A 1.00 mL sample of red blood cells containing chromium-51 as a tracer was injected into a patient. After several hours, a 5.00 mL sample of blood was drawn and its activity compared to the activity of the injected tracer sample. If the collected sample activity was 0.10% of the original tracer, calculate the total blood volume of the patient (see the Chemistry in Action 'Medical Uses of Radioactivity,' p. 338).

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Hey, everyone. We're told that a five millimeter sample of radionuclide labeled red blood cells is delivered to a patient. The initial activity of the sample is 4.6 militaries. After some time, two mL of the patient's blood was collected and found to have an activity of 1.67 times 10 to the negative. Third military's estimate the total blood fall volume of the patient. For this question, the blood volume can be estimated by comparing the concentration of labeled cells in the blood versus the labeled cells injected. The injected labeled cells will be circulated and distributed throughout the body which is similar to our isotope dilution. So to answer this question, we need to use the following equation where we have the radio activity of our sample, which is signified by our is equal to the radio activity of our tracer times the mass or volume of our sample, which is signified by w which is then divided by the mass or volume of our system plus the mass or the volume of our tracer. So here our system is going to be our blood and they also gave us the volume. So let's go ahead and rewrite our equation. So we have the radio activity of our sample is equal to the radio activity of our tracer. And we're going to multiply this by the volume of our sample and then divided by the volume of our blood. Now a key thing to note here is that we're going to assume that the volume of our blood is much greater than the volume of our tracer, which means that our volume of our tracer will be negligible. So we remove that from our equation. Now, looking back at our question, it wants us to estimate the total blood volume. So we have to go ahead and rearrange to solve for our volume of our blood to solve the volume of our blood. We're going to take the radio activity of our tracer times the volume of our sample dividing by the radio activity of our sample. So let's go ahead and plug in our values. We were told that the radio activity of our tracer is 4.6 militaries. And we're going to multiply this by the volume of our sample, which is 2.0 mL. We're then going to divide this all by the radio activity of our sample, which was said to be 1.67 times 10 to the negative third militaries. Now, when we calculate this out and cancel out our units, we end up with a volume of ml which we can then convert into leaders by using dimensional analysis. And we know that one male leaders contain 10 to the negative 3rd leaders. So this gets us to 5.5 L as our final total blood volume. Now, I hope this made sense and let us know if you have any questions.

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Key Concepts

Radioactive Tracers

Dilution Principle

Activity Measurement