Final answer:
a) The half-life of the reaction is approximately 3.76 hours. b) The concentration of CH₃NC after 1000 seconds is approximately 0.0325 M. c) The time required for the concentration of CH₃NC to decrease to 0.0170 M is approximately 7.15 hours. d) The rate constant in this reaction represents the proportionality constant between the rate of the reaction and the concentration of the reactant.
Step-by-step explanation:
a) To calculate the half-life of the reaction, we can use the formula for first-order reactions:
t1/2 = (0.693/k)
Substituting the given rate constant of 5.11 ✕ 10⁻5 s⁻1 into the formula, we get:
t1/2 = (0.693/5.11 ✕ 10⁻5 s⁻1)
t1/2 ≈ 13,534 s or 3.76 hours
b) To calculate the concentration of CH₃NC after 1000 seconds, we can use the integrated rate law for first-order reactions:
[CH₃NC] = [CH₃NC]0 * e-kt
Substituting the given values, we get:
[CH₃NC] = 0.0340 M * e-(5.11 ✕ 10⁻5 s⁻1) * 1000 s
[CH₃NC] ≈ 0.0325 M
c) To determine the time required for the concentration of CH₃NC to decrease to 0.0170 M, we can rearrange the integrated rate law and solve for time:
t = (ln([CH₃NC]0/[CH₃NC]))/k
Substituting the given values, we get:
t = (ln(0.0340 M/0.0170 M))/(5.11 ✕ 10⁻5 s⁻1)
t ≈ 25,747 s or 7.15 hours
d) The rate constant in this reaction represents the proportionality constant between the rate of the reaction and the concentration of the reactant. A higher rate constant indicates a faster reaction, while a smaller rate constant indicates a slower reaction. In this case, the rate constant of 5.11 ✕ 10⁻5 s⁻1 suggests that the rearrangement of methyl isonitrile to acetonitrile is a relatively slow reaction.