To** identify the order** with **respect to the reactant, A: **

General form of **Rate Law**:

$\overline{){\mathbf{rate}}{\mathbf{}}{\mathbf{law}}{\mathbf{=}}{\mathbf{k}}{\left[\mathbf{A}\right]}^{{\mathbf{x}}}{\left[\mathbf{B}\right]}^{{\mathbf{y}}}}$

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:

**t**_{1/2 }= 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.**

**2nd order reaction**→ the**half-life t**on initial concentration_{1/2}depends**[A]**_{0}_{$\mathbf{\uparrow}{\mathbf{t}}_{\mathbf{1}\mathbf{/}\mathbf{2}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{k}{\left[A\right]}_{\mathbf{0}}\mathbf{\downarrow}}$}**second-order reaction**

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.

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