Study Notes for PHRM 6201: Physical Pharmacy - Chemical Kinetics and Shelf-life

Rate of Chemical Reactions

• Rate = K: A fundamental concept in kinetics where the constant K relates to the concentration of reactants and products over time.
• Kinetics Equation:
  • $[A]' + \frac{1}{2} = \ln^{2}$
  • $K<em>1 C = C</em>0 e^{kt}$
  • $\text{In } C_0 - kt$

Overview of PHRM 6201: Physical Pharmacy

• Main Topics:
  • Chemical Kinetics
  • Shelf-life considerations
  • Relationships of rate constants with pH and temperature

Objectives for Week 14

• Understand and identify the order of reactions.
• Apply zero-order and first-order kinetics to calculate half-life and shelf-life.
• Understand the relationship between rate constants and solution pH and temperature.

Chemical Kinetics

• Definition: Chemical kinetics studies the rate of chemical reactions, focusing on degradation reactions associated with drug instability.
  • Key Points:
  • Rate of disappearance of reactants.
  • Rate of formation of products.
  • Predicts product shelf-life or expiry dates (e.g., using the ASAP model).
  • Recommends proper storage conditions to better protect drug products.

General Rate Expression

• The rate can be expressed for each reactant and product:

  • Loss of Reactants: Negative rate for disappearing reactants.
  • Formation of Products: Positive rate for forming products.

• Example Rate Expression: For a general reaction:
  $aA + bB + cC \rightleftharpoons dD + eE$

  • Rate =
    $-\frac{d[A]}{a \cdot dt} = -\frac{d[B]}{b \cdot dt} = \frac{d[D]}{d \cdot dt} = \frac{d[E]}{e \cdot dt}$

Specific Rate Expressions

• Isomerization Reaction:
  $A \rightleftharpoons D$

  • Rate = $-\frac{dA}{dt} = \frac{dD}{dt}$

• Dimerization Reaction:
  $2A \rightleftharpoons D$

  • Rate = $-\frac{dA}{2dt} = \frac{dD}{dt}$
  • Relevant for cases where two reactant molecules combine.

Law of Mass Action

• The rate of a chemical reaction is proportional to the product of the molar concentrations of the reactants, each raised to a power equal to the number of molecules undergoing the reaction.
  • General form:
    $Rate = k[A]^a[B]^b[C]^c$
  • Rate Constant (k): Directly influences the speed of the reaction.

Order of Reaction

• Definition: The overall order of a reaction is the sum of the exponents in the rate expression.
  • For a reaction:
    $Rate = k[A]^a[B]^b[C]^c$
  • Overall order of reaction = $a + b + c$
  • Order of reaction for each reactant is defined as its exponent in the rate expression (e.g., order for A is a, for B is b).

Example Rate Expressions for Reactions

• Degradation Reaction:

  • $A \rightleftharpoons B + C$
  • Rate = $-\frac{d[A]}{dt} = k[A]$
  • Order for A is 1 (first-order).

• Dimerization Reaction of 2A:

  • Rate = $-\frac{d[A]}{2dt} = k[A]^2$
  • Order for A is 2 (second-order).

• General Reaction A + B → C:

  • Rate = $-\frac{d[A]}{dt} = -\frac{d[B]}{dt} = k[A][B]$
  • Overall reaction order is 2, with orders for A and B both equal to 1.

The Rate of Chemical Degradation

• Focus on Reactants: Chemical kinetics considers degradation based on reactant concentrations, not products.
• Equation for Rate:
  $Rate = -\frac{dA}{adt} = -\frac{dB}{b dt} = k[A]^a[B]^b$

First-Order Kinetics

• Describes reactions where the reaction rate is directly proportional to the concentration of one reactant (the active drug).
  • Rate Law:
    $ext{Rate} = -\frac{dA}{dt} = k_1[A]$
  • Half-life expression:
    $t<em>{1/2} = \frac{0.693}{k</em>1}$
  • Concentration over time:
    $C<em>t = C</em>0 e^{-k_1 t}$
  • Relates initial concentration (C0) to concentration after time t (Ct).

Half-Life of a Drug Example

• A drug is found to isomerize in an oral solution with a half-life of 4 hours.
  • Given: Initial concentration 5 mg/mL.
  • Find: Amount remaining after 6 hours in 100 mL solution.

Second-Order Reactions

• Characterized by reactions where the rate of reaction is proportional to the square of the concentration of the reactant.
  • Rate Law:
    $Rate = -\frac{dC}{dt} = k_2[C]^2$
  • Half-life expression:
    $t<em>{1/2} = \frac{1}{k</em>2 C_0}$
  • Concentration over time:
    $\frac{1}{C(t)} = \frac{1}{C<em>0} + k</em>2 t$

Pseudo Zero-Order Reactions

• Appears in scenarios where reaction rate is constant and independent of reactant concentration.
  • Rate Law:
    $Rate = -\frac{dC}{dt} = k_0$
  • Concentration relation:
    $C(t) = C<em>0 - k</em>0 t$
  • Half-life expression:
    $t<em>{1/2} = \frac{C</em>0}{2k_0}$

Predicting Shelf-Life with Kinetics

• Shelf-Life Definition: The time during which the drug potency drops to 90%, or equivalently, the period where there is 10% degradation.
  • Factors Influencing Shelf-Life:
  • Order of reaction.
  • Rate constant (k).
• Shelf-Life Calculations: Utilizing different orders of reactions to find the remaining concentration or time at which degradation is acceptable.

Factors Determining Shelf-Life

• Pseudo-Zero Order Kinetics: Dependent on initial concentrations and rate constant.
• First-Order Kinetics: Also dependent on the first-order rate constant.
• Other Influences: Temperature must be considered as it affects reaction kinetics.

Activated Complex Theory

• States that reactant molecules must be activated with sufficient kinetic energy to exceed the activation energy for the reaction to proceed.
  • Activation energies can be lowered by catalysts, which do not alter the overall energy change of the reaction.

Arrhenius Equation

• Establishes the relationship between the rate constant and activation energy, along with temperature.
  • General Form:
    $k<em>{app} = A e^{-\frac{E</em>a}{RT}}$
  • Where:
    • A = Frequency factor.
    • $E_a$ = Activation energy.
    • R = Universal gas constant.
    • T = Absolute temperature in Kelvin.

Arrhenius Plot

• A log-linear representation of the Arrhenius equation where ln k is plotted against 1/T.
  • Slope: $-\frac{E_a}{R}$
  • Intercept: $ext{ln } A$
  • Indicates the relationship of increased temperature to increased degradation rates.

Accelerated Stability Studies

• Involve subjecting drug products to increased temperatures and humidity to generate significant degradation products.
• Allows for determination of the rate constant at these elevated conditions.

Hydrolysis Reactions

• Hydrolysis reactions can be catalyzed by acids or bases and are pH-dependent.
  • Acid-Catalyzed Hydrolysis:
    $Rate = k[H^+][C]$
  • Base-Catalyzed Hydrolysis:
    $Rate = k[OH^-][C]$

Conclusion and Assignments

• Required readings and studies revolve around the chemical kinetics in drug formulations and stability assessments.
• Refer to Textbook: Chapters 11, pages 221-234 for detailed insights into chemical kinetics and reactions relevant to pharmacology.