Phosphine is a poisonous gas responsible in part for the odor of decaying fish.

Question

Phosphine is a poisonous gas responsible in part for the odor of decaying fish. Concentrations below 8 parts per million (ppm) are detectable by the human olfactory system. Higher concentrations cause weakness, vertigo, convulsions, coma, or even death. At 680ºC, pure phosphine decomposes into phosphorus and hydrogen gas with a half-life that is independent of the initial phosphine pressure. This half-life at 680ºC is 35.0(s). How long would it take for 96.88% of the phosphine to decompose at 680ºC?

Explanation / Answer

ln (Nt/No) = -kt . . .where Nt/No is the fraction of radioactive nuclei remaining. If 96.88% has decomposed, then 3.12% remains, or 0.0312.

t 1/2 = 0.693 / k
k = 0.693 / t 1/2 = 0.693 / 35.0 s) = 0.0198 s^-1

The integrated rate law for a first order reaction is:

ln (A) - ln (Ao) = -kt

ln (A/Ao) = - kt

ln (3.12/100) = -0.0198 t

-3.467 = - 0.0198 t

t = 175 sec.

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