Drug X becomes ineffective after 50% decomposition. The original concentration of drug in a bottle was 16 mg/mL which…

Step 1 — Find the rate constant k

For first-order kinetics: $\ln\left(\dfrac{[A]_0}{[A]_t}\right) = kt$

At $t = 12$ months: $[A]_0 = 16$ mg/mL, $[A]_t = 4$ mg/mL

$\ln\left(\dfrac{16}{4}\right) = k \times 12$

$\ln 4 = 12k$

$k = \dfrac{\ln 4}{12} = \dfrac{2\ln 2}{12} = \dfrac{\ln 2}{6}$ months$^{-1}$

Step 2 — Find expiry time (50% decomposition)

Drug becomes ineffective after 50% decomposition, i.e., when $[A]_t = 0.5 \times [A]_0 = 8$ mg/mL.

This is the half-life $t_{1/2}$:

$t_{1/2} = \dfrac{\ln 2}{k} = \dfrac{\ln 2}{\ln 2 / 6} = 6$ months

Answer: 6 months → Option (d)

Key Points to Remember:

• First-order: $\ln([A]_0/[A]_t) = kt$
• $k = \ln 4/12 = \ln 2/6$ months$^{-1}$
• Expiry = $t_{1/2} = \ln 2/k = 6$ months
• Alternatively: $t_2/t_1 = (C_1/C_2)^n$ for $n=1$: $t_2 = (4/8)^1 \times 12 = 6$ months