Question

radioactive decay of iodine - 125

1. what is the half - life of iodine?
2. what percent of iodine has decayed if 5 half - lives have passed?
3. if three half - lives have passed, how many grams of iodine - 125 remain undecayed?
4. if 130 days have passed approximately how many grams of iodine - 125 will remain?
5. a sample contains 200 grams of a substance with a half - life of 8 hours. how much will remain after 24 hours?
6. a 50 - gram sample of iodine - 131 decays to 6.25 grams. if the half - life is 8 days, how many days have passed?
7. a sample decays from 500 grams to 62.5 grams in 12 hours. what is the half - life?
8. a sample is reduced to 25% of its original amount in 10 minutes. what is the half - life of the substance?

Explanation:

Step1: Determine half - life from graph (for iodine - 125)

From the graph, when the mass of iodine - 125 reduces from 600 grams to 300 grams, the time elapsed is 60 days. So the half - life of iodine - 125 is 60 days.

Step2: Calculate decay percentage after 5 half - lives

The formula for the remaining amount after $n$ half - lives is $N = N_0\times(\frac{1}{2})^n$. The amount decayed is $1 - (\frac{1}{2})^n$. For $n = 5$, the amount decayed is $1-(\frac{1}{2})^5=1-\frac{1}{32}=\frac{31}{32}=96.875\%$.

Step3: Calculate remaining amount after 3 half - lives

Using $N = N_0\times(\frac{1}{2})^n$, if we assume the initial amount $N_0$ is some amount (let's say we start with 1 unit for simplicity), after $n = 3$ half - lives, $N=1\times(\frac{1}{2})^3=\frac{1}{8}$ of the original amount remains. If we had an initial amount, say $m_0$ grams, the remaining amount $m = m_0\times\frac{1}{8}$. But without an initial amount given in this part of the question related to iodine - 125 in context, if we assume we started with 600 grams (from the graph's starting point), then $m = 600\times\frac{1}{8}=75$ grams.

Step4: Estimate remaining amount at 130 days

From the graph, at 130 days, we can estimate the mass of iodine - 125 remaining by looking at the $y$ - value corresponding to $x = 130$ days on the graph. It is approximately 150 grams.

Step5: Calculate remaining amount for a different substance

The number of half - lives $n=\frac{24}{8}=3$. Using $N = N_0\times(\frac{1}{2})^n$, with $N_0 = 200$ grams, $N=200\times(\frac{1}{2})^3=200\times\frac{1}{8}=25$ grams.

Step6: Calculate time passed for iodine - 131

We know $N = N_0\times(\frac{1}{2})^n$. Given $N_0 = 50$ grams, $N = 6.25$ grams. So $6.25=50\times(\frac{1}{2})^n$. Then $(\frac{1}{2})^n=\frac{6.25}{50}=\frac{1}{8}$, so $n = 3$. Since the half - life is 8 days, the time passed $t=3\times8 = 24$ days.

Step7: Calculate half - life of a sample

We know $N = N_0\times(\frac{1}{2})^n$. Given $N_0 = 500$ grams, $N = 62.5$ grams. So $62.5=500\times(\frac{1}{2})^n$. Then $(\frac{1}{2})^n=\frac{62.5}{500}=\frac{1}{8}$, so $n = 3$. The time passed is 12 hours. If $n$ half - lives occur in 12 hours, then the half - life $T=\frac{12}{3}=4$ hours.

Step8: Calculate half - life of a sample

If a sample is reduced to 25% of its original amount, $N = 0.25N_0$. Using $N = N_0\times(\frac{1}{2})^n$, we have $0.25N_0=N_0\times(\frac{1}{2})^n$. So $(\frac{1}{2})^n = 0.25=\frac{1}{4}$, and $n = 2$. Since this occurs in 10 minutes, the half - life $T=\frac{10}{2}=5$ minutes.

Answer:

1. 60 days
2. 96.875%
3. 75 grams
4. 150 grams
5. 25 grams
6. 24 days
7. 4 hours
8. 5 minutes