Q1. The blood volume in a cancer patient was measured by injecting 5.0 mL of Na2SO4(aq) labeled with 35S (t1/2 = 87.4 d). The activity of the sample was 300 µCi. After 22 min, 12.9 mL of blood was withdrawn from the man and the activity of that sample was found to be 0.75 µCi. Report the blood volume of the patient.

I listed below already about where I am confused....I have been doing these problems by converting the t1/2 into minutes and plugging everything in to N=Ne^-kt....but I think I am confusing some variables because my answers are a bit off... Same applies for Q2.

A radioactive sample contains 3.25 1018 atoms of a nuclide that decays at a rate of 3.4 1013 disintegrations per 26 min.
(a) What percentage of the nuclide will have decayed after 159 d?
%

(b) How many atoms of the nuclide will remain in the sample?
atoms

(c) What is the half-life of the nuclide?
days

I don't need the answers handed to me...I just want to know how to do it. As before, I listed my work already....I don't know if I am making mistakes in converting or if I need to use N=Ne^-kt in a different way than I already described. Thanks for the help...

I figured out the first one...still struggling with the 2nd...

Amountleft= Originalamount*e^-kt

-Rate=-k*original amount*e^-kt

3.4E13/26=3.25E18(k)e^k0

k= 4.023E-7 /min or 5.74E-4/day

amount left/original amount= e^-kt
= e^(-5.74E-4 *159.21E-2)
fraction left=9.21E-1=.921
fraction decayed=.079
percent decayed=7.9

halflife:

5.74E-4= ln2/thalflife
thalfflife= 1207 days

To solve these radioactive decay problems, you are on the right track by using the equation N = N₀ * e^(-kt), where N is the final number of atoms, N₀ is the initial number of atoms, k is the decay constant, and t is the time.

Let's go through the steps for each question:

Q1. Blood Volume Calculation:
First, we need to determine the decay constant (k) for the given 35S isotope. The half-life is given as t₁/₂ = 87.4 d.

Step 1: Convert the half-life to minutes:
t₁/₂ = 87.4 d * 24 hours/d * 60 minutes/hour ≈ 125,760 minutes

Step 2: Calculate the decay constant:
k = ln(2) / t₁/₂

Once you have obtained the value of k, you can proceed with the rest of the calculations.

Step 3: Calculate the initial number of atoms (N₀):
N₀ = (activity of sample / specific activity) * volume injected

Step 4: Calculate the final number of atoms (N):
N = (activity of sample / specific activity) * volume withdrawn

Step 5: Use the equation N = N₀ * e^(-kt) to relate N, N₀, and t and solve for t.

Now, substitute the values into the equation and solve for t:
N = N₀ * e^(-kt)

Remember, we are solving for the blood volume of the patient, so the concentration of the radioactive sample in the blood is constant.

Q2. Radioactive Decay Problems:
In this problem, you are given the initial number of atoms, the decay rate, and the time.

(a) To find the percentage of the nuclide that will have decayed after 159 days, you can follow the same steps mentioned above. Convert the given half-life to minutes, calculate the decay constant, and use the equation N = N₀ * e^(-kt) to find the final number of atoms (N). The percentage decayed will be (N₀ - N) / N₀ * 100.

(b) To find the number of atoms remaining in the sample, simply calculate N using the equation N = N₀ * e^(-kt).

(c) To find the half-life of the nuclide (t₁/₂), you can rearrange the equation N = N₀ * e^(-kt) to solve for t₁/₂. Rearranging the equation gives t₁/₂ = ln(2) / k.

Remember to convert the time units as necessary (e.g., days to minutes) and substitute the values into the equations to find the required quantities.

By following these steps, you should be able to solve these radioactive decay problems and get accurate answers.