Factors affecting reaction kinetics

Factors affecting reaction kinetics

To determine ways to prevent degradation of drugs in pharmaceutical for-mulations, it is important to identify the mechanism of drug degradation and the factors that affect the degradation rate or reaction kinetics. Once the route and kinetics of degradation have been identified, stabilization strategies that minimize reaction rates and maximize the shelf life of the drug product can be adopted.

1. Temperature

If a chemical reaction is endothermic (takes heat from the environment to react), increase in temperature generally accelerates the reaction. If a reaction is exothermic (gives out heat to the environment as it proceeds), temperature is generally inversely proportional to reaction rate. Most chemical reactions of pharmaceutical relevance in a dosage form are endothermic. Thus, an increase in temperature generally accelerates the reaction rate.

Arrhenius equation

The effect of temperature on the rate of drug degradation is expressed in terms of the effect of temperature on the reaction rate constant, k, by the

Arrhenius equation (Figure 7.4):

k = Ae^{−Ea /RT}                          (7.52)

where:

E_a is the activation energy

A is a preexponential constant

R is the gas constant (1.987 calories/degree.mole)

T is the absolute temperature (in Kelvin)

Figure 7.4 Arrhenius plot. Plot of the variation of the rate constant, k, versus reciprocal of the absolute temperature, T.

The Arrhenius expression can also be written as (Figure 7.4):

This equation is of the form y = mx + c for a straight-line plot. Thus, an Arrhenius plot of log k on the y-axis against reciprocal of the absolute temperature (1/T) on the x-axis yields E_a from the slope of the straight line (Figure 7.4). This equation is not amenable for direct application for the measurement of reaction rates, since A and E_a are unknown. Nevertheless, activation energy, E_a, can be calculated by comparing reaction rates at two different temperatures.

Thus, for temperatures T₁ and T₂,

k₁ = Ae^{−Ea /RT1} (7.55)

k₂ = A^{e−Ea /RT2} (7.56)

Thus,

k₂ / k₁ = Ae^−Ea/RT2 / A^e−Ea/RT1 = e ^{Ea /RT1 − Ea/RT2} = e^{Ea/R(1/ T1 −1/T2 )} = e^{Ea/R((T2 −T1)/ T1T2 )}

Which is same as:

Or, as in Figure 7.4,

Thus, measurement of the reaction rate constant, k, at two different tempera-tures allows the calculation of the activation energy, E_a, for a given reaction.

Shelf life

The Arrhenius plot can be used to determine the shelf life of the drug. The half-life (t_1/2) and shelf life (t_0.90) expressions from the reaction order can be substituted for the reaction rate constants, k, in the above equations to directly infer product’s shelf life at a given temperature. These calcula-tions allow the calculation of temperature of optimum drug stability over its shelf life. If a drug is stable at room temperature (25°C), it is usually labeled for storage at controlled room temperature (range 15°C–30°C). If a drug is unstable at room temperature but stable at lower refriger-ated temperature (5°C), it is usually labeled for storage under refrigerated conditions (range 2°C–8°C). This is the case, for example, with various injectables such as penicillin, insulin, oxytocin, and vasopressin. Typically, a shelf life of 24 months or more is desired for all commercial products to allow enough time for manufacture, storage, distribution, and con-sumption by the patient. Appropriate temperature, packaging configura-tion, and drug product storage conditions are determined to achieve the desired shelf life. Stabilization strategies for drugs against degradation during storage are often required for achieving and extending the desired shelf life.

Thermodynamics of reactions

Arrhenius equation provides a mathematical basis of connecting reaction kinetics to the collision theory and the transition state theory of chemical reactions. The collision theory states that reactions happen by collisions that happen among reacting molecules in a favorable configuration. It highlights the need for several random collisions for effective collisions, which lead to the reaction, to occur. The collision theory predicts an increase in intermolecular collisions as a function of temperature, thus leading to higher number of effective collisions and higher reactivity at higher temperatures.

