Predicting shelf life: the Arrhenius equation and accelerated stability testing

The Arrhenius equation, k = A·e^−Ea/RT, describes how a reaction's rate constant k rises with temperature T. It explains why labs keep peptides in a freezer rather than on a shelf, and how a manufacturer can state a shelf life of two years without waiting two years. This guide explains it in plain terms, shows how accelerated stability studies use it, and sets out where its estimates stop being reliable.

Terms are defined in the glossary.

Why temperature speeds up breakdown

Molecules in a sample are always moving and colliding. Most collisions do nothing. A reaction happens only when molecules meet with enough energy to break and re-form bonds. That minimum energy is the activation energy.

Raising the temperature does more than speed molecules up a little. It sharply increases the share of collisions that have enough energy to react. That is why a small change in temperature can make a large change in how fast something breaks down.

The reactions that matter for stored peptides are covered in shelf life and stability: hydrolysis of peptide bonds, oxidation of methionine and cysteine, deamidation of asparagine and glutamine, and aggregation. Each has its own activation energy, so each responds to temperature differently. For rate constants, reaction order and half-lives in general, see rates of reaction.

The Arrhenius equation

Svante Arrhenius proposed the equation in 1889. It is written:

k = A · e^{−Ea / RT}

To convert Celsius to kelvin, add 273.15. So 25 °C is 298.15 K, 5 °C is 278.15 K and −20 °C is 253.15 K. See temperature scales. The kelvin scale starts at absolute zero, so it never goes negative, which the equation needs (BIPM).

The important part is the exponent. Because temperature sits in the denominator of an exponent, the rate changes exponentially with temperature, not in a straight line.

Comparing two temperatures

You rarely need A. To compare the rate at two temperatures, divide one form of the equation by the other:

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

This shows how many times faster a reaction runs at T₂ than at T₁, given only the activation energy.

Plotting ln k against 1/T gives a straight line with a slope of −Ea/R. This is an Arrhenius plot. Measuring the rate at three or more temperatures and fitting the line is how activation energy is found in practice.

How to calculate activation energy from two temperatures

If you know the rate at two temperatures, rearrange the two-temperature form to give the activation energy:

Ea = R × ln(k₂ / k₁) ÷ (1/T₁ − 1/T₂)

Worked example: a material loses 0.2% purity a month at 25 °C and 1.0% a month at 40 °C.

1. Convert to kelvin: T₁ = 298.15 K, T₂ = 313.15 K.
2. Ratio of rates: k₂ / k₁ = 1.0 ÷ 0.2 = 5, and ln 5 = 1.609.
3. 1/T₁ − 1/T₂ = 0.0033540 − 0.0031934 = 0.0001607 per kelvin.
4. Ea = 8.314 × 1.609 ÷ 0.0001607 ≈ 83,000 J/mol, or about 83 kJ/mol.

Two points always fit a straight line, so this gives no check that the reaction really follows the equation. Three or more temperatures, fitted as an Arrhenius plot, are more reliable.

Q10: the rule of thumb

Q10 is the factor by which a rate increases for a 10 °C rise. It follows from the Arrhenius equation once you pick an activation energy. Between 20 and 30 °C, around room temperature:

Many degradation reactions of drugs and peptides in solution have activation energies in the range of roughly 50–120 kJ/mol. That is where the common rule of thumb comes from: every 10 °C of cooling slows breakdown by a factor of two to four.

How accelerated stability testing works

An accelerated study turns the equation around. Instead of waiting for slow breakdown at the storage temperature, it measures fast breakdown at higher temperatures and works back.

1. Store samples at several raised temperatures, for example 25, 40 and 50 °C.
2. Test them at intervals with a stability-indicating method, usually HPLC to track purity and new impurity peaks, with mass spectrometry to identify them.
3. Work out a rate at each temperature from how fast purity falls.
4. Fit an Arrhenius plot to find the activation energy.
5. Extrapolate to the storage temperature to estimate how long the material stays within specification.

