Question

Half-life $(T_{1/2})$ of a radioisotope is related to decay constant $(\lambda)$:

• A. $T_{1/2} = \frac{0.693}{\lambda}$
• B. $T_{1/2} = e^{-\lambda}$
• C. $T_{1/2} = -\lambda N dt$
• D. $T_{1/2} = -\lambda e^{-t}$

Hint:

Important relation: - $T_{1/2} = \frac{0.693}{\lambda}$ - Mean life $= \frac{1}{\lambda}$

Answer 1

The Correct Option is A

Concept: Radioactive decay follows an exponential law: \[ N = N_0 e^{-\lambda t} \] where:
• $N_0$ = initial number of nuclei
• $\lambda$ = decay constant The half-life is the time required for the quantity to reduce to half its initial value.

Step 1: Apply half-life condition.
At $t = T_{1/2}$: \[ N = \frac{N_0}{2} \]

Step 2: Substitute in decay equation.
\[ \frac{N_0}{2} = N_0 e^{-\lambda T_{1/2}} \]

Step 3: Simplify.
\[ \frac{1}{2} = e^{-\lambda T_{1/2}} \]

Step 4: Take natural logarithm.
\[ \ln\left(\frac{1}{2}\right) = -\lambda T_{1/2} \]

Step 5: Final expression.
\[ T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda} \]