Extrapolating a Shelf Life: A Reconstituted Antibiotic Suspension

Test first order: is ln(fraction remaining) linear in t?

t / d	Fraction	ln(fraction)	Δln per 14 d
0	1.000	0	-
14	0.880	−0.1278	0.1278
28	0.774	−0.2562	0.1284
42	0.681	−0.3840	0.1278

The increments are constant to within 0.5 %, so a plot of ln c against t is a straight line through the origin → first order.

k = 0.1278 / 14 = 9.13 × 10⁻³ day⁻¹

Why zero order fails: a zero-order reaction would lose a constant amount per unit time (constant differences in percentage remaining). The observed drops are 12.0, 10.6 and 9.3 percentage points - falling steadily, not constant. A constant-amount model cannot fit all four points.

t90 and t½:

• t90 = −ln(0.90)/k = 0.1054 / 9.13 × 10⁻³ = 11.5 days
• t½ = 0.693 / 9.13 × 10⁻³ = 75.9 days

So at 25 °C the product falls below the 90 % specification after about 11–12 days - already shorter than the 14-day refrigerated label, which is only defensible because storage at 2–8 °C is much slower.

Get k at 40 °C the same way.

ln 0.794 = −0.2307 (7 d); ln 0.630 = −0.4620 (14 d) Increments: 0.2306 and 0.2313 → consistent first order. k₄₀ = 0.2310 / 7 = 3.30 × 10⁻² day⁻¹

Arrhenius: Ea = R ln(k₂/k₁) / (1/T₁ − 1/T₂)

• k₂/k₁ = 3.30 × 10⁻² / 9.13 × 10⁻³ = 3.615; ln = 1.285
• 1/298.15 − 1/313.15 = 3.354 × 10⁻³ − 3.193 × 10⁻³ = 1.606 × 10⁻⁴ K⁻¹
• Ea = 8.314 × 1.285 / 1.606 × 10⁻⁴ = 6.65 × 10⁴ J mol⁻¹ ≈ 66.5 kJ mol⁻¹

Extrapolate to 4 °C (277 K):

ln(k₂₇₇/k₂₉₈) = −(Ea/R)(1/277 − 1/298.15) = −(66 500/8.314)(3.610 × 10⁻³ − 3.354 × 10⁻³) = −8000 × 2.561 × 10⁻⁴ = −2.049

k₂₇₇ = 9.13 × 10⁻³ × e^(−2.049) = 9.13 × 10⁻³ × 0.129 = 1.18 × 10⁻³ day⁻¹

t90 = 0.1054 / 1.18 × 10⁻³ = 89 days ≈ 3 months

Assumptions being made: that the reaction is first order at both temperatures and stays so at 4 °C; that a single activation energy applies over a 36 K extrapolation; and that the same degradation route (hydrolysis) dominates throughout. None of these is tested by two runs at two temperatures.

Advise against relabelling on this evidence. Keep 14 days and generate real-time data.

The calculation is arithmetically sound but answers the wrong question. Chemical potency is only one of several shelf-life-determining attributes, and here it is probably not the limiting one.

Why the extrapolation is unsafe:

• Two points, 36 K apart. Arrhenius plots are curved when the rate-determining step changes with temperature. Hydrolysis, oxidation and photolysis have very different Ea values; cooling may switch which one dominates, so the low-temperature prediction can be wrong in either direction.
• The accelerated run is short and deep. It reaches 37 % degradation in 14 days, well outside the 10 % window being modelled, and degradation products may catalyse further change.
• The product is not sterile. It is an unpreserved aqueous multi-dose suspension. Microbial contamination and preservative efficacy in a bottle opened repeatedly by a patient is the usual reason for a 7–14 day limit, and no chemical kinetics calculation addresses it.
• Dose uniformity and redispersibility are not tested by a potency assay; a settled suspension that cannot be resuspended fails even if the assay is 95 %.
• Real-world temperature excursions. t90 at 25 °C is only ~11 days, so several hours at room temperature each day erodes the margin. The kinetic model assumes isothermal storage, which the patient's kitchen does not provide.

What would justify an extension: real-time stability at 5 ± 3 °C in the marketed container out to the proposed 3 months, with potency, pH, appearance, redispersibility and dose uniformity at each point; a preservative efficacy test and microbial challenge; and in-use studies mimicking patient handling. Only then would a 3-month claim be a data-based judgement rather than an extrapolation.