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I understand that ionic compounds dissociate when dissolved in water. Why don't covalent compounds do the same?

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    $\begingroup$ Some of them do. For example, $\ce{HCl}$ that is absolutely covalent in gas phase almost completely dissociates in diluted solutions. $\endgroup$ – permeakra Aug 22 '16 at 4:27
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Many 'covalent' molecules do dissociate in water, HCl (as pointed out in the comments), phenol, acetic acid, for example, whereas some 'ionic' compounds do not to any appreciable extent, e.g. silver chloride, lead sulfate.
The general reason is that the 'system', that is molecule plus solvent, will move into its lowest energy state. This means that if breaking the weakest bond and solvating the ions produced, is of lowest energy then this will occur.
In the case of HCL, the polar water molecules can solvate making $\ce{H3O^+_{aq} and Cl^-_{aq}}$ where both ions are surrounded by a few water molecules. Water has a very high dielectric constant (relative permittivity) which means that electric fields are 'damped', i.e they are attenuated to short range so that oppositely charged ions can easily separate and hence exist independently in solution.
With carboxylic acids the reasoning is similar, the weakest (O-H) bond dissociates, and the ions are solvated. However, dissociation may not be 100% indicating that the energy gain on dissociating is not that great. In insoluble salts, the lattice binding energy and also bond energy is too great compared to the energy gain expected upon solvation and they remain as molecules in a solid.
Organic molecules with strong bonds e.g. cyclohexane, benzene, with only C-C and C-H, hardy dissociate in water. The bonds are too strong compared to energy gain on solvation after dissociation.
(If dissociation produced radicals, there will be no effective solvation in a polar solvent so would remain at high energy and hence unlikely to occur)

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It all depends on the type of covalent compound used.If the compound used is polar covalent in nature(like HCl,as pointed out) the partially negatively charged Oxygen attracts the partially positively charged Hydrogen from HCl.

Hence, if a compound is polar covalent in nature,i.e., it shows charge separation,it will dissolve or dissociate in water otherwise not.

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  • $\begingroup$ Then why does Carbon-di-oxide which is non-polar, react with H2O, giving H+ and HCO3- ions $\endgroup$ – DJphy Feb 8 '20 at 16:07

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