Question

If the half-life (t_1/2) for a first order reaction is 1 minute, then the time required for 99.9% completion of the reaction is closest to ______.

पर्याय

• 2 minutes

• 4 minutes

• 5 minutes

• 10 minutes

Answer 1

If the half-life (t_1/2) for a first order reaction is 1 minute, then the time required for 99.9% completion of the reaction is closest to 10 minutes.

Explanation:

The rate constant for a first-order reaction is related to its half-life by the formula:

k = `0.693/(t_(1//2))`

= `0.693/1`

= 0.693 min⁻¹

The integrated rate law for a first-order reaction is:

t = `2.303/k log(([A]_0)/([A]))`  ...(i)

For 99.9% completion, the remaining concentration [A] is 0.1% of the initial concentration [A]₀. This can be expressed as:

= `([A]_0)/([A])`

= `100/0.1`

= 1000

= 10³

Substitute the values into the time equation (i):

`t_(99.9%) = 2.303/k log_10(10^3)` 

= `(2.303 xx 3)/k`    ...(∵ log 10³ = 3)

= `6.909/k`

Using k = 0.693

= `6.909/0.693`

= 9.97 ≈ 10 minutes