Why is #CH_3OH# miscible in water whereas #CH_3(CH_2)_6OH# is not?
Well, methanol is 1/2 a water molecule, and should show solubility in water; heptanol is not half of a water molecule, and would show solubility in aprotic solvents.
The longer the alkyl chain, the less soluble in water is the alcohol. Methanol and ethanol are infinitely miscible in water; propanol and butanol have limited solubility. Of course, if we add salts to the water phase, the alcohol may separate.
The longer the alkyl chain, the more important (and the greater the magnitude of) the dispersion forces that operate between the chains. For long alkyl chains, dispersion forces are the most important intermolecular force. Please check the boiling points of homologous alcohols. As chain length increases, hydrogen bonding ceases to be the dominant intermolecular force. Hydrogen bonding with water thus becomes less viable, and solubility of the long-chain alcohol IN WATER decreases.
As another effect to consider, you should know that ethanol and hexanes are MISCIBLE (i.e one is soluble in the other in any proportion); methanol and hexanes are IMMISCIBLE. How can you develop the ideas here to rationalize this phenomenon?