If the half-life (t₁/₂) for a first order reaction is 1 minute, then the time required for 99.9% completion of the reaction is close to:

CONCEPT:

First Order Reaction and Half-Life

• For a first-order reaction, the relationship between the time required for a certain fraction of the reaction to complete and its half-life is given by:

  t = (2.303 / k) x log(1 / (1 - fraction completed))

• The half-life (t₁/₂) for a first-order reaction is related to the rate constant k by the formula:

  t₁/₂ = 0.693 / k

• Using the above equations, the time required for 99.9% completion of the reaction (fraction completed = 0.999) can be calculated.

EXPLANATION:

• From the formula for half-life:

  t₁/₂ = 0.693 / k

  • k = 0.693 / t₁/₂
  • k = 0.693 / 1
  • k = 0.693 min⁻¹
• The time required for 99.9% completion can be calculated using the formula:

  t = (2.303 / k) x log(1 / (1 - fraction completed))

  • t = (2.303 / 0.693) x log(1 / (1 - 0.999))
  • t = (2.303 / 0.693) x log(1 / 0.001)
  • t = (2.303 / 0.693) x log(1000)
  • t = (2.303 / 0.693) x 3
  • t ≈ 10 minutes

Therefore, the time required for 99.9% completion of the reaction is approximately 10 minutes.