Iodine-131 decays to xenon-131 by a first order process with a rate constant of 0.138 per day. How many days does it take for 90% of a sample to decay?

1. Use the first order integrated law in the form ln([A]0 / [A]) = kt. 2. When 90 percent has decayed, 10 percent remains, so [A]0 / [A] = 10. 3. Taking the logarithm gives ln 10 = 2.303. 4. Solve for the time: t = 2.303 / 0.138 per day. 5. The division yields 16.7 days. 6. Units check: a pure number divided by per day leaves days. 7. The value 5.02 days is the half-life, 0.693 / k, which removes only 50 percent, not 90. 8. The value 6.52 days divides the fraction 0.90 by k, skipping the logarithm entirely. 9. The value 50.2 days treats a factor of ten as ten half-lives, when it actually takes about 3.3 of them. _Source: OpenStax Chemistry (CC BY 4.0), Ch 12 "Kinetics", section 12.4 Integrated Rate Laws_