In isotope dilution, a known amount of an unusual isotope (called the spike) is added to an unknown as an internal standard for quantitative analysis. After the mixture has been homogenized, some of the element of interest must be isolated. The ratio of the isotopes is then measured. From this ratio, the quantity of the element in the original unknown can be calculated. Natural vanadium has atom fractions ⁵¹V = 0.9975 and ⁵⁰V = 0.0025. The atom fraction is defined as

A spike enriched in ⁵⁰V has atom fractions ⁵¹V = 0.639 1 and ⁵⁰V = 0.360 9.
(a) Let isotope A be ⁵¹V and isotope B be ⁵⁰V. Let Ax be the atom fraction of isotope A (= atoms of A/[atoms of A + atoms of B]) in an unknown. Let Bx be the atom fraction of B in an unknown. Let As and Bs be the corresponding atom fractions in a spike. Let Cx be the total concentration of all isotopes of vanadium (μmol/g) in the unknown and let Cs be the concentration in the spike. Let mx be the mass of unknown and ms be the mass of spike. After mx grams of unknown are mixed with ms grams of spike, the ratio of isotopes in the mixture is found to be R. Show that

(b) Solve Equation A for Cx to show that

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