This worksheet provides six practice problems for calculating radioactive half-life using various isotopes.

Let's solve each of these radioactive decay problems step by step using the concept of half-life and the exponential decay formula:

The general formula for radioactive decay is:

$$
N(t) = N_0 \times \left(\frac{1}{2}\right)^{t / T}
$$

Where:
- $ N(t) $ = amount remaining after time $ t $
- $ N_0 $ = initial amount
- $ T $ = half-life
- $ t $ = elapsed time

---

1. Zn-71: Half-life = 2.4 minutes, Initial = 100.0 g, Time = 7.2 minutes

We need to find how much remains after 7.2 minutes.

First, calculate number of half-lives:
$$
\text{Number of half-lives} = \frac{7.2}{2.4} = 3
$$

So, after 3 half-lives:
$$
N = 100.0 \times \left(\frac{1}{2}\right)^3 = 100.0 \times \frac{1}{8} = 12.5 \text{ grams}
$$

✔ Answer: 12.5 grams

---

2. Os-182: Half-life = 21.5 hours, Initial = 10.0 g, After 3 half-lives

After exactly 3 half-lives:
$$
N = 10.0 \times \left(\frac{1}{2}\right)^3 = 10.0 \times \frac{1}{8} = 1.25 \text{ grams}
$$

✔ Answer: 1.25 grams

---

3. Isotope: Half-life = 36.0 hours, Initial = 100.0 g, Final = 6.25 g, Find time

We use:
$$
6.25 = 100.0 \times \left(\frac{1}{2}\right)^{t / 36}
$$

Divide both sides by 100:
$$
0.0625 = \left(\frac{1}{2}\right)^{t / 36}
$$

Now recognize that:
$$
0.0625 = \frac{1}{16} = \left(\frac{1}{2}\right)^4
$$

So:
$$
\left(\frac{1}{2}\right)^{t / 36} = \left(\frac{1}{2}\right)^4 \Rightarrow \frac{t}{36} = 4 \Rightarrow t = 144 \text{ hours}
$$

✔ Answer: 144 hours

---

4. Radon-222: Half-life = 3.8 days, Initial = 100 g, Time = 15.2 days

Number of half-lives:
$$
\frac{15.2}{3.8} = 4
$$

So:
$$
N = 100 \times \left(\frac{1}{2}\right)^4 = 100 \times \frac{1}{16} = 6.25 \text{ grams}
$$

✔ Answer: 6.25 grams

---

5. Carbon-14: Half-life = 5,730 years, Initial = 70 mg, Time = 17,190 years

Number of half-lives:
$$
\frac{17,190}{5,730} = 3
$$

So:
$$
N = 70 \times \left(\frac{1}{2}\right)^3 = 70 \times \frac{1}{8} = 8.75 \text{ mg}
$$

✔ Answer: 8.75 mg

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6. Au-198: Initial = 5000 g, Final = 156.25 g, Time = 20 days, Find half-life

Use the formula:
$$
156.25 = 5000 \times \left(\frac{1}{2}\right)^{20 / T}
$$

Divide both sides by 5000:
$$
\frac{156.25}{5000} = 0.03125 = \left(\frac{1}{2}\right)^{20 / T}
$$

Now express 0.03125 as a power of 1/2:
$$
0.03125 = \frac{1}{32} = \left(\frac{1}{2}\right)^5
$$

So:
$$
\left(\frac{1}{2}\right)^{20 / T} = \left(\frac{1}{2}\right)^5 \Rightarrow \frac{20}{T} = 5 \Rightarrow T = \frac{20}{5} = 4 \text{ days}
$$

✔ Answer: 4 days

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✔ Final Answers:

1. 12.5 grams
2. 1.25 grams
3. 144 hours
4. 6.25 grams
5. 8.75 mg
6. 4 days

Let me know if you'd like a visual explanation or graph!