---
title: "Isotope dilution"
chunk: 2/2
source: "https://en.wikipedia.org/wiki/Isotope_dilution"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T03:44:05.958864+00:00"
instance: "kb-cron"
---

  
    
      
        
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    {\displaystyle 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 RAB:

  
    
      
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    {\displaystyle 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 nA, ur(nA) = u(nA)/nA:

  
    
      
        
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    {\displaystyle 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}}
  

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

  
    
      
        
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    {\displaystyle 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):

  
    
      
        
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    {\displaystyle 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 Komori et al. and 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 ==
The 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*). First proposed in the 1940s and further developed in the 1950s, reverse isotope dilution remains an effective means of characterizing a labeled material.

Reverse isotope dilution analysis of the enriched analyte:

  
    
      
        
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    {\displaystyle 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:

  
    
      
        
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    {\displaystyle 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):

  
    
      
        
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    {\displaystyle 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:

  
    
      
        
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    {\displaystyle 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 the 1970s and developed in 2002.

== Calculations using calibration curve ==
Many analysts do not employ analytical equations for isotope dilution analysis. Instead, they rely on building a calibration curve from mixtures of the natural primary standard (A*) and the isotopically enriched standard (the spike, B). Calibration curves are obtained by plotting measured isotope ratios in the prepared blends against the known ratio of the sample mass to the mass of the spike solution in each blend. Isotope dilution calibration plots sometimes show nonlinear relationships and in practice polynomial fitting is often performed to empirically describe such curves.
When calibration plots are markedly nonlinear, one can bypass the empirical polynomial fitting and employ the ratio of two linear functions (known as Padé approximant) which is shown to describe the curvature of isotope dilution curves exactly.

== See also ==
Standard addition
Internal standard
Mass spectrometry
Mark and recapture
Lincoln index

== References ==

== Further reading ==