In: Chemistry
A particular reactant decomposes with a half-life of 119 s when
its initial concentration is 0.324 M. The same reactant decomposes
with a half-life of 233 s when its initial concentration is 0.165
M.
Determine the reaction order?
What is the value and unit of the rate constant for this
reaction?
Let the reaction follow a simple nth order rate law:
rate = k∙[A]ⁿ
Half-life t₁₂ initial concentration [A]₀ and rate constant k for
such a reaction are related as:
t₁₂ = (2ⁿ⁻¹ - 1) / ( (n - 1)∙k∙[A]₀ⁿ⁻¹ )
except the particular case of first order reactions, i.e. n=1, in
which half-life does not depend on initial concentration:
t₁₂ = ln(2)/k
Apparently your reaction is not a first order reaction. When you
combine the constant factors in the relation above to a constant K,
you can see that half-life of a non-first order reaction is
inversely proportional to initial concentration raised to the power
(n-1):
t₁₂ = K/[A]₀ⁿ⁻¹
with K=(2ⁿ⁻¹ - 1)/((n - 1)∙k)
K cancels out when you take the ratio of the two given
half-lifes:
t₁₂₍₂₎ / t₁₂₍₁₎ = (K/[A]₀₍₂₎ⁿ⁻¹) / (K/[A]₀₍₁₎ⁿ⁻¹) =
([A]₀₍₁₎/[A]₀₍₂₎)ⁿ⁻¹
to find the exponent (n-1) take logarithm
ln(t₁₂₍₂₎/t₁₂₍₁₎) = ln(([A]₀₍₁₎/[A]₀₍₂₎)ⁿ⁻¹) = (n -
1)∙ln([A]₀₍₁₎/[A]₀₍₂₎)
=>
n - 1 = ln(t₁₂₍₂₎/t₁₂₍₁₎) / ln([A]₀₍₁₎/[A]₀₍₂₎)
= ln(233s / 119s) / ln(0.324M / 0.165M )
= 0.995
≈ 1
=>
n = 2
With known order n we can compute k from given half-life and
initial concentration.
For a second order reaction half-life is given by:
t₁₂ = (2²⁻¹ - 1) / ( (2 - 1)∙k∙[A]₀²⁻¹ ) = 1/(k∙[A]₀)
Hence
k = 1/(t₁₂∙[A]₀)
= 1/(119s ∙ 0.324M)
= 2.593×10⁻² M⁻¹s⁻¹
= 0.02593 M-1s-1