Question

The rate of disintegration of a radioactive element at any time is proportional to its mass at that time, where \(k\) \((k > 0)\) is the constant of proportionality. The time during which an original mass of 1.5 gm will disintegrate to a mass of 0.5 gm is \(\ldots\)

• A. \(klog3\)
• B. \(\frac{1}{k}log3\)
• C. \(klog5\)
• D. \(\frac{1}{k}log5\)

Hint:

Solve dm/dt = -km to get m = m0 e^(-kt), then set m = m0/3.

Answer 1

The Correct Option is B

Step 1: Use the exponential solution directly:
Mass follows $m = m_0e^{-kt}$ for decay with constant $k$.

Step 2: Substitute the values:
$0.5 = 1.5\,e^{-kt}$, so $e^{kt} = \frac{1.5}{0.5} = 3$.

Step 3: Take logarithms:
$kt = \log 3$, so $t = \frac{\log 3}{k}$.

Step 4: Units check:
The constant $k$ has units of 1/time, so $t$ must be a number divided by $k$. That matches (B) and rules out (A) and (C), where $k$ is multiplied.

Final Answer:
$t = \frac1k\log 3$, option (B). \[ \boxed{\frac{1}{k}\log 3 \text{ (B)}} \]