# Isotope dilution

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Isotope dilution analysis is a method of analyzing chemical substances in analytical chemistry. In its most simple conception, the method of isotope dilution comprises the addition of isotopically-enriched substance to the analyzed sample. Isotope dilution method is classified as the method of internal standard, whereby the standard (isotopically-enriched form of analyte) is added directly to the sample. In addition, unlike traditional analytical methods which rely on signal intensity, isotope dilution employs signal ratios. Owing to both of these advantages, method of isotope dilution is regarded among chemistry measurement methods of the highest metrological standing.[1]

## Early history

The Hungarian chemist George de Hevesy was awarded the Nobel Prize in Chemistry for development of radiotracer method, which is a forerunner of isotope dilution

Analytical application of the radiotracer method is forerunner of isotope dilution. This method was developed in the early 20th century by George de Hevesy for which he was awarded the Nobel Prize in Chemistry for 1943.

An early application of isotope dilution in the form of radiotracer method was determination of the solubility of lead sulphide and lead chromate in 1913 by George de Hevesy and Friedrich Adolf Paneth.[2] In 1930s, US biochemist David Rittenberg pioneered the use of isotope dilution in biochemistry enabling detailed studies of cell metabolism.[3]

## Tutorial example

Tutorial illustration of isotope dilution analysis with fish counting in lakes

Isotope dilution can be effectively explained using mark and recapture method from biology - a method commonly used in ecology to estimate the population size of fish.

Isotope dilution can be likened to Lincoln-Petersen method. Assume that the number of fish in a pond is to be determined. Five fish are added to the pond during the first visit (nB = 5). On the second visit, a number of fish is captured and one observes that the ratio of native-to-labeled is 10:1. From here, we can estimate the original number of fish in the pond, nA:

$n_\mathrm{A} = n_\mathrm{B} \times \frac{10}{1} = 50$

This is a simplified view of isotope dilution yet it illustrates the salient features of isotope dilution. A more complex situation arises when the distinction between labeled and unlabeled fish becomes fuzzy. This can occur, for example, when the lake already contains a small number of labeled fish from the previous field experiments. In such situation, the following expression can be employed:

$n_\mathrm{A} = n_\mathrm{B} \times \frac{R_\mathrm{B}-R_\mathrm{AB}}{R_\mathrm{AB}-R_\mathrm{A}} \times \frac{1+R_\mathrm{A}}{1+R_\mathrm{B}}$

where RA is the ratio of the native-to-labeled fish in lake, RB is the ratio of the native-to-labeled fish in the lot of nB marked fish which are added to the pond, and RAB is the ratio of the native-to-labeled fish captured during the second visit.

## Single dilution method

Consider a natural analyte rich in isotope iA (denoted as A), and the same analyte, enriched in isotope jA (denoted as B). Then, the obtained mixture is analyzed for the isotopic composition of the analyte, RAB = n(iA)AB/n(jA)AB. If the amount of the isotopically-enriched substance (nB) is known, the amount of substance in the sample (nA) can be obtained:[4]

$n_\mathrm{A} = n_\mathrm{B} \frac{R_\mathrm{B}-R_\mathrm{AB}}{R_\mathrm{AB}-R_\mathrm{A}} \times \frac {x(^{j}\mathrm{A})_\mathrm{B}}{x(^{j}\mathrm{A})_\mathrm{A}}$

Here, RA is the isotope amount ratio of the natural analyte, RA = n(iA)A/n(jA)A, RB is the isotope amount ratio of the isotopically-enriched analyte, RB = n(iA)B/n(jA)B, RAB is the isotope amount ratio of the resulting mixture, x(jA)A is the isotopic abundance of the minor isotope in the natural analyte, and x(jA)B is the isotopic abundance of the major isotope in the isotopically-enriched analyte.

For elements with only two stable isotopes, such as boron, chlorine, or silver, the above single dilution equation simplifies to the following:

$n_\mathrm{A} = n_\mathrm{B} \frac{R_\mathrm{B}-R_\mathrm{AB}}{R_\mathrm{AB}-R_\mathrm{A}} \times \frac {1+R_\mathrm{A}}{1+R_\mathrm{B}}$

In a typical gas chromatography analysis, isotopic dilution can decrease the uncertainty of the measurement results from 5% to 1%. It can also be used in mass spectrometry (commonly referred to as isotopic dilution mass spectrometry or IDMS), in which the isotopic ratio can be determined with precision typically better than 0.25%.[5]

## Double dilution method

Single dilution method requires the knowledge of the isotopic composition of the isotopically-enriched analyte (RB) and the amount of the enriched analyte added (nB). Both of these variables are hard to establish since isotopically-enriched substances are generally available in small quantities of questionable purity. As a result, before isotope dilution is performed on the sample, the amount of the enriched analyte is ascertained beforehand using isotope dilution. This preparatory step is called the reverse isotope dilution and it involves a standard of natural isotopic-composition analyte (denoted as A*).

Reverse isotope dilution analysis of the enriched analyte:

$n_\mathrm{B} = n_\mathrm{A*} \frac{R_\mathrm{A*}-R_\mathrm{A*B}}{R_\mathrm{A*B}-R_\mathrm{B}} \times \frac {x(^{j}\mathrm{A})_\mathrm{A*}}{x(^{j}\mathrm{A})_\mathrm{B}}$

Isotope dilution analysis of the analyte:

$n_\mathrm{A} = n_\mathrm{B} \frac{R_\mathrm{B}-R_\mathrm{AB}}{R_\mathrm{AB}-R_\mathrm{A}} \times \frac {x(^{j}\mathrm{A})_\mathrm{B}}{x(^{j}\mathrm{A})_\mathrm{A}}$

Since isotopic composition of A and A* are identical, combining these two expressions eliminates the need to measure the amount of the added enriched standard (nB):

$n_\mathrm{A} = n_\mathrm{A*} \frac{R_\mathrm{A*}-R_\mathrm{A*B}}{R_\mathrm{A*B}-R_\mathrm{B}} \times \frac{R_\mathrm{B}-R_\mathrm{AB}}{R_\mathrm{AB}-R_\mathrm{A}}$

Double dilution method can be designed such that the isotopic composition of the two blends, A+B and A*+B, is identical, i.e., RAB = RA*B. This condition of exact-matching double isotope dilution simplifies the above equation significantly:[6]

$n_\mathrm{A} = n_\mathrm{A*} \; (R_\mathrm{A*B}=R_\mathrm{AB} \and R_\mathrm{A*} = R_\mathrm{A})$

## Triple dilution method

To avoid contamination of the mass spectrometer with the isotopically-enriched spike, an additional blend of the primary standard (A*) and the spike (B) can be measured instead of measuring the enriched spike (B) directly. This approach was first put forward in 1970s and developed in 2002.[7]