): Calibration Factor

Calibration Factor#

(2)#\[f_{\mathrm{calib}}=\frac{V_{\mathrm{calib}}*\chi _{\mathrm{calib}}}{A_{\mathrm{calib}}-A_{\mathrm{blank}}}\]

\(f_{\mathrm{calib}}\): Calibration Factor
\(V_{\mathrm{calib}}\): Volume of calibration gas samples
\(\chi _{\mathrm{calib}}\): Certified amount fraction of calibration gas
\(A_{\mathrm{calib}}\): Peak area of calibration gas measurement
\(A_{\mathrm{blank}}\): Possible blank area value determined in zero gas measurements

Carbon Response Factor Equation#

(3)#\[C_{\mathrm{resp}}=\frac{A_{\mathrm{calib}}-A_{\mathrm{blank}}}{C_{\mathrm{num}}*y*V_{\mathrm{calib}}*\chi _{\mathrm{calib}}}=\frac{1}{C_{\mathrm{num}}*y*f_{\mathrm{calib}}}\]

\(C_{\mathrm{resp}}\): Carbon Response Factor (CRF)
\(A_{\mathrm{calib}}\): Peak area of calibration gas measurement
\(A_{\mathrm{blank}}\): Possible blank area value determined in zero gas measurements
\(C_{\mathrm{num}}\): Carbon number
\(y\): ECN contribution
\(V_{\mathrm{calib}}\): Volume of calibration gas samples
\(\chi _{\mathrm{calib}}\): Certified amount fraction of calibration gas
\(f_{\mathrm{calib}}\): Calibration Factor

Amount fraction using CRF#

(4)#\[\chi _{\mathrm{sample}}=\frac{A_{\mathrm{sample}}-A_{\mathrm{blank}}}{V_{\mathrm{sample}}*C_{\mathrm{num}}*y*\overline{C}_{\mathrm{resp}}}\]

\(\chi _{\mathrm{sample}}\): Amount fraction of a substance in a sample
\(A_{\mathrm{sample}}\): Peak area of sample measurement
\(A_{\mathrm{blank}}\): Possible blank area value determined in zero gas measurements
\(V_{\mathrm{sample}}\): Volume of measured samples
\(C_{\mathrm{num}}\): Carbon number
\(y\): ECN contribution
\(\overline{C}_{\mathrm{resp}}\): Mean carbon response factor
Use in the General Mean CRF calculation

Retrieval of blank area from amount fraction#

(5)#\[A_{\mathrm{blank}}=\frac{\chi _{\mathrm{blank}}}{\left(\frac{V_{\mathrm{calib}}}{V_{\mathrm{sample}}}\chi _{\mathrm{calib}}+\chi _{\mathrm{blank}}\right)}A_{\mathrm{calib}}\]

\(A_{\mathrm{blank}}\): Possible blank area value determined in zero gas measurements
\(\chi _{\mathrm{blank}}\): Blank amount fraction
\(\chi _{\mathrm{calib}}\): Certified amount fraction of calibration gas
\(A_{\mathrm{calib}}\): Peak area of calibration gas measurement
\(V_{\mathrm{calib}}\): Volume of calibration gas samples
\(V_{\mathrm{sample}}\): Volume of measured samples
\(f_{\mathrm{calib}}\): Calibration Factor

Derivation#

\[ \begin{align}\begin{aligned}\begin{split}\chi _{\mathrm{sample}}&=\frac{A_{\mathrm{sample}}-A_{\mathrm{blank}}}{V_{\mathrm{sample}}}*f_{\mathrm{calib}} \\ \mathrm{In\ the\ case\ of\ blanks\ transforms\ to:\ } \\ \chi _{\mathrm{blank}}&=\frac{A_{\mathrm{blank}}}{V_{\mathrm{sample}}}*f_{\mathrm{calib}} \\ f_{\mathrm{calib}}&=\frac{V_{\mathrm{calib}}*\chi _{\mathrm{calib}}}{A_{\mathrm{calib}}-A_{\mathrm{blank}}} \\ \mathrm{Merging\ the\ equations\ together:\ }\end{split}\\\begin{split}\chi _{\mathrm{blank}}&=\frac{V_{\mathrm{calib}}}{V_{\mathrm{sample}}}\frac{A_{\mathrm{blank}}}{A_{\mathrm{calib}}-A_{\mathrm{blank}}}*\chi _{\mathrm{calib}} \\ \chi _{\mathrm{blank}}*\left(A_{\mathrm{calib}}-A_{\mathrm{blank}}\right)&=\frac{V_{\mathrm{calib}}}{V_{\mathrm{sample}}}*A_{\mathrm{blank}}*\chi _{\mathrm{calib}} \\ \chi _{\mathrm{blank}}*A_{\mathrm{calib}}&=\left(\frac{V_{\mathrm{calib}}}{V_{\mathrm{sample}}}*\chi _{\mathrm{calib}}+\chi _{\mathrm{blank}}\right)*A_{\mathrm{blank}} \\ \Rightarrow A_{\mathrm{blank}}=\frac{\chi _{\mathrm{blank}}}{\left(\frac{V_{\mathrm{calib}}}{V_{\mathrm{sample}}}\chi _{\mathrm{calib}}+\chi _{\mathrm{blank}}\right)}A_{\mathrm{calib}}\end{split}\end{aligned}\end{align} \]

Retrieval of blank area from amount fraction for substances not in the calibration gas#

(6)#\[A_{\mathrm{blank}}=\chi _{\mathrm{blank}}*V_{\mathrm{sample}}*C_{\mathrm{num}}*y*\overline{C}_{\mathrm{resp}}\]

\(A_{\mathrm{blank}}\): Possible blank area value determined in zero gas measurements
\(\chi _{\mathrm{blank}}\): Blank amount fraction
\(V_{\mathrm{sample}}\): Volume of measured samples
\(C_{\mathrm{num}}\): Carbon number
\(y\): ECN contribution
\(\overline{C}_{\mathrm{resp}}\): Mean carbon response factor

Derivation#

\[\begin{split}\chi _{\mathrm{sample}}&=\frac{A_{\mathrm{sample}}-A_{\mathrm{blank}}}{V_{\mathrm{sample}}*C_{\mathrm{num}}*y*\overline{C}_{\mathrm{resp}}} \\ \mathrm{In\ the\ case\ of\ blanks\ transforms\ to:\ } \\ \chi _{\mathrm{blank}}&=\frac{A_{\mathrm{blank}}}{V_{\mathrm{sample}}*C_{\mathrm{num}}*y*\overline{C}_{\mathrm{resp}}} \\ \Rightarrow A_{\mathrm{blank}}=\chi _{\mathrm{blank}}*V_{\mathrm{sample}}*C_{\mathrm{num}}*y*\overline{C}_{\mathrm{resp}}\end{split}\]