• Decrease font size
  • Return font size to normal
  • Increase font size
U.S. Department of Health and Human Services

Food

  • Print
  • Share
  • E-mail

Elemental Analysis Manual: Section 3.4 Special Calculations

Version 1 (June 2008) Authors: Stephen G. Capar
William R. Mindak
Susan C. Hight

Table of Contents

3.4.1 FORTIFICATION RECOVERY

3.4.2 OTHER RECOVERY

3.4.3 DILUTION FACTOR

3.4.4 GRAVIMETRIC STANDARD SOLUTION PREPARATION

3.4.5 PERCENT DIFFERENCE

3.4.6 MASS CORRECTION FACTOR

GLOSSARY


3.4.1 FORTIFICATION RECOVERY 

The marginal method of calculating percent recovery is used for fortification recovery calculations1.

(1) Fortified analytical portion (FAP)

EAM Equation 3.4.1.1a to calculate Fortified analytical portion (FAP) % recovery

where
Cf = concentration of element measured in FAP (mg/kg)
Cu = concentration of element measured in UAP (mg/kg)
Ca = calculated mass fraction of element added in FAP (mg/kg)
and

EAM Equation 3.4.1.1b to calculate Ca - calculated mass fraction of element added in FAP (mg/kg)

where
Cs = concentration of element in fortification solution (mg/L)
Vs = volume of fortification solution added to analytical portion (L)
m = mass of analytical portion (kg)
MCF = mass correction factor (1 if water or other solvent not added to aid homogenization)
Note: Cu = 0 if concentration of element measured in UAP is less than zero. Mass of fortification solution added assumed to be negligible compared with m.

(2) Fortified method blank (FMB)

EAM Equation 3.4.1.2 to calculate Fortified method blank (FMB) recovery %

where
Cf = concentration of element measured in FMB (µg/L)
MBK = Batch MBK (µg/L) [see §4.0.2.7]
Ca = calculated concentration of element added in FMB (µg/L)

(3) Fortified analytical solution (FAS)

EAM Equation 3.4.1.3 to calculate Fortified analytical solution (FAS)

where
Sf = concentration of element measured in FAS (µg/L)
Su = concentration of element measured in unfortified analytical solution (µg/L)
Sa = concentration of element added to FAS (µg/L)
Note: Su equals 0 if concentration of element measured in unfortified analytical solution is less than zero.

3.4.2 OTHER RECOVERY 

(1) Reference material (RM)

EAM Equation 3.4.2.1 to calculate Reference material (RM) recovery %

where
R = analytical result for RM (mg/kg)
T = reference value concentration of RM (mg/kg)

(2) Check solution recovery (Independent check solution or Continuing calibration verification)

EAM Equation 3.4.2.2. to calculate Check solution recovery (Independent check solution or Continuing calibration verification)

where
R = analytical result for check solution (µg/L)
T = concentration of check solution as prepared (µg/L)
Note: Concentration units may be different than given above but must be the same for R and T.

3.4.3 DILUTION FACTOR 

Dilution factor (DF) — factor by which concentration in a diluted analytical solution is multiplied to obtain concentration in the analytical solution.

(1) Volumetric dilution (Type A)

Diluting a volume portion of initial solution to a final volume.

EAM Equation 3.4.3.1 to caclutate diluting a volume portion of initial solution to a final volume

where
Vi = portion of initial solution (L)
Vf = final volume (L)

(2) Volumetric dilution (Type B)

Mixing a volume portion of initial solution with a volume of diluent of the same matrix.

EAM Equation 3.4.3.2 to calculate Mixing a volume portion of initial solution with a volume of diluent of the same matrix

where
Vi = portion of initial solution (L)
Vd = volume of diluent (L)

(3) Gravimetric dilution

Gravimetric dilution — practice of quantitatively preparing dilute solutions from more concentrated ones by combining known mass of solution and diluent. A mass of initial solution is mixed with diluent and final mass is measured.