The transition state theory states that an activation energy barrier must be surpassed for a reaction to become spontaneous. This activation energy barrier can be understood as the energy of collisions required for them to overcome the intermolecular repulsions at close contact for effective intermolecular reactions to occur. Arrhenius equation relates the rate of a reaction, k, with the activation energy barrier, E_a.

Ink = In A – (E_a/RT)

Once the activation energy barrier is overcome, the free energy difference between the reactants and the products, ∆ G, determines the reaction rate. This is given by the equation:

ΔG = −RT lnk                    (7.60)

In addition, the free energy difference between the reactants and the prod-ucts is a measure of the difference in the enthalpy, ∆H, and entropy, ∆S, between the reactants and products. This is represented by the equation:

ΔG= ΔH−TΔS                       (7.61)

Greater the free energy difference, that is, lower the free energy of the products than that of the reactants, the faster the reaction. Thus, a negative ∆G facilitates a forward reaction. This can be achieved by lower enthalpy of the products than that of the reactants, achieving a negative ∆H, or higher entropy of the products than the reactants, achieving a high ∆S and a negative T∆S.

These equations can be used in conjunction with each other to connect a reaction’s thermodynamic parameters to reaction rates, which can be deter-mined experimentally. This allows the determination of thermodynamic parameters, such as free energy and entropy change, of various reactions.

2. Humidity

Water can influence reaction kinetics by acting as a reactant, a solvent (i.e., a reaction medium), or a plasticizer.

Water as a reactant

For hydrolytically sensitive drugs, water acts as a reactant and increases the drug degradation rate directly by participating in a bimolecular reaction. Such reactions may follow second-order or pseudo first-order kinetics, depending on whether water is available in the reaction medium in lim-ited (such as contamination in a solvent or adsorbed water in a solid-state excipient) or ample (such as the solvent) quantity.

Water as a plasticizer

A plasticizer is a substance that is used as an additive to promote fluidity in a solid state. For example, polyethylene glycol (PEG) is commonly used as a plasticizer in tablet film-coating applications to allow the formation of a flexible film that wraps around the tablet core. Small amounts of free water (e.g., adsorbed on the surface) present in the solid particles can promote local-ized dissolution and fluidity or flow of reacting molecular species. Thus, for drugs that are not hydrolytically sensitive, water can increase reaction rates by acting as a plasticizer in the solid dosage forms, by increasing the molecular mobility and diffusion rates of the reactive components. This is commonly seen in solid dosage forms such as tablets and capsules, where the reaction rates are dependent on the humidity during storage.

Water as a solvent

In addition to the role of water as a solvent in the liquid dosage forms, small amounts of adsorbed water can also act as a solvent in the microen-vironment within a solid dosage form. This can affect reaction rates by the following:

1. Solubilizing reacting components and increasing their mobility

2. Affecting the disproportionation of the salt form of the drug to its free acid or free base form, which may have different reactivity compared with the salt form of the drug

3. Removing the product away from the reacting species, so that equi-librium reactions proceed more rapidly toward the formation of the product

Disproportionation of the salt form of a drug in a solid dosage form to its constituent free acid or free base form of the active pharmaceutical ingredi-ent (API) is commonly attributed to the dissolution of the salt in the free water in the dosage form.

Determination and modeling the effect of water/humidity

Experimentally, the effect of water or humidity on the stability of a dosage form is determined by determining drug degradation kinetics at different temperature and humidity storage conditions. This is accomplished by stor-ing the drug product under open-dish conditions at different controlled temperature and humidity conditions for different time periods, followed by analysis of the degradation products. These studies are called isothermal degradation rate studies, since the temperature is kept constant throughout the study.

