Isotope Dilution Is A Mass Spectrometric Technique In Which

Isotope dilution is a mass spectrometric technique in whic

Isotope dilution is a mass spectrometric technique where a known amount of an unusual isotope, called the spike, is added to an unknown sample as an internal standard for quantitative analysis. The technique involves measuring the ratio of isotopes in the mixture, which allows calculation of the original element's concentration in the unknown sample. For the case of vanadium isotopes, natural atom fractions are 51V = 0.9975 and 50V = 0.0025, with the atom fraction of 51V defined as: atom fraction of 51V = atoms of 51V / (atoms of 51V + atoms of 50V). A spike enriched in 50V has atom fractions of 51V = 0.6391 and 50V = 0.3609. The problem involves deriving expressions to relate the isotope ratios before and after mixing, exploiting the known atom fractions and concentrations to determine the unknown vanadium concentration in a sample.

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Isotope dilution mass spectrometry (IDMS) is a highly precise and accurate analytical technique used for quantitative elemental analysis. Its core principle involves introducing a known quantity of an isotopically enriched standard—referred to as the spike—into an unknown sample, then measuring the resulting isotope ratios to deduce the original concentration of the element. When applied to vanadium analysis, natural isotopic compositions and specially enriched spikes provide the basis for precise calculations essential in various fields, including environmental monitoring, metallurgy, and quality control.

Theoretical Framework of Isotope Dilution

In isotope dilution, the key quantities are the atom fractions of isotopes A (51V) and B (50V) before and after mixing. Let Ax represent the atom fraction of isotope A in the unknown sample, and Bx for isotope B. Similarly, As and Bs denote the atom fractions in the spike. The total concentration of vanadium in the unknown and spike are represented by Cx and Cs respectively. When m_x grams of the unknown are mixed with m_s grams of the spike, the isotope ratios in the mixture, denoted by R, can be modeled mathematically.

The isotope ratio R for the mixture is defined as the ratio of isotope A to B, which can be expressed as:

R = (Atoms of A in mixture)/(Atoms of B in mixture)

Considering the contributions from the unknown and spike, the detailed derivation leads to an expression for R:

R = (A_x ≠ m_x C_x + A_s ≠ m_s C_s) / (B_x ≠ m_x C_x + B_s ≠ m_s C_s)

Normalizing and rearranging the equation yields a formula for the total concentration C_x in the unknown sample, based on known parameters and measured isotope ratios. The resulting formula simplifies to:

Cx = (Cs ⋅ (R – B_s) ⋅ (A_s – B_s)) / (A_x – B_x) ⋅ (R – A_s) )

Deriving the Expression for Cx

The derivation begins by expressing the measured isotope ratio R in terms of the contributions from the unknown and spike. Using the known atom fractions, the equation can be solved for Cx:

Cx = [ (Cs ⋅ (R – B_s)) / (A_x – B_x) ] ⋅ (A_s – B_s) / (R – A_s)

This expression relates the unknown sample’s concentration directly to the known quantities of the spike and the isotope ratio measurements, providing a practical formula for quantification in analytical applications.

Application to a Vanadium Sample

Given a sample of crude oil containing an unknown concentration of vanadium, and a spike enriched in 50V, the concentration Cx can be calculated using the derived formula. Using the provided data, including the mass of the sample, the spike concentration, isotope fractions, and the measured isotope ratio, the calculation proceeds as follows:

The measured isotope ratio R = 10.545 thus allows solving for Cx. Plugging in the known values, the calculation yields the vanadium concentration in the crude oil in mol/g, which can be critical for compliance with environmental regulations or for assessing the purity of the sample.

Conclusion

Isotope dilution mass spectrometry offers unparalleled accuracy in elemental quantification, leveraging the distinct isotopic signatures and precise measurements of isotope ratios. The mathematical derivations underpinning the technique are essential for translating isotope ratio data into meaningful concentration values. Its application to vanadium analysis demonstrates its utility in real-world analytical chemistry, ensuring reliable data for industrial and environmental assessments.

References

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