# Elemental Analysis Manual: Section 3.4 Special Calculations

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

3.4.2 OTHER RECOVERY

3.4.3 DILUTION FACTOR

3.4.5 PERCENT DIFFERENCE

GLOSSARY

### 3.4.1 FORTIFICATION RECOVERY

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

(1) Fortified analytical portion (FAP)

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

where
Cs = concentration of element in fortification solution (mg/L) volume of fortification solution added to analytical portion (L) mass of analytical portion (kg) 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)

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

(3) Fortified analytical solution (FAS)

where
Sf = concentration of element measured in FAS (µg/L) concentration of element measured in unfortified analytical solution (µg/L) 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)

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

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

where
R = analytical result for check solution (µg/L) 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.

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

(2) Volumetric dilution (Type B)

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

where
Vi = portion of initial solution (L) 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.

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

(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:

where
Vi1 = portion of initial solution for first dilution (L) final volume for first dilution (L) portion of first diluted solution for second dilution (L) final volume for second dilution (L) portion of second diluted solution for third dilution (L) 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:

where
Sf = concentration of final solution (mg/L) concentration of initial solution (mg/L) density of final solution (g/mL) density of initial solution (g/mL) mass of final solution (g) portion of initial solution dispensed (g)

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

Estimation of mass needed to obtain desired concentration:

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

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

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

### 3.4.5 PERCENT DIFFERENCE

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

where
C1 = concentration of first measurement concentration of second measurement

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

where
C1 = known concentration 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.

where
ms = mass of analytical sample homogenized (g) 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 ,  Gaithersburg, MD, USA, Appendix D: Guidelines for Collaborative Study Procedures To Validate Characteristics of a Method of Analysis.

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