The theory in which state space an internal energy $U(S,V,\{N_i\})$ is part of the axioms.

(In statistical physics, $U(S,V,\{N_i\})$ is only derived, see e.g. Statistical internal energy.)

$U, S, V, \{N_i\}$ are all the extensive quantities.

Legendre transformation of $U$ w.r.t. $S$ gives what is called $F$ for $U$ with $T$ for $S$.

Thus, here, $T:=\frac{\partial U}{\partial S}$.

Legendre transformation of $U$ w.r.t. $V$ gives what is called $H$ for $U$ with $-p$ for $V$.

Thus, here, $p:=-\frac{\partial U}{\partial V}$.

Legendre transformation of $F$ or $H$ w.r.t. the second extensive variable ($V$ resp. $S$), gives what is called $G$.

We also define $\mu_i:=\frac{\partial U}{\partial N_i}$.

Volume in a 3D world is easy to grasp.

The laws of thermodynamics for equilibration of system-parameters for systems in contact make the intensive quantities $T$ and $p$ experimentally accessible quantities.

Equations of state are relations $V(p,T,\{n_i\})$, determined by $U(S,V,\{n_i\})$.

The simplest nontrivial systems would be a linear,

$U\sim c^S T\ \Leftrightarrow\ T=\frac{U}{c^S}$

or

$U\sim c^V p\ \Leftrightarrow\ p=\frac{U}{c^V}$.

but for this we need characteristic constants with $[c^S]=[S]$ resp. $[c^V]=[V]$.

For ideal gas the former relation holds and the pressure equation is a variant of the latter, namely one where $c^V$ is replaced with $V$. That form of the energy, $U=p\cdot V$, can and probably should be understood from the microscopic perspective given by Statistical Physics of particles.

The fact that pressure and volume are then reciprocal, $p=\frac{c^ST}{V}$, leads to the occurrence of all the logarithms that are so typical for phenomenological thermodynamics (see below). E.g. in computing change of energies, we often must integrate external variables against internal ones.

$(\Delta G)_{T,\{N_i\}}:=\int_{p_2}^{p_1}V(T,p)\,{\mathrm d}p=c^S T\ln(\frac{p_2}{p_1})$

$\implies p_1 = p_2\cdot\exp\left(-\dfrac{(\Delta G)_{T,\{N_i\}}}{c^S T}\right)$.

This sort of “concentration varies with $\exp(-E/k_BT)$” equation also pops up often in chemistry and electronics (Goldmann equation, Nernst equation, diode current-voltage-characteristics,…)

Among many others, the Van der Waals gas is a gas model defined by an equation of state that generalizes ideal gas via two constants.

Experimentally, deviations from the ideal gas are often captured by measuring the fugacity coeffients, which are the deviations from the naive ideal gas pressure (see Fugacity).

Wikipedia: Fugacity

Notes on physical theories . note, Statistical internal energy