In the first order decomposition of acetone at 500°c, ch3och3→ productit is found that the concentration of acetone is 0.0300 m after 200 min and 0.0200 m after 400 min. calculate the rate constant, the half life, and the initial concentration.

For a first-order decomposition, the formula is: ln (x0 / x) = kt

At t = 200, x = 0.0300 M: ln (x0 / 0.03) = 200k

At t = 400, x = 0.0200 M: ln (x0 / 0.02) = 400k

Multiplying the first equation by 2: 2ln (x0 / 0.03) = 400k, and 400k is equivalent to the second equation, so:

2ln (x0 / 0.03) = ln (x0 / 0.02)

(x0 / 0.03) ^2 = x0 / 0.02

x0 = 0.045 M (initial concentration)

Substituting into the 1st equation: ln (0.045 / 0.03) = 200k

k = 0.0020273 (rate constant)

The half life can be found when x = 0.5x0:

ln (x0 / 0.5x0) = 0.0020273t

ln (2) = 0.0020273t

t = 341.90 minutes (half-life)