32.4: Isotope Dilution Methods

Another important radiochemical method for the analysis of nonradioactive analytes is isotope dilution. An external source of analyte is prepared in a radioactive form with a known activity, \(A_T\), for its radioactive decay—we call this form of the analyte a tracer. To prepare a sample for analysis we add a known mass of the tracer, w_T, to a portion of sample that contains an unknown mass, w_x , of analyte. After homogenizing the sample and tracer, we isolate w_A grams of analyte by using a series of appropriate chemical and physical treatments. Because these chemical and physical treatments cannot distinguish between radioactive and nonradioactive forms of the analyte, the isolated material contains both. Finally, we measure the activity of the isolated sample, A_A. If we recover all the analyte—both the radioactive tracer and the nonradioactive analyte—then A_A and \(A_T\) are equal and w_x = w_A – w_T. Normally, we fail to recover all the analyte. In this case \(A_A\) is less than \(A_T\), and

\[A_{A}=A_{T} \times \frac{w_{A}}{w_{x}+w_{T}} \label{13.8} \]

The ratio of weights in Equation \ref{13.8} accounts for any loss of activity that results from our failure to recover all the analyte. Solving Equation \ref{13.8} for w_x gives

\[w_{x}=\frac{A_{T}}{A_{A}} w_{A}-w_{T} \label{13.9} \]

How we process the sample depends on the analyte and the sample’s matrix. We might, for example, digest the sample to bring the analyte into solution. After filtering the sample to remove the residual solids, we might precipitate the analyte, isolate it by filtration, dry it in an oven, and obtain its weight.

Given that the goal of an analysis is to determine the amount of nonradioactive analyte in our sample, the realization that we might not recover all the analyte might strike you as unsettling. A single liquid–liquid extraction rarely has an extraction efficiency of 100%. One advantage of isotope dilution is that the extraction efficiency for the nonradioactive analyte and for the tracer are the same. If we recover 50% of the tracer, then we also recover 50% of the nonradioactive analyte. Because we know how much tracer we added to the sample, we can determine how much of the nonradioactive analyte is in the sample.

Example 32.4.1

The concentration of insulin in a production vat is determined by isotope dilution. A 1.00-mg sample of insulin labeled with ¹⁴C that has an activity of 549 cpm is added to a 10.0-mL sample taken from the production vat. After homogenizing the sample, a portion of the insulin is separated and purified, yielding 18.3 mg of pure insulin. The activity for the isolated insulin is measured at 148 cpm. How many mg of insulin are in the original sample?

Solution

Substituting known values into Equation \ref{13.8} gives

\[w_{x}=\frac{549 \text{ cpm}}{148 \text{ cpm}} \times 18.3 \text{ mg}-1.00 \text{ mg}=66.9 \text{ mg} \text { insulin } \nonumber \]

Equation \ref{13.8} and Equation \ref{13.9} are valid only if the tracer’s half-life is considerably longer than the time it takes to conduct the analysis. If this is not the case, then the decrease in activity is due both to the incomplete recovery and the natural decrease in the tracer’s activity. Table 32.4.1 provides a list of several common tracers for isotope dilution.

Table 32.4.1 . Common Tracers for Isotope Dilution

isotope	half-life
^3H	12.5 years
^14C	5730 years
^32P	14.3 days
^35S	87.1 days
^45Ca	152 days
^55Fe	2.91 years
^60Co	5.3 years
^131I	8 days

An important feature of isotope dilution is that it is not necessary to recover all the analyte to determine the amount of analyte present in the original sample. Isotope dilution, therefore, is useful for the analysis of samples with complex matrices, where a complete recovery of the analyte is difficult.