# What Is Isotope Dilution?

Figure 2.5 places isotope dilution under the second option for using the IS mode of instrument calibration. The principal EPA methods that require *isotope dilution mass spectrometry’* (IDMS) as the means to calibrate determinative techniques such as: GC-MS, GC-MS-MS, LC-MS-MS, and ICP-MS are shown in Figure 2.5. Other analytical methods that rely on isotope dilution as the chief means to calibrate and to quantitate are liquid scintillation counting and various radioimmunoassay techniques that are outside the scope of this book.

TEQA can be implemented using isotope dilution. The unknown concentration of an element or compound in a sample can be found by knowing only the natural isotopic abundance (atom fraction of each isotope of a given element) *and*, after an enriched isotope of this element has been added and equilibrated, by measuring this altered isotopic ratio in the spiked or diluted mixture. This is the simple yet elegant conceptual framework for isotope dilution as a quantitative tool.

# Can a Fundamental Quantification Equation Be Derived from Simple Principles?

Yes, indeed, and we proceed to do so now. The derivation begins by first defining this altered and measured ratio of isotopic abundance after the enriched isotope (spike or addition of labeled analog) has been added to a sample and equilibrated. Only two isotopes of a given element are needed to provide quantification. Fassett and Paulsen showed how isotope dilution is used to determine the concentration at trace levels for vanadium in crude oil.^{12} We use their illustration to develop the principles that appear below.

Let us start by defining *R _{m}* as the measured ratio of each of the two isotopes of a given element in the spiked unknown. The contribution made by

^{50}P appears in the numerator, and that made by

*appears in the denominator. Fassett and Paulsen obtained this measured ratio from mass spectrometry. Mathematically stated,*

^{51}V

The amount of ^{50}V in the unknown sample can be found as a product of the concentration of vanadium in the sample as the ^{50}V and the weight of sample. This is expressed as follows:

The natural isotopic abundances for the element vanadium are 0.250% as ^{50}V and 99.750% as ^{51}V, so that/^{51} = 0.9975'^{2} for the equations that follow.

Equation (2.6) can be abbreviated and is shown rewritten as follows:

In a similar manner, we can define the amount of the higher isotope of vanadium in the unknown as follows:

Equation (2.7) and Equation (2.8) can also be written in terms of the respective amounts of the 50 and 51 isotopes in the enriched spike. This is shown as follows:

Equation (2.5) can now be rewritten using the symbolism defined by Equation (2.7) to Equation (2.10) and generalized for the first isotope of the /th analyte (/, 1) and for the second isotope of the /'th analyte (/, 2) according to

where

*R _{m}* = isotope ratio (dimensionless number) obtained after an aliquot of the unknown sample has been spiked and equilibrated by the enriched isotope mix. This is

*measurable*in the laboratory using a determinative technique such as mass spectrometry. The ratio could be found by taking the ratio of peak areas at different quantitation ions (quant ions or Q ions) if GC-MS was the determinative technique used.

[/wit J ^{= natun>}l abundance (atom fraction) of the /th element of the first isotope in the unknown sample. This is known from tables of isotopic abundance.

[/wt-J = natural abundance (atom fraction) of the /th element of the second isotope in the unknown sample. This is known from tables of isotopic abundance.

*\_fspike* J ^{= natura}l «abundance (atom fraction) of the /th element of the first isotope in the spiked sample. This is known from tables of isotopic abundance.

*\_f 1'pike* J ^{=} natural abundance (atom fraction) of the /th element of the second isotope in the spiked sample. This is known from tables of isotopic abundance.

*C' _{unk}=* concentration [pmol/g, pg/g] of the /th element or compound in the unknown sample. This is unknown;

*the goal of isotope dilution is to find this value.*

*C‘ _{spjke}* = concentration [pmol/g, pg/g] of the /th element or compound in the spike. This is known.

* W„_{nk}* = weight of unknown sample in g. This is measurable in the laboratory.

IV = weight of spike in g. This is measurable in the laboratory.

Equation (2.11), the more general form, can be solved algebraically for *C‘ _{mk}* to yield the quantification equation:

Equation (2.12) also appears as the quantification equation for IDMS in Table 2.2. We proceed now to consider the use of isotopically labeled organic compounds in IDMS. Returning again to Figure 2.5, we find the use of IDMS as a means to achieve TEQA when a GC- MS is the determinative technique employed. Methods that determine polychloro-dibenzo- dioxins (PCDDs), polychloro-dibenzo-difurans (PCDFs), and coplanar polychlorinated biphenyls (cp-PCBs) require IDMS. IDMS coupled with the use of high-resolution GC-MS represents the most rigorous and highly precise trace organics analytical techniques designed to conduct TEQA known today. The author recently adapted Equation (2.12) as it relates to more accurately quantifying trace concentration levels of cyanide ion (CN ) in whole blood.^{13} The determinative technique (see Chapter 4) used in this application was automated static headspace GC-MS. This paper describes an application of isotope dilution applied to *enviro-health* TEQA.

