The radioactive isotope `""^(32)P` decays by first order kinetics and has a half-life of 14.3 days. How long does it take for 95.0% of a given sample of 32 p to decay?

Options

A. 21 days
B. 42 days
C. 62 days
D. 80 days

Explanation

To solve the problem of how long it takes for 95.0% of a given sample of the radioactive isotope \(^{32}P\) to decay, we will follow these steps:

### Step 1: Determine the decay constant (k)
The decay constant \(k\) for a first-order reaction can be calculated using the half-life (\(t_{1/2}\)) formula:
\[ k = \frac{0.693}{t_{1/2}} \]
Given that the half-life \(t_{1/2} = 14.3\) days, we can substitute this value into the equation:
\[ k = \frac{0.693}{14.3} \approx 0.0485 \text{ days}^{-1} \]

### Step 2: Set up the first-order kinetics equation
For a first-order reaction, the relationship between the initial concentration (\(A_0\)), the concentration at time \(t\) (\(A\)), and the amount that has decayed (\(x\)) can be expressed as:
\[ t = \frac{2.303}{k} \log \left( \frac{A_0}{A_0 - x} \right) \]
Here, since we want to find the time taken for 95% decay, we can express \(x\) as:
\[ x = 0.95 A_0 \]
Thus, \(A_0 - x = A_0 - 0.95 A_0 = 0.05 A_0\).

### Step 3: Substitute values into the equation
Now we can substitute \(A_0\) and \(A_0 - x\) into the equation:
\[ t = \frac{2.303}{k} \log \left( \frac{A_0}{0.05 A_0} \right) \]
This simplifies to:
\[ t = \frac{2.303}{k} \log \left( \frac{1}{0.05} \right) = \frac{2.303}{k} \log (20) \]

### Step 4: Calculate the logarithm
Next, we calculate \(\log(20)\):
\[ \log(20) = \log(2 \times 10) = \log(2) + \log(10) = 0.3010 + 1 = 1.3010 \]

### Step 5: Substitute \(k\) and \(\log(20)\) into the time equation
Now we can substitute \(k\) and \(\log(20)\) into the equation for \(t\):
\[ t = \frac{2.303}{0.0485} \times 1.3010 \]

### Step 6: Calculate \(t\)
Now we perform the calculation:
\[ t = \frac{2.303 \times 1.3010}{0.0485} \approx \frac{2.993}{0.0485} \approx 61.82 \text{ days} \]

### Step 7: Round the answer
Rounding \(61.82\) days gives us approximately \(62\) days.

### Final Answer
Therefore, it takes approximately **62 days** for 95.0% of the sample of \(^{32}P\) to decay.

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