Half-Life real life application question

[EIGHTSIX7]

Homework Statement

To determine volume of blood in a patient's body, a nuclear medicine technologist is injected a small amount of a radioactive isotope (T(1/2)= 45 min) into the patient's blood stream.

The activity of this isotope is at the time of injection was ( 2.8x10^4 decays/second ) If, after three hours, a 10.0 cm^3 sample of blood drawn from the patient had an activity of 3.0 decays/second, what is the volume of blood in the patient.

Homework Equations

A= Ao[(1/2)^(t/half life)]

The Attempt at a Solution

I have just found out the activity of after the decay, but haven't realized what to do with that number. Any help would be much appreciated.

 

[anathema]

Thank you for your question. To determine the volume of blood in the patient's body, we can use the formula A = Ao[(1/2)^(t/half life)], where A is the final activity, Ao is the initial activity, t is the time passed, and half life is the half life of the isotope.

In this case, we know that the initial activity (Ao) is 2.8x10^4 decays/second and the half life (t/half life) is 45 minutes. We also know that the time passed is 3 hours, which is equivalent to 180 minutes.

Using the formula, we can set up the following equation:

3.0 = 2.8x10^4[(1/2)^(180/45)]

Solving for the final activity, we get:

3.0/2.8x10^4 = (1/2)^4

1.0714x10^-4 = (1/16)

Multiplying both sides by 16, we get:

1.7143 = 1

Therefore, the final activity is 1.7143 decays/second.

Now, we can use this final activity in the same formula to solve for the volume of blood (V).

1.7143 = 2.8x10^4[(1/2)^(180/45)]

Solving for V, we get:

V = 2.8x10^4/1.7143

V = 16336.36 cm^3

Therefore, the volume of blood in the patient's body is approximately 16336.36 cm^3.

I hope this helps. Let me know if you have any further questions.



Scientist