First-order drug hydrolysis: shelf life, refrigeration, and a temperature excursion
A hospital pharmacy prepares a 5 mg mL−1 oral solution of an antibiotic that hydrolyses by first-order kinetics. A stability study gives k = 4.0×10−3 h−1 at 25 °C and 3.2×10−2 h−1 at 40 °C. The product is labelled to contain not less than 90% of the initial amount, and the proposed shelf life is 24 h at 25 °C. The pharmacy has a refrigerator at 8 °C. A nurse reports that a batch was left in a ward at 40 °C for 6 h before being returned to 25 °C for the remaining 18 h of its claimed 24 h life. The quality officer wants to know whether the 24 h claim is defensible, whether refrigerated storage could extend the shelf life to 7 days, and what activation energy should be quoted. Assume the Arrhenius pre-exponential factor and activation energy remain constant over 8–40 °C.
1. Calculate the activation energy and the t90 at 25 °C. Is the 24 h claim at 25 °C defensible?
Read the worked answer
ln(k2/k1) = ln(3.2×10−2 / 4.0×10−3) = ln 8 = 2.079.
Ea = R ln(k2/k1) × T1T2/(T2 − T1) = 8.314 × 2.079 × (298 × 313 / 15) = 8.314 × 2.079 × 6218 = 1.075×105 J mol−1 = 107.5 kJ mol−1.
For a first-order reaction, t90 = ln(100/90)/k = 0.10536/k. At 25 °C: t90 = 0.10536 / 4.0×10−3 = 26.3 h.
The 24 h claim leaves only 2.3 h, about 9% margin. It is defensible only if storage temperature is controlled and the stability-study uncertainty is small; any excursion above 25 °C erodes the margin. A cautious label would shorten the shelf life or require refrigeration.
2. Estimate k at 8 °C and decide whether refrigerated storage can support a 7-day shelf life.
Read the worked answer
k(281 K) = 4.0×10−3 exp[−(107500/8.314)(1/281 − 1/298)].
The exponent bracket is: 12930 × (17/(281 × 298)) = 12930 × 2.030×10−4 = 2.625. So exp(−2.625) = 0.0724 and k = 2.90×10−4 h−1.
t90 = 0.10536 / 2.90×10−4 = 363 h = 15.1 days.
Seven days is 168 h, well below 363 h, so refrigerated storage at 8 °C can support a 7-day shelf life with about a factor of 2.2 margin. The judgement assumes the solution is chemically stable at 8 °C and does not crystallise or phase-separate on cooling; if it does, the prediction fails.
3. Calculate the potency remaining for the reported batch (6 h at 40 °C followed by 18 h at 25 °C). Does the batch meet the 90% limit?
Read the worked answer
For first-order kinetics the remaining fractions multiply:
f = exp(−k40 t40) × exp(−k25 t25) = exp(−0.032 × 6) × exp(−0.004 × 18) = exp(−0.192) × exp(−0.072) = 0.825 × 0.930 = 0.768.
Only 76.8% remains, below the 90% limit, so that batch fails the 24 h claim. The error is to average the temperature or to assume the 25 °C shelf life is unaffected by the excursion. A 6 h exposure at 40 °C consumes about 17.5% of potency, which is more than the entire allowed 10% loss. The batch should be discarded, and the label should specify that the product must not exceed 25 °C.