EAM Equation 3.4.3.3 to calculate Gravimetric dilution-practice of quantitatively preparing dilute solutions from more concentrated ones by combining known mass of solution and diluent

where
mi = portion of initial solution (g)
mf = final mass (g)
di = density of initial solution (g/mL)
df = density of final solution (g/mL)
When densities of initial solution and diluent are the same then

EAM Equation 3.4.3.3b to calculate Gravimetric dilution - When densities of initial solution and diluent are the same

(4) Serial dilution

Individual volumetric or gravimetric DFs are multiplied to obtain an overall DF for solutions produced by serial dilution. For example, the DF for an initial solution diluted sequentially by three Type A volumetric dilutions is calculated as follows:

EAM Equation 3.4.3.4 showing Individual volumetric or gravimetric DFs are multiplied to obtain an overall DF for solutions produced by serial dilution

where
Vi1 = portion of initial solution for first dilution (L)
Vf1 = final volume for first dilution (L)
Vi2 = portion of first diluted solution for second dilution (L)
Vf2 = final volume for second dilution (L)
Vi3 = portion of second diluted solution for third dilution (L)
Vf3 = final volume of third dilution (L)

3.4.4 GRAVIMETRIC STANDARD SOLUTION PREPARATION 

To perform gravimetric standard solution preparation the density of the initial solution (e.g., stock standard) must be known and is provided by most commercial manufactures. The density of the final solution must also be known and can be assumed the same as the diluent, which can be determined easily by measuring the mass of diluent in a tared volumetric flask. Dispense a mass (0.1-1.0 g) of initial solution to nearest 0.0001 g in a tared, clean plastic bottle. Add diluent so that final solution mass (100-270 g) provides the required concentration. The concentration of each element in the final solution is calculated as follows:

EAM Equation 3.4.4a to calculate the concentration of each element in the final solution

where
Sf = concentration of final solution (mg/L)
Si = concentration of initial solution (mg/L)
df = density of final solution (g/mL)
di = density of initial solution (g/mL)
mf = mass of final solution (g)
mi = portion of initial solution dispensed (g)

When densities of the initial and final solutions are the same then

Equation 3.4.4b to calculate the concentration of each element in the final solution when the densities of initial and final solutions are equal

Estimation of mass needed to obtain desired concentration:

The following equation provides the mass needed to obtain a desired concentration.

EAM Equation 3.4.4c to calculate the estimation of mass needed to obtain desired concentration

where
Sf* = desired concentration of final solution (mg/L)
Si = concentration of initial solution (mg/L)
di = density of initial solution (g/mL)
Vf = approximate desired volume of final solution (mL)
mi = portion of initial solution (g)

For example, if the approximate desired volume is 0.1 L and the desired final concentration is 5 mg/L, then for a 1,000 mg/L stock solution with a density of 1.009 then

EAM Equation 3.4.4d to calculate the analyte concentration in the final solution (Sf) which is calculated based on the exact mass of the initial solution taken

This mass, within about 10%, is used to prepare the final solution. The analyte concentration in the final solution (Sf) is calculated based on the exact mass of the initial solution taken. Continuing the example above, if the densities of the initial and final solutions are equal and a 0.5548 g portion of initial solution was used and the mass of the final solution was 102.5250 g then

EAM Equation 3.4.4e to calculate the estimation of mass needed to obtain desired concentration

3.4.5 PERCENT DIFFERENCE 

(1) Relative percent difference (RPD) of two measurements

EAM Equation 3.4.5.1 to calculate the relative percent difference (RPD) of Two Measurements

where
C1 = concentration of first measurement
C2 = concentration of second measurement

(2) Percent difference (PD) of a known and calculated value

EAM Equation 3.4.5.2 to calculate the percent difference (PD) of a known and calculated value

where
C1 = known concentration
C2 = calculated concentration

3.4.6 MASS CORRECTION FACTOR (MCF) 

Factor applied to analytical portion mass to account for water (or other solvent) added to aid homogenization of analytical sample.

Equation 3.4.6 to calculate factor applied to analytical portion mass to account for water (or other solvent) added to aid homogenization of analytical sample

where
ms = mass of analytical sample homogenized (g)
mw = mass of reagent water (or other solvent) added to aid homogenization (g)

REFERENCES

  1. Official Methods of Analysis of AOAC INTERNATIONAL (2005) 18th Ed., AOAC International disclaimer icon,  Gaithersburg, MD, USA, Appendix D: Guidelines for Collaborative Study Procedures To Validate Characteristics of a Method of Analysis.