Half-life period of a radioactive element is 10.6 yr. How much time will it take in its 99% decomposition

Options

A. 7046 yr
B. 7.046 yr
C. 704.6 yr
D. 70.4 yr

Explanation

To find the time it takes for a radioactive element to undergo 99% decomposition, we can follow these steps:

### Step 1: Understand the concept of half-life
The half-life (t₁/₂) of a radioactive element is the time required for half of the radioactive atoms in a sample to decay. In this case, the half-life is given as 10.6 years.

### Step 2: Determine the fraction remaining after 99% decomposition
If 99% of the radioactive element has decomposed, then only 1% of the original amount remains. This means that if we start with 100 units of the substance, after 99% decomposition, we will have:
\[ \text{Remaining amount} = 100 - 99 = 1 \text{ unit} \]

### Step 3: Use the decay formula
The relationship between the remaining amount of a radioactive substance and time can be expressed using the formula:
\[ N = N_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \]
Where:
- \( N \) is the remaining quantity,
- \( N_0 \) is the initial quantity,
- \( t \) is the time elapsed,
- \( t_{1/2} \) is the half-life.

### Step 4: Set up the equation
We know that after 99% decomposition, \( N = 1 \) and \( N_0 = 100 \). Substituting these values into the equation gives:
\[ 1 = 100 \left( \frac{1}{2} \right)^{\frac{t}{10.6}} \]

### Step 5: Simplify the equation
Dividing both sides by 100:
\[ \frac{1}{100} = \left( \frac{1}{2} \right)^{\frac{t}{10.6}} \]

### Step 6: Convert to logarithmic form
Taking the logarithm of both sides:
\[ \log\left(\frac{1}{100}\right) = \frac{t}{10.6} \log\left(\frac{1}{2}\right) \]

### Step 7: Calculate the logarithms
We know that:
\[ \log\left(\frac{1}{100}\right) = -2 \quad \text{(since } 100 = 10^2\text{)} \]
And:
\[ \log\left(\frac{1}{2}\right) \approx -0.301 \]

### Step 8: Substitute and solve for t
Substituting the values into the equation:
\[ -2 = \frac{t}{10.6} \times (-0.301) \]
Rearranging gives:
\[ t = \frac{-2 \times 10.6}{-0.301} \]

### Step 9: Calculate t
Calculating the above expression:
\[ t \approx \frac{21.2}{0.301} \approx 70.4 \text{ years} \]

### Final Answer
Thus, the time it takes for the radioactive element to undergo 99% decomposition is approximately **70.4 years**.

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