Integrated Rate Law for First Order Reaction

Integrated Rate Law Equation

• Functional Form:
  • (\ln [A]_t - \ln [A]_0 = -kt)
  • Where:
    • ([A]_t) = concentration at time (t)
    • ([A]_0) = initial concentration
    • (k) = rate constant
    • (t) = time
• Log Property:
  • (\ln \left( \frac{[A]_t}{[A]_0} \right) = -kt)

Exponentiation to Solve

• Exponentiate both sides:
  • (\frac{[A]_t}{[A]_0} = e^{-kt})
• Multiply by initial concentration:
  • ([A]_t = [A]_0 e^{-kt})

Graphical Representation

• Exponential Decay Graph:
  • ([A]_t) on y-axis
  • (t) on x-axis
  • Key Points:
    • Time (t = 0): ([A]_t = [A]_0)
    • As (t \to \infty): ([A]_t \to 0)

Concept of Half-Life

Definition

• Half-Life ((t_{1/2})):
  • Time it takes for concentration to decrease to half its initial value.

Calculating Half-Life

• Set ([A]_t = \frac{[A]0}{2}) when (t = t{1/2})
• Substitute into the equation:
  • (\frac{[A]_0}{2} = [A]0 e^{-kt{1/2}})
• Simplification:
  • (\frac{1}{2} = e^{-kt_{1/2}})
• Take natural log:
  • (\ln \frac{1}{2} = -kt_{1/2})
  • Solve for (t_{1/2}):
    • (t_{1/2} = \frac{-\ln(\frac{1}{2})}{k})
    • (\ln(0.5) = -0.693)
    • (t_{1/2} = \frac{0.693}{k})

Properties

• Constant Half-Life:
  • Independent of initial concentration ([A]_0).
  • Example: If half-life is 10 seconds, remains constant regardless of initial particle count.

Practical Example

• Starting with 8 particles:
  • Half-life = time to reduce to 4 particles = 10 seconds.
• Subsequent Reductions:
  • From 4 to 2 particles, another 10 seconds.
  • From 2 to 1 particle, another 10 seconds.
• Conclusion:
  • Consistency of half-life across different initial concentrations demonstrates the characteristic behavior of first order reactions.