Standard conditions

The ICH stability guideline, written for medicines but widely used as a reference, sets standard storage conditions (ICH Q1A(R2)):

Note the humidity column. For a dry powder, moisture can matter as much as temperature, so the conditions control both. See humidity, dew point and condensation.

A worked example

Suppose an accelerated study finds a material loses 1% of its purity per month at 40 °C, and the measured activation energy is 80 kJ/mol. Using the two-temperature form of the equation:

• At 25 °C the rate is about 4.7 times slower: roughly 0.2% a month. Losing 2% would take about 9 months.
• At 5 °C it is about 10 times slower again: roughly 0.02% a month. Losing 2% would take about 8 years.
• At −20 °C the arithmetic gives about 240 years.

The last figure shows the method's limit. Nobody would state a 240-year shelf life. Extrapolating that far beyond the tested range assumes the same reaction, with the same activation energy, controls breakdown across 60 °C, and in a frozen solid it usually doesn't.

These numbers are an illustration of the method, not a prediction for any product.

Where the estimates go wrong

The Arrhenius approach assumes one well-behaved reaction. Real samples, especially freeze-dried powders, break that assumption in several ways.

• The reaction changes with temperature. One reaction may dominate at 40 °C and a different one at 5 °C. An estimate built on the first says nothing reliable about the second.
• Dry solids change physically. A lyophilised cake is a glassy solid. Above its glass transition temperature it softens, molecules move more freely, and breakdown speeds up much faster than the equation predicts. Results from above that temperature don't carry over to storage below it.
• Moisture interferes. Warmer samples can take up or lose water differently. Since water drives hydrolysis and increases molecular movement in the solid, a change in water content can outweigh the effect of temperature. See hygroscopic powders.
• Aggregation doesn't follow simple kinetics. Molecules clumping together can speed up as it goes, or happen in bursts.
• Small changes are hard to measure. If purity falls by only 0.2% over a study, the result may sit within the method's normal spread. See accuracy, precision and significant figures.

This is why the ICH guideline treats accelerated data as supporting evidence. A shelf life rests on real-time data at the intended storage condition, and accelerated results are used to support it and to judge the effect of short excursions.

Mean kinetic temperature: judging temperature swings

Storage temperatures are never perfectly steady. A sample might spend a few days in transit or sit in a room that warms in the afternoon. Mean kinetic temperature (MKT) uses the Arrhenius equation to express a varying temperature as the single steady temperature that would cause the same amount of breakdown.

Because breakdown rises exponentially with temperature, warm spells count for more than cool ones. For example, five days at 20 °C and two days at 30 °C average 22.9 °C. The MKT, using the standard assumed activation energy of 83.144 kJ/mol, is about 24.1 °C. The warm days pull it up.

MKT is used to judge whether a shipment or storage period stayed within acceptable limits. It doesn't make a long excursion harmless. It gives a fairer way to compare one temperature history with another. For why freeze-dried peptides can travel without cold packs, see why lyophilised peptides ship at room temperature.

What this means for storage

The equation explains the usual storage advice:

• Colder is slower. Each step down, from room temperature to a fridge to a freezer, slows chemical breakdown by a large factor.
• Dry matters as much as cold. Keeping a powder dry keeps it below its glass transition and starves hydrolysis of water.
• Short warm spells add up. They are weighted more heavily than their length suggests, which is why vials shouldn't sit on a warm bench longer than needed.

For practical storage, see how to store freeze-dried peptides and freezer temperatures for sample storage.

What a shelf life on our products means

A certificate of analysis describes a lot at the time it was tested. It doesn't predict a shelf life by itself, and we don't extrapolate one from the Arrhenius equation. We don't state a shelf life for a product unless there is stability data to support it. Each product page lists the recommended storage conditions for that compound. Every lot has its own COA, published on the product page and on that lot's own page. See Quality.