Aparticular reactant decomposes with a half‑life of 113 s when its initial concentration is 0.331 m. the same reactant decomposes with a half‑life of 243 s when its initial concentration is 0.154 m. calculate the rate constant (k) and reaction order?

The reaction is second-order, and k = 0.0267 L mol^-1 s^-1

Explanation:

Step 1: Data given

The initial concentration is 0.331 M

half‑life time =  113 s

The same reactant decomposes with a half‑life of 243 s when its initial concentration is 0.154 M.

Step 2: Determine the order

The reaction is not first-order because the half-life of a first-order reaction is independent of the initial concentration:

t½ = (ln(2))/k

Calculate k for the two conditions given:

⇒ 113 s with initial concentration is 0.331 M

t½ = ([A]0)/2k

113 s = (0.331 M)/2k

k = 0.00146 mol L^-1 s^-1

⇒ 243 s with an initial concentration is 0.154 M

t½ = ([A]0)/2k

243 s = (0.154 M)/2k

k = 0.000317 mol L^-1 s^-1

The values of k are different, so that rules out zero-order.

Step 3: Calculate if it's a second-order reaction

For a second-order reaction, the half-life is given by the expression

t½ = 1/((k*)[A]0))

Calculate k for the two conditions given:

⇒ 113 s when its initial concentration is 0.331 M

t½ = 1/((k*)[A]0))

113 s = 1/(k*(0.331 M))

k = 1/((0.331 M)*(113 s)) = 0.0267 L mol^-1 s^-1

⇒ 243 s when its initial concentration is 0.154 M

t½ = 1/((k*)[A]0))

243 s = 1/(k*(0.154 M))

k = 1/((0.154 M)*(243 s)) =  0.0267 L mol^-1 s^-1

The values of k are the same, so the reaction is second-order, and k = 0.0267 L mol^-1 s^-1

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