How long (in years) does it take DDT in a soil sample to decrease from 295 ppbm to 14 ppbm (part per billion by mass)?

Question

A biochemist studying the breakdown of the insecticide DDT finds that it decomposes by a first-order reaction with a half-life of 12.1 years. How long (in years) does it take DDT in a soil sample to decrease from 295 ppbm to 14 ppbm (part per billion by mass)? Enter to 1 decimal place.

Answer 1

Since the decomposition is first order, we can either use the 1st order integrated rate law or a simplified version. The simplified version is FR = 0.5ⁿ

FR = fraction remaining = 14 parts / 295 parts = 0.0475

n = number of half lives that have elapsed = ?

0.0475 = 0.5ⁿ

Take log of both sides...

-1.34 = -0.301 n

n = 4.45 half lives have elapsed

Time = 4.45 half lives x 12.1 years/half life = 53.9 years