Characteristic function (probability theory)

Basic principle of isotope dilution
Adding of an isotopically altered standard to the sample changes the natural isotopic composition of the analyte. By measuring the resulting isotopic composition, it is possible to calculate the amount of the analyte present in the sample.

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.

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. In 1930s, US biochemist David Rittenberg pioneered the use of isotope dilution in biochemistry enabling detailed studies of cell metabolism.

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 (n_B = 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, n_A:

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

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 R_A is the ratio of the native-to-labeled fish in lake, R_B is the ratio of the native-to-labeled fish in the lot of n_B marked fish which are added to the pond, and R_AB is the ratio of the native-to-labeled fish captured during the second visit.

Applications

Isotope dilution is almost exclusively employed with mass spectrometry in applications where high-accuracy is demanded. For example, all National Metrology Institutes rely significantly on isotope dilution when producing certified reference materials. In addition to high-precision analysis, isotope dilution is applied when low recovery of the analyte is encountered. In addition to the use of stable isotopes, radioactive isotopes can be employed in isotope dilution which is often encountered in biomedical applications.

Single dilution method

Isotope dilution notation
Name	Symbol
Analyte	A
Isotopic standard (Spike)	B
Analyte + Spike	AB

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, R_AB = n(ⁱA)_AB/n(^jA)_AB. If the amount of the isotopically-enriched substance (n_B) is known, the amount of substance in the sample (n_A) can be obtained:

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

Here, R_A is the isotope amount ratio of the natural analyte, R_A = n(ⁱA)_A/n(^jA)_A, R_B is the isotope amount ratio of the isotopically-enriched analyte, R_B = n(ⁱA)_B/n(^jA)_B, R_AB 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%.

Optimum composition of the blend

In a simplified manner, the uncertainty of the measurement results is largely determined from the measurement of R_AB:

$u(n_{\mathrm {A} })^{2}\propto \left({\frac {\partial {n_{\mathrm {A} }}}{\partial R_{\mathrm {AB} }}}\right)^{2}u(R_{\mathrm {AB} })^{2}=n_{\mathrm {A} }^{2}{\frac {(R_{\mathrm {A} }-R_{\mathrm {B} })^{2}}{(R_{\mathrm {A} }-R_{\mathrm {AB} })^{2}(R_{\mathrm {AB} }-R_{\mathrm {B} })^{2}}}u(R_{\mathrm {AB} })^{2}$

From here, we obtain the relative uncertainty of n_A, u_r(n_A) = u(n_A)/n_A:

$u_{\mathrm {r} }(n_{\mathrm {A} })^{2}\propto {\frac {(R_{\mathrm {A} }-R_{\mathrm {B} })^{2}}{(R_{\mathrm {A} }-R_{\mathrm {AB} })^{2}(R_{\mathrm {AB} }-R_{\mathrm {B} })^{2}}}u(R_{\mathrm {AB} })^{2}$

Lowest relative uncertainty of n_A corresponds to the condition when the first derivative with respect to R_AB equals zero. In addition, it is common in mass spectrometry that u(R_AB)/R_AB is constant and therefore we can replace u(R_AB) with R_AB. These ideas combine to give

$u_{\mathrm {r} }(n_{\mathrm {A} })_{\mathrm {min} }\mapsto \partial \left({\frac {(R_{\mathrm {A} }-R_{\mathrm {B} })}{(R_{\mathrm {A} }-R_{\mathrm {AB} })(R_{\mathrm {AB} }-R_{\mathrm {B} })}}R_{\mathrm {AB} }\right)/\partial R_{\mathrm {AB} }=0$

Solving this equation leads to the optimum composition of the blend AB, i.e., the geometric mean between the isotopic compositions of standard (A) and spike (B):

$R_{\mathrm {AB} }={\sqrt {R_{\mathrm {A} }R_{\mathrm {B} }}}$

This simplified equation was first proposed by De Bievre and Debus numerically and later by Riepe and Kaiser analytically. It has been noted that this simple expression is only a general approximation and it does not hold, for example, in the presence of Poisson statistics or in the presence of strong isotope signal ratio correlation.

Double dilution method

Single dilution method requires the knowledge of the isotopic composition of the isotopically-enriched analyte (R_B) and the amount of the enriched analyte added (n_B). 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*).

Isotope dilution notation
Name	Symbol
Analyte	A
Natural standard	A*
Isotopic standard (Spike)	B
Analyte + Spike	AB
Standard + Spike	A*B

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 (n_B):

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

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

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

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.

See also

• Standard addition
• Internal standard
• Mass spectrometry
• Mark and recapture
• Lincoln index

References

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