Final answer:
To find the time when 10.5% of (CH₃)₃CBr remains, one must use the first-order integrated rate law. The calculation involves taking the natural log of the remaining fraction of the initial concentration and dividing by the rate constant. It takes approximately 59.95 hours for 10.5% to remain.
Step-by-step explanation:
To calculate the time at which 10.5% of the initial concentration of (CH₃)₃CBr remains using the given first-order rate law, we utilize the integrated first-order rate equation: ln([A]t/[A]0) = -kt.
In this case, [A]t/[A]0 is 0.105 since 10.5% of the initial concentration remains.
Rearranging the equation for t gives us t = -ln([A]t/[A]0) / k.
Plugging in the values, we have t = -ln(0.105) / 0.038 hour⁻¹.
By doing the calculations, we find:
t = -ln(0.105) / 0.038 hour⁻¹
t ≈ 59.95 hours
Therefore, it takes approximately 59.95 hours for 10.5% of the initial concentration of (CH₃)₃CBr to remain.