# What is Organics IDMS?

Organics IDMS exploits the excellent specificity afforded by MS and particularly MS-MS determinative techniques by utilizing 2H-, 13C-, or 37C1- isotopically labeled organic compounds as internal standards. The identical unlabeled organic compound is quantitated against its isotopic analog. These labeled analogs are also added to environmental samples or to human specimens to conduct *enviro-chemical* or *enviro-health* TEQA.

Isotopically labeled analogs are structurally identical except for the substitution of ^{2}H for 1H, ^{13}C for 12C, or ^{37}C1 for 35C1. A plethora of isotopically labeled analogs are now available for most priority pollutants or persistent organic pollutants (POPs) that are targeted analytes in various EPA methods. To illustrate, molecular structures for the priority pollutant phenanthracene and its deuterated isotope, symbolized by 2H, or D, are shown below:

Polycyclic aromatic hydrocarbons (PAHs), of which phenanthracene is a member, have abundant molecular ions in electron-impact MS. The molecular weight for phenanthracene is 178, while that for the deuterated isotopic analog is 188 (phen-dlO). If phenanthracene is diluted with itself, and if an aliquot of this mixture is injected into a GC-MS, the native and deuterated forms can be distinguished at the same gas chromatographic retention time by monitoring the mass-to- charge ratio, abbreviated *m/z,* at 178 and then at 188. Refer back to Table 1.11 whereby phen-d 10 is used as an IS to quantitate all of the analytes listed when implementing EPA Method 8270C. Contrast this with the ultimate goal of IDMS, just discussed, in which an isotopically labeled organic compound is needed for each and every targeted organic compound! Isotopically labeled organic reference standards are very expensive!

## How Does the SA Mode of Instrument Calibration Work?

The SA mode is used primarily when there exists a significant matrix interference and where the concentration of the analyte in the unknown sample is appreciable. SA becomes a calibration mode of choice when the analyte-free matrix cannot be obtained for the preparation of standards for ES. However, for each sample that is to be analyzed, a second so-called standard added or spiked sample must also be analyzed. This mode is preferred when trace metals are to be determined in complex sample matrices such as wastewater, sediments, and soils. If the analyte response is linear within the range of concentration levels anticipated for samples, it is not necessary to construct a multipoint calibration. Only two samples need to be measured—the unspiked and spiked samples.

# Can We Derive a Quantification Equation for SA?

Yes, indeed. We proceed to do so now. Assume that *C' _{unk}* represents the ultimate goal of TEQA, i.e., the concentration of the /th analyte, such as a metal in the unknown environmental sample or human specimen. Also assume that

*C*represents the concentration of the /'th analyte in a spike solution. After an aliquot of the spike solution has been added to the unknown sample, an instrument response of the /'th analyte for the standard added sample, denoted as

_{jke}*R*'

_{SA}whose concentration must be

*C'*is measured. Knowing only the instrument response for the unknown,

_{SA}*R'*and the instrument response for the standard added,

_{llllk }*R'*can be found. Mathematically, let us prove this. The proportionality constant

_{SA},C‘_{unk}*к*must be the same between the concentration of the /'th analyte and the instrument response, such as a peak area in atomic absorption spectroscopy. The following four relationships must be true:

Solving Equation (2.15) for *R' _{spike}* and substituting this into Equation (2.14) leads to the following ratio:

Solving Equation (2.17) for c, yields the quantification equation

For real samples that may have nonzero blanks, the concentration of the /'th analyte in an unknown sample, *C‘ _{mk}* can be found knowing only the measurable parameters

*R'*and

_{SA}*R'*and instrument responses in blanks along with the known concentration of single standard added or spike concentration

_{unk}*R'*according to

_{spike}

where *R,i_ _{unk}* represents the instrument response for a blank that is associated with the unknown sample.

*R'i,i_*is the instrument response for a blank associated with the spike solution and accounts for any contribution that the spike makes to the blank. Equation (2.19) is listed in Table 2.2 as the quantification equation for SA.

_{spi}k_{e}If a multipoint calibration is established using SA, the line must be extrapolated across the ordinate (у-axis) and terminate on the abscissa (x-axis). The value on the abscissa that corresponds to the amount or concentration of unknown analyte yields the desired result. Students are asked to create a multipoint SA calibration to quantitate both Pb and anionic surfactants in Chapter 5. Contemporary software for graphite furnace atomic absorption spectroscopy (GFAA) routinely incorporates SA as well as ES modes of instrument calibration. Autosamplers for GFAA can be programmed to add a precise aliquot of a standard solution containing a metal to an aqueous portion of an unknown sample that contains the same metal at an unknown concentration.

Most comprehensive treatments of various analytical approaches utilizing SA as the principal mode of calibration can be found in an earlier published paper.^{14}