It has always bugged me that tables for water (and other) properties have the capability to look up internal energy as a function of both temperature and pressure. If we limit the discussion to liquid below the saturation temperature, then what is the qualitative argument to say that $u(T)$ is inaccurate and that the multivariate function $u(P,T)$ is needed?
From Wikipedia Internal Energy:
In thermodynamics, the internal energy is the total energy contained by a thermodynamic system. It is the energy needed to create the system, but excludes the energy to displace the system's surroundings, any energy associated with a move as a whole, or due to external force fields.
I understand that internal energy is not fully a proxy for temperature, so what thermodynamic property could we define (in $J/kg$) that would be a fully 1-to-1 relationship with temperature with no influence from pressure? If a liquid was fully incompressible would internal energy then not be a function of pressure?
If my physics understanding is correct, temperature has a definition that stems from the concept of thermal equilibrium. Quantitatively, I thought that temperature was proportional to the average kinetic energy the molecules, but I doubt that as well (in fact, I think this is wrong). The zeroth law of thermodynamics is necessary for formally defining temperature but it, alone, is not sufficient to define temperature. My own definitions for temperature and internal energy do not have the rigor to stand up to scrutiny. What qualitative arguments can fix this?
- $u$ - internal energy
- $P$ - pressure
- $T$ - temperature