To determine which reaction is an example of an Arrhenius base, we need to understand the definition of an Arrhenius base. An Arrhenius base is a substance that increases the concentration of [tex]\(\text{OH}^-\)[/tex] ions in an aqueous solution.
Let's analyze each reaction step-by-step:
1. Reaction 1:
[tex]\[
\text{NaOH (s)} \rightarrow \text{Na}^+(\text{aq}) + \text{OH}^-(\text{aq})
\][/tex]
Sodium hydroxide ([tex]\(\text{NaOH}\)[/tex]) dissociates in water to produce [tex]\(\text{Na}^+\)[/tex] and [tex]\(\text{OH}^-\)[/tex] ions. The release of [tex]\(\text{OH}^-\)[/tex] ions in the solution is a clear characteristic of an Arrhenius base.
2. Reaction 2:
[tex]\[
\text{HCl (g)} + \text{H}_2\text{O (l)} \rightarrow \text{H}_3\text{O}^+(\text{aq}) + \text{Cl}^-(\text{aq})
\][/tex]
Hydrochloric acid ([tex]\(\text{HCl}\)[/tex]) reacts with water to form [tex]\(\text{H}_3\text{O}^+\)[/tex] and [tex]\(\text{Cl}^-\)[/tex] ions. This reaction produces hydronium ions ([tex]\(\text{H}_3\text{O}^+\)[/tex]), which indicates it's an acid, not a base.
3. Reaction 3:
[tex]\[
\text{CH}_3\text{COOH (aq)} + \text{H}_2\text{O (l)} \rightarrow \text{H}_3\text{O}^+(\text{aq}) + \text{CH}_3\text{COO}^-(\text{aq})
\][/tex]
Acetic acid ([tex]\(\text{CH}_3\text{COOH}\)[/tex]) reacts with water to form [tex]\(\text{H}_3\text{O}^+\)[/tex] and [tex]\(\text{CH}_3\text{COO}^-\)[/tex] ions. Similar to Reaction 2, the production of [tex]\(\text{H}_3\text{O}^+\)[/tex] ions indicates that acetic acid is also an acid.
4. Reaction 4:
[tex]\[
\text{NH}_3 (\text{aq}) + \text{HC}_2 \text{H}_3 \text{O}_2 (\text{aq}) \rightarrow \text{NH}_4^+ (\text{aq}) + \text{C}_2 \text{H}_3 \text{O}_2^- (\text{aq})
\][/tex]
Ammonia ([tex]\(\text{NH}_3\)[/tex]) reacts with acetic acid ([tex]\(\text{HC}_2\text{H}_3\text{O}_2\)[/tex]) to produce ammonium ([tex]\(\text{NH}_4^+\)[/tex]) and acetate ([tex]\(\text{C}_2\text{H}_3\text{O}_2^-\)[/tex]) ions. This reaction involves a weak base ([tex]\(\text{NH}_3\)[/tex]) and a weak acid ([tex]\(\text{HC}_2\text{H}_3\text{O}_2\)[/tex]), but it does not explicitly produce [tex]\(\text{OH}^-\)[/tex] ions.
Given this analysis, the reaction that displays an example of an Arrhenius base is:
[tex]\[
\text{NaOH (s)} \rightarrow \text{Na}^+(\text{aq}) + \text{OH}^-(\text{aq})
\][/tex]
Therefore, the correct reaction is the first one. This corresponds to the index [tex]\(0\)[/tex] in the list provided.
