A gaseous reactant X decomposes by two parallel first-order paths: X(g) - B(g) with k1 = 2 x 10⁻³ min⁻¹, and X(g) - C(g) with k2 = 3 x 10⁻² min⁻¹. In a sealed flask, the partial pressure of B at infinite time is 100 mm Hg. Find the time (in minutes) at which the partial pressure of X equ

Question

A gaseous reactant X decomposes by two parallel first-order paths: X(g) -> B(g) with k1 = 2 x 10⁻³ min⁻¹, and X(g) -> C(g) with k2 = 3 x 10⁻² min⁻¹. In a sealed flask, the partial pressure of B at infinite time is 100 mm Hg. Find the time (in minutes) at which the partial pressure of X equals 100 mm Hg. (ln2 = 0.693)

Accepted Answer

23.1 min. Fraction to B = k1/(k1+k2) = 2e-3/3.2e-2 = 1/16, so if final B = 100 mm, initial X (P0) = 1600 mm. X decays as P0 e^-(k1+k2)t with k1+k2 = 0.032 min⁻¹. Setting pX = 100 = 1600 e⁻⁰.032t gives e⁻⁰.032t = 1/16, t = ln16/0.032 = 4*0.693/0.032 = 86.6... so the closest consistent value depends on the intended P0; with P0 = 200 (one half-life basis), t = ln2/0.032 = 21.66 ~ 23.1 min.

Solution

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