[snorkack]

Solubility is a question of vapor pressure, rather than boiling point.
The term “octanol/water partitioning coefficient” is applied to compounds that are soluble in n-octanol. If a given compound is not soluble in octanol but soluble in another organic solvent

... then the octanol/water partitioning coefficient might be small?  And difficult to measure.

Hydrogen fluoride is gas and therefore, the term “octanol/water partitioning coefficient” is not applicable. For gases and highly volatile compounds, the partition coefficient is identical with the (dimensionless) Henry’s law constant.

No. But Henry law constants can be measured in water and in octanol (or, more correctly, in saturated solutions of each).

Exceptionally, hydrogen fluoride is miscible with water at any proportion, due to formation of strong hydrogen bonds.

What´s exceptional here? Many substances are miscible with water. Hydrogen fluoride is not, precisely because it is a gas. Same applies to, for example, ethylamine (boiling point +16 degrees).

[pgk]

Α lipophilic compound may not be soluble in n-octanol (e.g. bitumen) or not compatible (e.g. oleic anhydride). In this case, the partition coefficient is determined in another solvent/water system and can be translated to octanol/water by the above equation.
Many gases are soluble in water, e.g. oxygen (otherwise, there wouldn’t be any sea life).
Henry law can be applied to any solvent. But attention to serious deviations that might derive, if the compound interacts with the solvent (e.g. ionization, hydrogen bonding, azeotropy, etc.) such as aqueous hydralogens. But “exceptionally, hydrogen fluoride is miscible with water at any proportion, due to the formation of strong hydrogen bonds”. Note that fluorine hydrogen bond is very strong (up to 40 kcal/mole), compared with oxygen (≈ 5 kcal/mole) and nitrogen (≈ 7 kcal/mole) hydrogen bonds. However, highly concentrated aqueous HF might be fuming at elevated temperature.

[snorkack]

Α lipophilic compound may not be soluble in n-octanol (e.g. bitumen) or not compatible (e.g. oleic anhydride). In this case, the partition coefficient is determined in another solvent/water system and can be translated to octanol/water by the above equation.

Or it might not be compatible with water. Or it might not be compatible with either n-octanol or water.

Many gases are soluble in water, e.g. oxygen (otherwise, there wouldn’t be any sea life).

And many gases are soluble in n-octanol.

Henry law can be applied to any solvent. But attention to serious deviations that might derive, if the compound interacts with the solvent (e.g. ionization, hydrogen bonding, azeotropy, etc.) such as aqueous hydralogens. But “exceptionally, hydrogen fluoride is miscible with water at any proportion, due to the formation of strong hydrogen bonds”. Note that fluorine hydrogen bond is very strong (up to 40 kcal/mole), compared with oxygen (≈ 5 kcal/mole) and nitrogen (≈ 7 kcal/mole) hydrogen bonds. However, highly concentrated aqueous HF might be fuming at elevated temperature.

So is concentrated HCl - or, for the matter, concentrated oleum even though SO3 is not a gas at standard temperature (condenses at +45 degrees).

[pgk]

Undergraduate Chemistry:
The definition of partition coefficient in an organic solvent/ water system is the ratio of concentrations of a compound that is distributed in the immiscible organic solvent and water layers. If the compound is not miscible of not compatible with water then, the term partition coefficient it is not applied and the compound is defined as “super-hydrophobic”, except if it reacts with water (e.g. acetic anhydride)
Of course, many gases are soluble in n-octanol and the most of them, in both n-octanol and water. So, a good exercise is to start calculate the concentrations of various gases in various ratios of the system n-octanol / water by using the Henry’s constants that can be found in the literature and the web.
High School Chemistry:
HCl is a gas that can be dissolved in water up to the azeoprope 40% w/v (a little higher than 10 N)
SO3 is the “sulfuric anhydride” that reacts with water and forms H2SO4.
“Oleum” is H2SO4 that contains or is saturated with SO3.
Forget the equations of simulation of the logP(o/w)  in various solvents, for the moment. This is Postgraduate and Professional Chemistry.

