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}
    • K1 C = C0 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{1/2} = \frac{0.693}{k1}
    • Concentration over time:
      Ct = C0 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{1/2} = \frac{1}{k2 C_0}
    • Concentration over time:
      \frac{1}{C(t)} = \frac{1}{C0} + k2 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) = C0 - k0 t
    • Half-life expression:
      t{1/2} = \frac{C0}{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{app} = A e^{-\frac{Ea}{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.