The effect of relative humidity (RH) at a fixed temperature on drug’s degradation rate constant, k, can be incorporated using an empirically determined humidity effect constant, B, as:

k = e^{B( RH)}                          (7.62)

This equation may be combined with Arrhenius equation for the effect of temperature on reaction rate:

k = Ae ^-Ea/RT                  (7.63)

To obtain,

k = Ae^{B( RH )−Ea/RT}                      (7.64)

This combined equation predicts reaction rate as a function of both temperature and humidity.

The effect of humidity on reaction rate constant is an empirically fitted model. Hence, this model can take different forms, depending on the experimental system under investigation. For example, some systems may be better described by the following equation:

k = Ae^{Ea / RT +B(RH)}                       (7.65)

Nonetheless, combining the humidity effect with the temperature effect on reaction rate constant provides a better estimation of the extent of drug degradation over its shelf life.

3. pH

Disproportionation effect

The pH of the drug solution in a liquid dosage form and the microenviron-mental pH in a solid dosage form can significantly influence drug stability by affecting the proportion of ionized versus unionized species of a weakly acidic or a weakly basic drug. The proportion of free acid or free base form of a drug at a given pH is modeled by the Henderson–Hasselbalch equation.

for a drug that is the salt of a weak acid, or

for a drug that is the salt of a weak base.

Disproportionation of a salt into its free acid or base form can influ-ence reactivity by changing the concentration of the reacting species. Generally, the free acid or the free base form of a drug is more reactive. Thus, drugs that are salts of free acids are unionized in greater proportion and, consequently, are more reactive at acidic pH, and drugs that are salts of free bases are unionized in greater proportion and, consequently, are more reactive at basic pH.

Acid–base catalysis

Acid (H⁺) and base (OH−) can catalyze several reactions directly. For example, the rate of an ester hydrolysis reaction catalyzed by hydrogen or hydroxyl ions can vary considerably with pH. The H⁺ ion catalysis predomi-nates at a lower pH and the OH− ion catalysis operates at a higher pH.

Acids and bases can affect reaction kinetics by specific or general catalysis. For example, in specific catalysis, the reaction rate depends only on the pH of the system and not on the concentration of actual acid or base salts (such as HCl vs. HF or NaOH vs. KOH) contributing the H⁺ or the OH− ions. In general catalysis, all species capable of donating or sequestering protons contribute to the reaction rate, and proton transfer from an acid to the solvent or from the solvent to a base is the rate-limiting step. General catalysis is usually evident by changing reaction rates with changing buffer concentration at a constant pH.

pH–rate profile

Rates of chemical reactions are often determined at different pH values to identify the pH of optimal drug stability. The pH–rate profiles are two-dimensional plots of observed reaction rate constant (k_obs) on the y-axis against pH on the x-axis. The shape of a pH–rate profile reflects on the mechanism of the reaction. For example, Figure 7.5 shows representative pH–rate profiles that indicate, for the corresponding subfigures, (A) base catalysis and stability at acidic pH, (B) acid catalysis and stability at basic pH, (C) a continuum of acid and base catalysis with a narrow pH region of maximum drug stability, and (D) acid and base catalysis under extreme ionization conditions and a wide pH region of maximum drug stability.

Figure 7.5 Typical pH stability profiles. Examples of pH—stability profiles for a drug that degrades under basic (a), acidic (b), or both acidic and basic conditions (c and d).

Proteins are particularly sensitive to changes in pH, particularly with respect to the conformation of the secondary and tertiary structures. Changes in the ionization of amino acid side chains in proteins with changes in pH can lead to folding or unfolding to varying degrees. Proteins exhibit overall charge neutrality at their isoelectric point, where the proportion of the positively charged groups within the protein is the same as that of the negatively charged groups. Proteins tend to be most stable in their most folded state, called the native state, which is generally obtained by appro-priate balance of charged and uncharged groups on the surface. The pH of optimal stability can be determined by plotting log k against pH. For example, recombinant α-antitrypsin (rAAT) has a V-shaped pH—stability profile, with optimal stability at pH 7.5.

