Question

If in the case of a radio-isotope, the value of half-life(T_1/2) and decay constant(λ) are identical in magnitude, then their value should be

• A. \(\frac{0.693}{2}\)
• B. (0.693)^1/2
• C. (0.693)²
• D. 0.693

Answer 1

The Correct Option is B

The correct answer is option (B): \( (0.693)^{1/2} \)

Explanation: 

For a first-order reaction, the relationship between the half-life \( t_{1/2} \) and the rate constant \( \lambda \) (also denoted as \( k \)) is given by:

\[ t_{1/2} = \frac{0.693}{\lambda} \]

Now, let’s say the value of half-life \( t_{1/2} \) is given as 1 unit of time. Then:

\[ 1 = \frac{0.693}{\lambda} \Rightarrow \lambda = 0.693 \]

But if we reverse the question and say:

Given that:

\[ 0.693 = \lambda^2 \]

Then, taking square root on both sides:

\[ \lambda = \sqrt{0.693} = (0.693)^{1/2} \]

Therefore, the value of \( \lambda \) is \( (0.693)^{1/2} \), and that’s why option (B) is correct.