# Problem: For each of the following cases, identify the order with respect to the reactant, A. Case (A ----&gt; product) 1. The half-life of A increases as the initial concentration of A decreases. 2. A twofold increase in the initial concentration of A leads to a fourfold increase in the initial rate. 3. A twofold increase in the initial concentration of A leads to a 1.41-fold increase in the initial rate. 4. The time required for [A] to decrease from [A]0 to [A]0/2 is equal to the time required for [A] to decrease from [A]0/2 to [A]0/4. 5. The rate of decrease of [A] is a constant.

###### FREE Expert Solution

To identify the order with respect to the reactant, A: → product

General form of Rate Law

k = rate constant
A & B = reactants
x & y = reactant orders

Integrated rate law for Zeroth order reaction:

$\overline{){\mathbf{\left[}\mathbf{A}\mathbf{\right]}}_{{\mathbf{t}}}{\mathbf{=}}{\mathbf{-}}{\mathbf{kt}}{\mathbf{+}}{\mathbf{\left[}\mathbf{A}\mathbf{\right]}}_{{\mathbf{0}}}}$

Integrated rate law for First order reaction:

• t1/2 = half-life

$\overline{){\mathbf{ln}}{\mathbf{\left[}\mathbf{A}\mathbf{\right]}}_{{\mathbf{t}}}{\mathbf{=}}{\mathbf{-}}{\mathbf{kt}}{\mathbf{+}}{\mathbf{ln}}{\mathbf{\left[}\mathbf{A}\mathbf{\right]}}_{{\mathbf{0}}}}$        $\overline{){{\mathbf{t}}}_{\mathbf{1}\mathbf{/}\mathbf{2}}{\mathbf{=}}\frac{\mathbf{ln}\mathbf{2}}{\mathbf{k}}}$

Integrated rate law for Second order reaction:

$\overline{)\frac{\mathbf{1}}{{\mathbf{\left[}\mathbf{A}\mathbf{\right]}}_{\mathbf{t}}}{\mathbf{=}}{\mathbf{kt}}{\mathbf{+}}\frac{\mathbf{1}}{{\mathbf{\left[}\mathbf{A}\mathbf{\right]}}_{\mathbf{0}}}}$  $\overline{){{\mathbf{t}}}_{\mathbf{1}\mathbf{/}\mathbf{2}}{\mathbf{=}}\frac{\mathbf{1}}{\mathbf{k}{\left[A\right]}_{\mathbf{0}}}}$

1. The half-life of A increases as the initial concentration of A decreases.

•  For a 2nd order reaction → the half-life t1/2 depends on initial concentration [A]0
• $\mathbf{↑}{\mathbf{t}}_{\mathbf{1}\mathbf{/}\mathbf{2}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{k}{\left[A\right]}_{\mathbf{0}}\mathbf{↓}}$
• second-order reaction ###### Problem Details

For each of the following cases, identify the order with respect to the reactant, A. Case (A ----> product)

1. The half-life of A increases as the initial concentration of A decreases.

2. A twofold increase in the initial concentration of A leads to a fourfold increase in the initial rate.

3. A twofold increase in the initial concentration of A leads to a 1.41-fold increase in the initial rate.

4. The time required for [A] to decrease from [A]0 to [A]0/2 is equal to the time required for [A] to decrease from [A]0/2 to [A]0/4.

5. The rate of decrease of [A] is a constant.