The average life of a radioactive element is 7.2 min. Calculate the time travel (in min) between the stages of 33.33% and 66.66% decay

Answer

To solve the problem of calculating the time interval between the stages of 33.33% and 66.66% decay for a radioactive element with an average life of 7.2 minutes, we can follow these steps:

### Step 1: Understand the relationship between average life and half-life
The average life (\( t_{avg} \)) of a radioactive substance is related to its half-life (\( t_{1/2} \)) by the formula:
\[ t_{avg} = 1.44 \times t_{1/2} \]

### Step 2: Substitute the given average life into the formula
We know that the average life is 7.2 minutes. We can substitute this value into the formula to find the half-life:
\[ 7.2 = 1.44 \times t_{1/2} \]

### Step 3: Solve for half-life
To find \( t_{1/2} \), we rearrange the equation:
\[ t_{1/2} = \frac{7.2}{1.44} \]
Calculating this gives:
\[ t_{1/2} = 5 \text{ minutes} \]

### Step 4: Calculate the time interval for decay stages
Now, we need to find the time taken for the decay from 33.33% to 66.66%.

- At 33.33% decay, 66.67% of the substance remains. This corresponds to one half-life, as it takes one half-life to go from 100% to 50%.
- At 66.66% decay, 33.34% of the substance remains. This corresponds to two half-lives, as it takes another half-life to go from 50% to 25%.

Thus, the time intervals can be calculated as follows:
- Time to go from 100% to 33.33% (which is one half-life): \( t_{1/2} = 5 \text{ minutes} \)
- Time to go from 100% to 66.66% (which is two half-lives): \( 2 \times t_{1/2} = 2 \times 5 = 10 \text{ minutes} \)

### Step 5: Calculate the time interval between the two stages
Now, to find the time interval between 33.33% and 66.66% decay:
\[ \text{Time interval} = \text{Time at 66.66%} - \text{Time at 33.33%} \]
\[ \text{Time interval} = 10 \text{ minutes} - 5 \text{ minutes} = 5 \text{ minutes} \]

### Final Answer
The time interval between the stages of 33.33% and 66.66% decay is **5 minutes**.

---