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From my understanding, for homologous alcohols, primary alcohols have the highest boiling points, because the OH group is the most accessible for other molecules to hydrogen bond to, therefore stronger intermolecular forces result in higher mp/bp.

One the other hand tertiary alcohols have the highest solubility in water because the packed structure of branches means the non-polar part of the molecule is condensed and therefore water near any part of the molecule can receive these hydrogen bonds. Doesn't this mean tertiary alcohols have the most accessible hydrogen bonds and therefore should have the highest melting point too?

Alternatively does this have to do with water being small and alcohols being larger therefore 'access' requirements to hydrogen bonding are different as larger alcohols need to be even closer than water to make a difference therefore the packing is less relevant as even being slightly away from the molecule means failure to access hydrogen bonds but water will be ok with being slightly away.

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The key concept here is that for solubility, you need to compare alcohols with the same number of carbon atoms. For example, the solubility order of isomers of butanol is as follows: $$\ce{CH3CH2CH2CH2OH < (CH3)2CHCH2OH < (CH3)3COH}$$

This difference in solubility follows the order 3°>2°>1° and can be explained by the branching concept that you have used in your question. However, If you compare alcohols with different number of carbon atoms, say butanol and methanol, then the alcohol with a smaller hydrocarbon part will be more soluble in water.

The boiling point of alcohols increases with the increase in the number of carbon atoms, as stronger van der waals forces start acting in between the molecules. For example, Phenol boils at around 180°C whereas methanol boils at around 60°C. For alcohols with the same number of carbon atoms, a more exposed hydroxyl group will result in stronger intermolecular H-bonding.

TL;DR: your comparision can only be applied for alcohols with the same number of carbon atoms

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