To determine which equation correctly expresses the self-ionization of water, we first need to understand what self-ionization (or autoionization) means. Self-ionization is a process where water molecules react with each other to form ions without the addition of any other substances.
The self-ionization of water can be represented by the following generic reaction:
[tex]\[ 2 \text{H}_2\text{O} \leftrightarrow \text{H}_3\text{O}^+ + \text{OH}^- \][/tex]
In this reaction, two water molecules react to form a hydronium ion ([tex]\(\text{H}_3\text{O}^+\)[/tex]) and a hydroxide ion ([tex]\(\text{OH}^-\)[/tex]).
Now, let's compare this ideal equation to the given options:
1. [tex]\( \text{H}_2\text{O} + \text{H}_3\text{O}^+ \leftrightarrow \text{H}_2\text{O} + \text{OH}^- \)[/tex]
This equation does not correctly represent self-ionization because it shows a reactant and product of water on both sides without forming the required hydronium and hydroxide ions from water alone.
2. [tex]\( \text{H}_2\text{O} + \text{H}_2\text{O} \leftrightarrow 2 \text{OH}^- \)[/tex]
This equation suggests that two water molecules react to form only hydroxide ions, which is incorrect, as it ignores the formation of hydronium ions.
3. [tex]\( \text{H}_2\text{O} + \text{H}_2\text{O} \leftrightarrow 2 \text{H}_3\text{O}^+ \)[/tex]
This equation suggests that two water molecules react to form only hydronium ions, which is incorrect, as it ignores the formation of hydroxide ions.
4. [tex]\( \text{H}_2\text{O} + \text{H}_2\text{O} \leftrightarrow \text{H}_3\text{O}^+ + \text{OH}^- \)[/tex]
This equation correctly represents the self-ionization of water where two water molecules react to form one hydronium ion and one hydroxide ion. This is consistent with the generic reaction mentioned earlier.
Thus, option 4:
[tex]\[ \text{H}_2\text{O} + \text{H}_2\text{O} \leftrightarrow \text{H}_3\text{O}^+ + \text{OH}^- \][/tex]
is the correct representation of the self-ionization of water.
So, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