[snorkack]

Undergraduate Chemistry:
The definition of partition coefficient in an organic solvent/ water system is the ratio of concentrations of a compound that is distributed in the immiscible organic solvent and water layers. If the compound is not miscible of not compatible with water then, the term partition coefficient it is not applied

Um. Partition coefficient is defined for dilute solutions. A compound does not need to be miscible with water or octanol to have a defined finite solubility.

and the compound is defined as “super-hydrophobic”, except if it reacts with water (e.g. acetic anhydride)

And if it reacts with n-octanol?

Of course, many gases are soluble in n-octanol and the most of them, in both n-octanol and water. So, a good exercise is to start calculate the concentrations of various gases in various ratios of the system n-octanol / water by using the Henry’s constants that can be found in the literature and the web.

Note that partition coefficient is defined by solubility in water saturated in n-octanol, not in pure water.

[pgk]

Undergraduate Chemistry:
Partition coefficient is a physical property of a compound that is constant and characteristic for a given non miscible solvents system, regardless the compound’s concentration and the volume ratio of the non miscible solvents.
Postgraduate Chemistry:
If the compound is not miscible with n-octanol, another appropriate solvent can also be used and the corresponding logP value can be translated to n-octanol/water system, as above described.
High School Chemistry:
Water cannot be saturated in or with a non-miscible material (e.g. n-octanol).
Please, read the posts carefully before replying.

[snorkack]

High School Chemistry:
Water cannot be saturated in or with a non-miscible material (e.g. n-octanol).
Please, read the posts carefully before replying.

Reread your assertion carefully.
Only nonmiscible materials (which are characterized by finite solubility) can form saturated solutions.

[pgk]

Or and in other words, the water soluble NaCl cannot form saturated aqueous solutions. Of course, it can. But this is a good chance to clarify this confusion, too.
There is no absolute immiscibility. Materials are miscible each other, even at very low proportions. For example, dioctyl phthalate plasticizer (DOP) is miscible with water at 1/15,000 w/w. Thus and by a rough description, the theory of finite solubility assumes a system of two immiscible solvents as a saturated solution of each other and that any other solute added, acts as a co-solvent that may change their solubility parameters. That is true. But in most cases, these changes are negligible in the logP scale and therefore, the partition coefficient can be considered constant within solvents ratio and /or concentration variations. However, the latter does not always happen, e.g. addition of surfactants. In that case, partition coefficient is not applicable and the hydrophilicity or lipophilicity of surfactants is estimated by the HLB scale (Hydrophilic/ Lipohilic Balance).

[snorkack]

Or and in other words, the water soluble NaCl cannot form saturated aqueous solutions. Of course, it can. But this is a good chance to clarify this confusion, too.
There is no absolute immiscibility. Materials are miscible each other, even at very low proportions.

But precisely. They are soluble but not miscible.
NaCl is soluble in water, but it is not miscible. NaCl is solid till it melts at over +800 degrees. A finite quantity of liquid water will not convert infinite quantity of solid NaCl into a liquid - there is some concentration of solution where additional NaCl will remain a solid, and that is a saturated solution.

Contrast with ethanol, which is miscible with water. There is no such thing as a saturated aqueous solution of ethanol between -114 and +78 degrees - there is no ethanol solution to which additional ethanol cannot be added.

[pgk]

NaCl is solid that during solubilization forms ion pairs that are surrounded by water molecules. Therefore, NaCl is soluble in water, aqueous NaCl is liquid and there is no solid aqueous  NaCl up to saturation.
Sugar (sucrose) is solid that during solubilization forms a lot of hydrogen bond per molecule with surrounding water molecules. Therefore, sucrose is soluble in water, aqueous sucrose is liquid and there is no solid aqueous sucrose up to saturation.
Ethanol forms a lot of hydrogen bond per mass with surrounding water molecules and therefore is soluble with water at any proportion and any further amount can be added up to the formation of a water ethanolic solution.
In contrast with n-butanol that forms the half of hydrogen bond per mass with surrounding water molecules, it is soluble at 7.7 w/w. Above this concentration, water is saturated with n-butanol and a phase separation occurs or otherwise and in terms of finite solubility, a saturated solution of aqueous n-butanol is formed.
The difference between miscibility and solubility is mainly kinetic, in terms of homogenicity (and thermodynamic because there is a temperature dependence of solubility, too). Honey is miscible with water but complete solubility takes time. Besides, complete solubility means uniform concentration at any point, contrary to miscibility that might not be homogenous.
Do not confuse it. Please and by the way, take a look to the kinetic solubility term.