Drug X becomes ineffective after 50% decomposition. The original concentration of drug in a bottle was 16mg/mL which becomes 4mg/mL in 12 months. The expiry time of the drug in months is _________
Assume that the decomposition of the drug follows first order kinetics.

In first-order kinetics, the concentration at time $$t$$ is given by $$ C = C_0 e^{-kt} $$.

Here the initial concentration is $$C_0 = 16$$ mg/mL and after $$t = 12$$ months the concentration is $$C = 4$$ mg/mL.

Substituting these values gives $$4 = 16 e^{-12k}$$, which implies $$e^{-12k} = \frac{1}{4}$$ and so $$k = \frac{\ln 4}{12} = \frac{2\ln 2}{12} = \frac{\ln 2}{6}$$.

The drug becomes ineffective after 50% decomposition, that is when $$C = \frac{C_0}{2} = 8$$ mg/mL.

The half-life for first-order kinetics is $$t_{1/2} = \frac{\ln 2}{k} = \frac{\ln 2}{\ln 2/6} = 6 \text{ months}$$, so the expiry time is 6 months.

The correct answer is Option 2: 6.