What is the probability that a radioactive atom having a mean life of 10 days decays during the fifth day?

Explanation

To find the probability that a radioactive atom with a mean life of 10 days decays during the fifth day, we can follow these steps:

### Step 1: Understand the relationship between mean life and decay constant
The mean life (\( \tau \)) of a radioactive substance is related to the decay constant (\( \lambda \)) by the formula:
\[ \tau = \frac{1}{\lambda} \]
Given that the mean life is 10 days, we can find \( \lambda \):
\[ \lambda = \frac{1}{\tau} = \frac{1}{10 \text{ days}} = 0.1 \text{ days}^{-1} \]

**Hint:** Remember that the decay constant is the reciprocal of the mean life.

### Step 2: Determine the probability of decay during the fifth day
The probability that a radioactive atom decays during a specific time interval can be calculated using the exponential decay formula. The probability of decay during the interval from \( t = 4 \) days to \( t = 5 \) days is given by:
\[ P(4 < t < 5) = P(t < 5) - P(t < 4) \]
Where:
- \( P(t < t_0) = 1 - e^{-\lambda t_0} \)

### Step 3: Calculate \( P(t < 5) \)
Using the decay constant \( \lambda = 0.1 \text{ days}^{-1} \):
\[ P(t < 5) = 1 - e^{-\lambda \cdot 5} = 1 - e^{-0.1 \cdot 5} = 1 - e^{-0.5} \]

### Step 4: Calculate \( P(t < 4) \)
Similarly, calculate for \( t = 4 \):
\[ P(t < 4) = 1 - e^{-\lambda \cdot 4} = 1 - e^{-0.1 \cdot 4} = 1 - e^{-0.4} \]

### Step 5: Find the probability of decay during the fifth day
Now, substitute the values into the probability formula:
\[ P(4 < t < 5) = (1 - e^{-0.5}) - (1 - e^{-0.4}) = e^{-0.4} - e^{-0.5} \]

### Step 6: Calculate the numerical values
Using a calculator or exponential tables:
- \( e^{-0.4} \approx 0.6703 \)
- \( e^{-0.5} \approx 0.6065 \)

Now, substitute these values:
\[ P(4 < t < 5) \approx 0.6703 - 0.6065 = 0.0638 \]

### Final Answer
The probability that the radioactive atom decays during the fifth day is approximately **0.0638** or **6.38%**.

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