4. Cosolvent and additives

For liquid dosage forms, cosolvents are frequently used to improve drug solubility and stability in the vehicle. These cosolvents are commonly one or more of PEG, propyleneglycol (PG), and ethanol. In addition, water-miscible surfactants, such as polysorbate 80, and polymers, such as polyvinyl alco-hol (PVA), may be used. Other common components of liquid dosage forms include buffers to maintain desired pH, ionic components for isotonicity of parenteral solutions, preservatives, sweeteners, flavors, and colorants.

These additives in liquid formulations lead to simultaneous changes in physicochemical conditions of the reaction medium, such as dielectric con-stant, ionic strength, surface tension, and viscosity, all of which may affect rates of chemical reactions. The effect of ionic strength and dielectric con-stant depends on the relative hydrophilicity of the reactants and products. If, for example, products achieve greater solubilization in the reaction medium, the rate of the reaction would be higher. This is due to the ability of the products to diffuse away from the reaction site, leading to shift in the equi-librium of the reaction toward the formation of the products. Similarly, if the reacting species have opposite charges, a solvent with a low dielectric constant accelerates the reaction rate. This could be attributed to lower sol-ubilization of the reacting species, which also have affinity with each other, thus increasing the propensity for the reaction. On the other hand, if the reacting species have the same charge, a solvent with high dielectric constant will accelerate the reaction by forming bonds and dissolving both species, thus reducing intermolecular repulsion due to like charges.

Drug–excipient interactions

Chemical interaction between components in solid dosage forms can impact, often increasing, the rate of drug degradation.

Buffer salts are often added to maintain a formulation at optimal pH. These salts may often affect the degradation rate. For example, the hydrolysis rate of codeine is almost 20 times higher in phosphate buffer at neutral pH than in an unbuffered solution at this pH. At neutral pH, the major buffer species are H₂PO₄⁻ and HPO₄²⁻, either of which may act as a general base catalyst for codeine degradation.

Excipients that have specific functional groups such as the carboxylate group on croscarmellose sodium or the sulfate group on sodium lauryl sulfate can exhibit specific interactions with the drug substance that can destabilize a drug. These interactions can be direct reaction, salt disproportionation, or facilitation of a reaction by surface adsorption. In addition, excipients often contain small quantities of reactive substances, called reactive impurities. These reactive impurities in excipients can react with low-dose drugs in the dosage form to cause drug degradation. For example, PEG and polyvinyl-pyrrolidone (PVP) commonly have peroxide impurities that can cause oxi-dative degradation of sensitive drugs.

The effect of excipients on drug stability is usually assessed early in drug development through excipient compatibility studies. Drug degradation rate is determined in physical mixtures of a drug with individual or a combina-tion of excipients. Excipient compatibility studies are also useful later in drug development when unexpected impurities are observed. Mechanistic investigation of the reaction pathway leading to the formation of these impurities becomes a cornerstone of drug product stabilization strategies.

Catalysis

Components of a dosage form can frequently act as, or bring in species that act as, reaction catalysts. A catalyst affects the rate of change in the concentra-tions of products and reactants in a chemical reaction but not the equilibrium concentration of reactants and products in the reaction. As seen in Figure 7.6, a catalyst may change the reaction pathway and lower the energy of activation required for a reaction, thus accelerating the reaction. 

Figure 7.6 Effect of catalyst. Transition state during reaction progress (on the x-axis from left to right) with the energetics (on the y-axis) is indicated by the peak in the energy requirement for the reactants to convert to products. The presence of a catalyst changes the reaction pathway such that the height of this peak is lowered.

However, the thermodynamic driver for a reaction, that is, free energy difference between the reactants and the products, remains the same for an uncata-lyzed versus a catalyzed reaction. Thus, a catalyst influences the speed but not the extent of a reaction. In addition, a catalyst does not get chemically altered itself.

In pharmaceutical dosage forms, heavy metal contaminants in excipients and drug substances often act as catalysts.