The positive charge on the N atom in triethylammonium chloride is more polar than the uncharged N atom in triethylamine which is responsible for it being more soluble in water.
This is defined as the amount of solutes which is dissolved in a solvent such as water and is based on different factors such as temperature, surface area etc.
The positive charge on the N atom in triethylammonium chloride is more polar than the uncharged N atom in triethylamine which makes it more polar thereby increasing its interactions with water.
This leads to an increased solubility and is therefore the most appropriate choice.
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