I know that if the Helmoltz free energy, $A$, is expressed as a function $A\sim A(N,V,T)$, then this function contains all thermodynamic information about the system. For instance, the pressure of the system is given by:

$$P(N,V,T) = -\frac{\partial A}{\partial V}\Bigg|_{n,T}$$

Indeed, if we imagine $N$ and $T$ are fixed, then this in effect gives us pressure as a function of volume:

$$P\sim f(V).$$

My question relates to the inverse of this equation. If we are given $P$ rather than $V$ (still with $T$ and $N$ fixed and known) then either numerically or analytically we can solve this to find $V$. But if the equation is complicated enough, there maybe multiple values of $V$ for a given $P$, $T$ and $N$.

This didn't seem too unreasonable to me (perhaps physically this represents a gas and a liquid co-existing at the same $T$, $P$ and $N$ at equilibrium), until I realised that, if we worked with the Gibbs free energy instead, and found values of $G(T,P,N)$, then in that case

$$V(T,P,N) = \frac{\partial G}{\partial P} \Bigg|_{N,T}$$

and this gives us an explicit, single-valued equation for $V$ as a function of $P$, $T$ and $N$ (though perhaps the inverse $P(V)$ is now multi-valued!?!.

What's going on here? One formulation of thermodynamics gives me an EOS in which there can exist multiple values of $V$ for a given $P$, and the other says I can have multiple values of $P$ for a given $V$!

I know that if the Helmholtz free energy, $A$, is expressed as a function $A\sim A(N,V,T)$, then this function contains all thermodynamic information about the system. For instance, the pressure of the system is given by:

$$P(N,V,T) = -\frac{\partial A}{\partial V}\Bigg|_{n,T}$$

Indeed, if we imagine $N$ and $T$ are fixed, then this in effect gives us pressure as a function of volume:

$$P\sim f(V).$$

My question relates to the inverse of this equation. If we are given $P$ rather than $V$ (still with $T$ and $N$ fixed and known) then either numerically or analytically we can solve this to find $V$. But if the equation is complicated enough, there maybe multiple values of $V$ for a given $P$, $T$ and $N$.

This didn't seem too unreasonable to me (perhaps physically this represents a gas and a liquid co-existing at the same $T$, $P$ and $N$ at equilibrium), until I realised that, if we worked with the Gibbs free energy instead, and found values of $G(T,P,N)$, then in that case

$$V(T,P,N) = \frac{\partial G}{\partial P} \Bigg|_{N,T}$$

and this gives us an explicit, single-valued equation for $V$ as a function of $P$, $T$ and $N$ (though perhaps the inverse $P(V)$ is now multi-valued?).

What's going on here? One formulation of thermodynamics gives me an EOS in which there can exist multiple values of $V$ for a given $P$, and the other says I can have multiple values of $P$ for a given $V$!

I know that if the Helmoltz free energy, $A$, is expressed as a function $A\sim A(N,V,T)$, then this function contains all thermodynamic information about the system. For instance, the pressure of the system is given by:

$$P(N,V,T) = -\frac{\partial A}{\partial V}\Bigg|_{n,T}$$

Indeed, if we imagine $N$ and $T$ are fixed, then this in effect gives us pressure as a function of volume:

$$P\sim f(V).$$

My question relates to the inverse of this equation. If we are given $P$ rather than $V$ (still with $T$ and $N$ fixed and known) then either numerically or analytically we can solve this to find $V$. But if the equation is complicated enough, there maybe multiple values of $V$ for a given $P$, $T$ and $N$.

This didn't seem too unreasonable to me (perhaps physically this represents a gas and a liquid co-existing at the same $T$, $P$ and $N$ at equilibrium), until I realised that, if we worked with the Gibbs free energy instead, and found values of $G(T,P,N)$, then in that case

$$V(T,P,N) = \frac{\partial G}{\partial V} \Bigg|_{N,T}$$

and this gives us an explicit, single-valued equation for $V$ as a function of $P$, $T$ and $N$ (though perhaps the inverse $P(V)$ is now multi-valued!?!.

What's going on here? One formulation of thermodynamics gives me an EOS in which there can exist multiple values of $V$ for a given $P$, and the other says I can have multiple values of $P$ for a given $V$!

I know that if the Helmoltz free energy, $A$, is expressed as a function $A\sim A(N,V,T)$, then this function contains all thermodynamic information about the system. For instance, the pressure of the system is given by:

$$P(N,V,T) = -\frac{\partial A}{\partial V}\Bigg|_{n,T}$$

Indeed, if we imagine $N$ and $T$ are fixed, then this in effect gives us pressure as a function of volume:

$$P\sim f(V).$$

My question relates to the inverse of this equation. If we are given $P$ rather than $V$ (still with $T$ and $N$ fixed and known) then either numerically or analytically we can solve this to find $V$. But if the equation is complicated enough, there maybe multiple values of $V$ for a given $P$, $T$ and $N$.

This didn't seem too unreasonable to me (perhaps physically this represents a gas and a liquid co-existing at the same $T$, $P$ and $N$ at equilibrium), until I realised that, if we worked with the Gibbs free energy instead, and found values of $G(T,P,N)$, then in that case

$$V(T,P,N) = \frac{\partial G}{\partial P} \Bigg|_{N,T}$$

and this gives us an explicit, single-valued equation for $V$ as a function of $P$, $T$ and $N$ (though perhaps the inverse $P(V)$ is now multi-valued!?!.

What's going on here? One formulation of thermodynamics gives me an EOS in which there can exist multiple values of $V$ for a given $P$, and the other says I can have multiple values of $P$ for a given $V$!