A sample of radioactive element has a mass of `10 g` at an instant `t=0`. The approximate mass of this element in the sample after two mean lives is

Options

A. `3.70 g`
B. `6.30 g`
C. `1.35 g`
D. `2.50 g`

Answer

To solve the problem of finding the approximate mass of a radioactive element after two mean lives, we can follow these steps:

### Step 1: Understand the Concept of Mean Life
The mean life (τ) of a radioactive substance is the average time it takes for half of the substance to decay. After one mean life, half of the original mass remains.

### Step 2: Determine the Initial Mass
We are given that the initial mass (m₀) of the radioactive element at time t=0 is 10 grams.

### Step 3: Calculate the Mass After One Mean Life
After one mean life (τ), the remaining mass can be calculated using the formula:
\[ m = m_0 \times e^{-\lambda t} \]
where \( \lambda \) is the decay constant and \( t \) is the time. However, for one mean life:
\[ m = \frac{m_0}{2} \]
So, after one mean life:
\[ m = \frac{10 \, \text{g}}{2} = 5 \, \text{g} \]

### Step 4: Calculate the Mass After Two Mean Lives
After two mean lives (2τ), we can again use the same concept:
\[ m = \frac{m_0}{2^2} \]
So, after two mean lives:
\[ m = \frac{10 \, \text{g}}{4} = 2.5 \, \text{g} \]

### Step 5: Approximate the Mass
To find the mass using the exponential decay formula:
\[ m = m_0 e^{-\lambda t} \]
For two mean lives (t = 2τ):
\[ m = 10 \, \text{g} \times e^{-2} \]
Calculating \( e^{-2} \) gives approximately 0.1353. Therefore:
\[ m \approx 10 \times 0.1353 \approx 1.353 \, \text{g} \]

### Final Result
The approximate mass of the radioactive element in the sample after two mean lives is approximately **1.35 grams**.

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