Arrhenius Equation and Stability Studies

Arrhenius equation 

The Arrhenius equation is the one which explains the temperature dependence of the reaction rate constant, and therefore, rate of a chemical reaction.

The equation was first proposed by the Dutch chemist J. H. van 't Hoff in 1884; five years later in 1889, the Swedish chemist Svante Arrhenius provided a physical justification and interpretation for it. 

Common sense and chemical intuition suggest that the higher the temperature, the faster a given chemical reaction will proceed. Quantitatively this relationship between the rate a reaction proceeds and its temperature is determined by the Arrhenius Equation.

At higher temperatures, the probability that two molecules will collide is higher. This higher collision rate results in a higher kinetic energy, which has an effect on the activation energy of the reaction. The activation energy is the amount of energy required to ensure that a reaction happens.

After observing that many chemical reaction rates depended on the temperature, Arrhenius developed this equation to characterize the temperature-dependent reactions.

The Factors

K : Chemical reaction rate and is unit of s^-1(for 1st order rate constant) or M^-1s^-1(for 2nd order rate constant)

A is pre-exponential factor or frequency factor and is realted to mollecular collisions Deals with the frequency of molecules that collide in the correct orientation and with enough energy to initiate a reaction. It is a factor that is determined experimentally, as it varies with different reactions.In unit of Lmol^-1s^-1 or M^-1s^-1(for 2^nd order rate constant) and s^-1(for 1^st order rate constant. Because frequency factor A is related to molecular collision, it is temperature dependent. Hard to extrapolate pre-exponential factor because lnk is only linear over a narrow range of temperature  

E_a: The activation energy is the threshold energy that the reactant(s) must acquire before reaching the transition state. Once in the transition state, the reaction can go in the forward direction towards product(s), or in the opposite direction towards reactant(s).A reaction with a large activation energy requires much more energy to reach the transition state.Likewise, a reaction with a small activation energy doesn't require as much energy to reach the transition state. In unit of kJ/mol. -Ea/RT resembles the Boltzmann distribution law.

R: The gas constant. Its value is 8.314 J/mol K.

T: The absolute temperature  in which the reaction takes place. In units of Kelvin (K).

Application of Arrhenius equation 

Application of the Arrhenius equation in pharmaceutical stability testing is straightforward. In the isothermal method, the system to be investigated is stored under several high temperatures with all other condi- tions  fixed. Excess thermal exposure accelerates the degradation and thus allows the rate constants to be determined in a shorter time period.

Most accelerated testing models are based on the Arrhenius equation

where k2 and k1 are rate constants at temperature T2 and T1, respectively; Ea is the activation energy; and R is the gas constant. Temperature is in kelvins.

This equation describes the relationship between storage temperature and degradation rate. Use of the Arrhenius equation permits a projection of stability from the degradation rates observed at high temperatures.

Activation energy, the independent variable in the equation, is equal to the energy barrier that must be exceeded for the degradation reaction to occur. When the activation energy is known (or assumed), the degradation rate at low temperatures may be projected from those observed at “stress” temperatures.

The relationship between drug concentration [D]after time (t) for a first-order equation is

where K1 is the reaction rate constant at a given temperature. Integrating from t = 0 to t = t, one obtains

where [D]₀ is the drug concentration at time 0. By substituting 0.90 [D]₀ = [D]₀ in the equation above, the time required to reach 90% of the original drug concentration can be shown to be

A common practice of manufacturers in pharmaceutical and diagnostic reagent industries is to utilize various shortcuts, e.g., bracket tables and the Q rule, to estimate product shelf life. These techniques (see below) share the advantage that decisions may be made by analyzing only a few stressed samples.

However, they are based on assumptions about the activation energy of product components and are valid only in so far as these assumptions are accurate. Whatever method is chosen, the validity of product stability projections depends on analytical precision, the use of appropriate controls within the experimental design, the assumptions embodied in the mathematical model, and the assumed or measured activation energy of product components.

Reaction rate is influenced by pH, tonicity, the presence of stabilizers, and so forth. Key product component( s) may degrade (or otherwise become unavailable) through multiple mechanisms . In complex chemical systems, therefore, minor variation in formulation can profoundly affect lot-to-lot stability and, indeed, the activation energy of product degradation. Consequently, shelf life projected from accelerated studies must be validated by appropriate real-time stability testing.

Rapid Techniques for Projecting Shelf Life

Examples of shortcuts for projecting product shelf life for storage at 5 #{1i7n6c}lCude the QRule and bracket table methods.

QRule.

 The Q Rule states that a product degradation rate decreases by a constant factor (Q₁₀) when the storage temperature is lowered by 10⁰C. value of Q₁₀ is typically set at either 2, 3, or 4 because these correspond to reasonable activation energies. For larger shifts in temperature, the rate constant changes exponentially with temperature, and is proportionalt  (Q₁₀)ⁿ where n equals the temperature change (⁰C) divided by 10. For example, the estimated decrease in degradation rate caused by lowering the storage temperature by 50 ⁰C(Q10)5 because n = 50/10 = 5 (see Table 1). This model falsely assumes that the value of Q does not vary with temperature. A more detailed treatment has been published.

A Q₁₀ value of 2 provides a conservative estimate, and results calculated with this value are considered probable. A Q₁₀ value of 4 is less conservative and yields results considered tobe possible. To illustrate the application of the Q Rule in predicting shelf life, assume that 90% of phencyclidine (PCP) is recovered after 26 days at 55₀C. stability of  PCP under refrigerated (5⁰C) conditions may be estimated [26 days/(Q10)5] as follows:

(a) PCP is probably stable for 832 days (2.3 years) if Q₁₀= 2.

(b) PCP may be stable for 6318 days (17 years) if = 3.

(c) PCP is possibly stable for 26 624 days (73 years) if Q₁₀ = 4.

Table 1 : Q10 Factors
Q10	Ea [Kcal/Mol]	(Q10)5 = Q50
2.0	12.2	32
3.0	19.4	243
4.0	24.5	1024

Bracket tables.

The bracket table technique assumes that, for a given analyte, the activation energy is between

two limits,e.g.,between 10 and 20 kcal. As a result, a table may be constructed showing “days of stress” at various stress temperatures. The use of a 10 to 20 kcal bracket table is reasonable, because broad experience indicates that most analytes and reagents of interest in pharmaceutical and clinical laboratories have activation energies in this range .

Because the bracket table provided in Table 2 does not specify stability requirements at a stress temperature of 55⁰C, conservative use of the table requires that projections be taken from the 47.5⁰C. The PCP stability assumed in the Q Rule illustration (26 days) exceeds the six-month stability requirement (16.6 days) for the 10 kcal model. Thus, the interpretation of the bracket table is that PCP is probably stable at 5⁰Cat least six months. Furthermore, because the assumed 26 days also exceeds the nine days stability required for the three-year 20 kcal model, it is possible that PCP is stable for at least three years.

Prudent use of either of these rapid techniques would dictate that data at three or four higher temperatures be incorporated into the projection of refrigerated shelf life. To evaluate the usefulness of the Q Rule and the bracket table in this example, one must determine the activation energy for PCP and then project the refrigerated shelf life by using the Arrhenius equation.