The formula $PV = nRT$ explains the relationship between pressure, volume and temperature in terms of the quantity of gas present in a container. I'm trying to understand how these are related once the gas reaches a liquid state. Can anybody shed some light for me?
This formula is strictly applicable for gases. So what happens to this formula is it is no longer valid. Provided the gas density is low, this law holds for any single gas or for any mixture of different gases. However it is interesting that for osmotic pressure of a liquid we have a similar formula $$\pi=CRT $$ where $\pi$ is osmotic pressure and $C$ is concentration.
C=n/V, so its basically the same formula PV=nRT
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The "Ideal Gas Law" (IGL) (pV = nRT) is an "equation of state" and any such equation is a relation between the macroscopical "state variables" of a matterial system in a given phase. Therefore, the form of a state equation depends on the system's phase. The basic assumption of the IGL is that the molecules do not interact by any other means besides point collisions; hence the "Ideal" specifier, since molecules do interact. The strength of the interaction, amongst other properties, determines the phase of the matter: very weak interactions for gaseous state, weak interaction for liquid state and strong for solid state. Since IGL assumes no interactions at all, then the material would never enter the liquid state.
However, in reality, the molecules of a gas do interact weakly and there are several other models which describe the gases better, e.g. "Van der Waals equation". Any such realistic equation would wave some "critical points", i.e. some values of the state variables that the equation becomes mathematically or physically inconsistent. Such critical points presage a phase transition. If the equation of state is realistic enough (e.g. "Peng–Robinson" equation of state), then it could be valid, in the same form, in the liquid phase also. If not, then it should be replaced with an appropriate one.
So, what exactly happens to an equation of state when changing phase is ambiguous; it depents on how much accuratelly it describes the physical reality and its accuracy depends on the assumptions that lead to the construction of the equation.
The ideal gas law applies in systems made of large number of particles (atoms or molecules) that are completely independent from each other as in a dilute gas. In that situation, the particles are able to move in all space directions without interfering with the others. That's an ideal situation, hence the name ideal gas. The law $$PV=nRT$$ is an immediate consequence of this independence. Note that the number of particles is $N=n\times\mathcal N_{\text A}$ and writing $k_{\text B}=R/\mathcal N_{\text A}$ the Boltzmann constant, the ideal gas law rewrites as $$PV=Nk_{\text B}T.$$ In this second expression, the number of particles appears, revealing that each particle contribute the same amount of energy, regardless of its mass, nature, etc.
In a liquid, the particles are not independent of each other, because it is more favorable for them to remain close to each other. The possibilities of movement for a particle are strongly reduced, the ideal gas law cannot be used anymore. Of course this remark also applies in solids, where the movements of the particles are even more restrained.
Remark inspired by the answer of @arutor egni.
In a dilute solution the particles in the liquid also have independent movements, if the solution is not too concentrated. The situation is actually the same as for the ideal gas and van t'Hoff's law $$\Pi V=Nk_{\text B}T$$ applies for the same reason. Note that $N$ here is logically the number of particles (polymers) and not the number of atoms constituing the polymer.
