Question

What is the shelf life of drug which decomposes with first order having reaction rate constant of 0.002 day\(^{-1}?\)

• A. \( 425 \text{ days} \)
• B. \( 525 \text{ days} \)
• C. \( 625 \text{ days} \)
• D. \( 725 \text{ days} \)

Hint:

For first-order reactions, shelf life is inversely proportional to the rate constant. A smaller rate constant leads to a longer shelf life.

Answer 1

The Correct Option is B

The shelf life ($t_{90}$) of a drug that follows first-order kinetics is the time required for 10% of the drug to decompose, leaving 90% of the original concentration. The formula for the shelf life of a first-order reaction is: $$ t_{90} = \frac{0.105}{k} $$ where $k$ is the first-order rate constant. Given $k = 0.002 \, \text{day}^{-1}$, we can calculate $t_{90}$: $$ t_{90} = \frac{0.105}{0.002 \, \text{day}^{-1}} = 52.5 \, \text{days} $$ There seems to be a discrepancy between the calculated value (52.5 days) and the provided correct answer (525 days). Let's double-check the formula and the interpretation. Re-evaluating the formula for $t_{90}$ for a first-order reaction: The amount of drug remaining at time $t$ is given by $C(t) = C_0 e^{-kt}$, where $C_0$ is the initial concentration. For shelf life $t_{90}$, $C(t_{90}) = 0.9 C_0$. So, $0.9 C_0 = C_0 e^{-kt_{90}}$ $0.9 = e^{-kt_{90}}$ Taking the natural logarithm of both sides: $\ln(0.9) = -kt_{90}$ $t_{90} = -\frac{\ln(0.9)}{k} = \frac{\ln(1/0.9)}{k} = \frac{\ln(1.111)}{k}$ $t_{90} = \frac{0.1054}{0.002 \, \text{day}^{-1}} \approx 52.7 \, \text{days}$ There still appears to be a significant difference from the provided answer of 525 days. It's possible there was a mistake in the question or the provided options/answer. However, based on standard first-order kinetics, the shelf life should be approximately 52.5 - 52.7 days. Given the choices, 525 days is the closest if we assume a rate constant of 0.0002 day$^{-1}$ instead of 0.002 day$^{-1}$. If the rate constant was indeed 0.0002 day$^{-1}$: $$ t_{90} = \frac{0.105}{0.0002 \, \text{day}^{-1}} = 525 \, \text{days} $$ Assuming there was a typo in the rate constant in the question, and it should have been 0.0002 day$^{-1}$, then 525 days would be the correct answer.