Answer:
(first choice)
<em>One difference between first- and second-order reactions is that </em><em><u>the half-life of a first-order reaction does not depend on [Ao]; the half-life of a second-order reaction does depend on [Ao].</u></em>
Explanation:
<u>1) First order reactions' model</u>
This is a brief deduction of the first order reactions' half-life
- Equation: aA +b B → cC + dD
- rate: r = - d[A]/dt = k[A]
- d[A]/[A] = kdt ⇒ - ln { [A]/[Ao] } = kt
t half-life = T
[A] = (1/2) [Ao] ⇒- ln { [A]/[Ao] } = - ln (1/2) = ln(2) = kT
⇒ T = ln(2) / T
<em>* The half-life of a first order reaction is a constant; it does not depend on the initial concentration of the reactants, it only depend on the rate constant.</em>
<u>2) Second order reaction's model:</u>
This is a brief deduction of the second order reactions' half-life
- Equation: aA +b B → cC + dD
- rate: r = - d[A]/dt = k[A]²
- d[A]/[A]² = kdt ⇒ 1/[Ao] - 1/[A] = kt
t half-life = T
[A] = (1/2) [Ao] ⇒ 1 / [Ao] - 1 / {2[Ao]} = 1 / {2[Ao]} = kT
⇒ T = 1 / {2k[Ao] }
<em> * The half-life of a second order reaction depends on the initial concentration and the rate constant.</em>
<u>3) Final conclusion:</u>
We have found that while the half life of a first order reaction is does not depend on the initial concentration, the half-life of a second order reaction does depends on the initial concentration. Hence, the correct answer to the question is:
<em>One difference between first- and second-order reactions is that </em><em><u>the half-life of a first-order reaction does not depend on [Ao]; the half-life of a second-order reaction does depend on [Ao].</u></em>
