Question

Half life of radio-active element is $1600$ years. The fraction of sample remains undecayed after $6400$ years will be

• A. \frac{1}{16}
• B. \frac{1}{4}
• C. \frac{1}{8}
• D. \frac{1}{24}

Hint:

The fraction of a radioactive substance remaining after $n$ half-lives is always $(1/2)^n$. Simply count how many times the half-life fits into the total time.

Answer 1

The Correct Option is A

Step 1: First, determine the number of half-lives $n$ that have passed in the given total time $t = 6400$ years, with a half-life $T_{1/2} = 1600$ years: \[ n = \frac{t}{T_{1/2 \] \[ n = \frac{6400}{1600} = 4 \text{ half-lives} \]
Step 2: The fraction of the radioactive sample remaining undecayed after $n$ half-lives is given by the formula: \[ f = \left(\frac{1}{2}\right)^n \]
Step 3: Substitute $n = 4$ into the formula: \[ f = \left(\frac{1}{2}\right)^4 = \frac{1}{2 \times 2 \times 2 \times 2} \] \[ f = \frac{1}{16} \]