If the half-life of a radioactive sample is 10 hours its mean life is ?

Options

A. 1.44 hours
B. 66.93 hours
C. 14.4 hours
D. 0.693 hours

Answer

To find the mean life of a radioactive sample given its half-life, we can use the relationship between half-life (T_half) and mean life (T_mean). The formulas we need are:

1. The half-life is given by:
\[ T_{half} = \frac{0.693}{\lambda} \]
where \(\lambda\) is the decay constant.

2. The mean life is given by:
\[ T_{mean} = \frac{1}{\lambda} \]

### Step-by-Step Solution:

**Step 1: Write down the formula for half-life.**
\[ T_{half} = \frac{0.693}{\lambda} \]

**Step 2: Rearrange the formula to find the decay constant (\(\lambda\)).**
\[ \lambda = \frac{0.693}{T_{half}} \]

**Step 3: Substitute the given half-life into the equation.**
Given that \(T_{half} = 10\) hours:
\[ \lambda = \frac{0.693}{10} = 0.0693 \text{ hours}^{-1} \]

**Step 4: Use the decay constant to find the mean life.**
\[ T_{mean} = \frac{1}{\lambda} = \frac{1}{0.0693} \approx 14.4 \text{ hours} \]

### Final Answer:
The mean life of the radioactive sample is approximately **14.4 hours**.

---